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SUPERSINGULAR MAIN CONJECTURES, GOLDFELD’S CONJECTURE
AND SYLVESTER’S CONJECTURE
DANIEL KRIZ
Abstract. We prove a p-converse theorem for elliptic curves E/Q with complex multiplication(over K) by OK , namely letting rp = corankZpSelp∞(E/Q), we show that rp ≤ 1 =⇒ rankZE(Q) =ords=1L(E, s) = rp and #X(E/Q) < ∞. In particular this has applications to two classicalproblems of arithmetic. First, it resolves Sylvester’s conjecture on rational sums of cubes, showingthat for all primes ` ≡ 4, 7, 8 (mod 9), there exists (x, y) ∈ Q⊕2 such that x3 + y3 = `. Second,combined with work of Smith, it resolves the congruent number problem in 100% of cases andestablishes Goldfeld’s conjecture on ranks of quadratic twists for the congruent number family. Themethod for showing the above p-converse theorem relies on new interplays between Iwasawa theoryfor imaginary quadratic fields at nonsplit primes and relative p-adic Hodge theory. In particular, weshow that a certain de Rham period qdR introduced by the author in previous work can be used toconstruct a certain analytic continuation of ordinary Serre-Tate expansions on the ordinary Igusatower to the infinite-level overconvergent locus. Using this coordinate, one can construct 1-variablemeasures for Hecke characters and GL2-newforms satisfying a new type of interpolation property.Moreover, one can relate the Iwasawa module of elliptic units to these anticyclotomic measures vianew “Coleman map”, which is roughly the qdR-expansion of the Coleman power series map.
Contents
1. Introduction 31.1. p-converse theorems 31.2. Outline of the argument 41.3. Relation to classical problems of arithmetic 81.4. Structure of the paper 92. Background 92.1. Coleman power series 102.2. Semi-local (log-)Coleman map 112.3. Functoriality of universal norms 122.4. Elliptic units and their Coleman power series 122.5. Measures 132.6. Review of construction of measures in the h = 1 case: idea 132.7. The Serre-Tate coordinate 142.8. Anticyclotomic measures via Serre-Tate coordinate in the h = 1 case 152.9. The h = 2 case: idea 152.10. Period sheaves 162.11. The fundamental de Rham period zdR and the coordinate qdR 162.12. qdR-expansions 182.13. Relation of qdR-expansions to classical Serre-Tate expansions 192.14. Overconvergent locus 232.15. The period Ωp 243. p-adic L-functions for GL1/K 243.1. Notation 243.2. Parametrizing anticyclotomic characters 25
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3.3. A transformation of Zp and twisted Fourier transforms of anticyclotomic characters 263.4. The p-adic L-function: statement of the theorem 283.5. Construction of the measure: setup 293.6. Construction of the local measures 303.7. The global measure 333.8. Interpolation and proof of Theorem 3.9 333.9. An interpolation property of a twist of Lλ 364. Anticyclotomic p-adic L-functions for GL2/Q 384.1. A “good” twist of λ 384.2. Anticyclotomic Rankin-Selberg p-adic L-functions 394.3. Form of the measure 394.4. Construction of the measure 394.5. Proof of Theorem 4.2 414.6. A factorization of measures and special values 415. Rubin-type main conjectures and explicit reciprocity laws 445.1. Review of the relevant Iwasawa modules 445.2. Rubin-type main conjectures (without p-adic L-functions) 455.3. Relating Rubin-type main conjectures to L-values and p-adic L-functions: outline 465.4. Choice of decomposition of module of semi-local universal norms: rank 1 case 465.5. Explicit reciprocity laws 475.6. Choice of decomposition of module of semi-local universal norms: a preview of the
rank 0 case 526. Rank 0 p-converse theorem 526.1. The Selmer group via local class field theory 526.2. Interlude: relating reciprocity maps to good reduction twists 556.3. The value on elliptic units 566.4. Description of the Selmer group via Wiles’s explicit reciprocity law 577. Rank 1 p-converse theorem 587.1. The rank 1 p-converse theorem for CM elliptic curves with p ramified in the CM field 587.2. Sylvester’s conjecture 627.3. Goldfeld’s conjecture for the congruent number family 628. Appendix: Elliptic units main conjecture via equivariant main conjecture 638.1. The cyclotomic algebraic µ-invariant 638.2. Computation of algebraic Iwasawa invariants: a valuation calculation 648.3. Computation of analytic Iwasawa invariants: application of analytic class number
formula and p-adic Kronecker limit formula 668.4. Relating to Kubota-Leopoldt p-adic L-functions 678.5. Vanishing of the µ-invariant 678.6. Proof of the Elliptic Units Main Conjecture from vanishing of the cyclotomic algebraic
µ-invariant 67References 68
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1. Introduction
1.1. p-converse theorems. One of the main motivational problems in modern number theory hasbeen the Birch and Swinnerton-Dyler (BSD) conjecture. Given an elliptic curve EQ, the conjecturestates
rankZE(Q) = ords=1L(E, s),
where L(E, s) is the L-function of E/Q. To date, the strongest result toward this conjecture is thefollowing implication which follows from the formula of Gross-Zagier ([21]) and Kolyvagin’s methodof Euler systems ([27]): for r = 0, 1,
(1) ords=1L(E, s) = r =⇒ rankZE(Q) = r,#X(E/Q) <∞
where X(E/Q) is the Shafarevich-Tate group. One can view this result as “one direction” of BSDin the rank r = 0, 1 case. A good approximation for rankZE(Q)) is the corank of the p∞-Selmergroup for any prime p: we have
rankZE(Q) ≤ corankZpSelp∞(E)
with equality if and only if #X(E/Q)[p∞] <∞ (which is of course conjectured to be true). Thus(1) implies
(2) ords=1L(E, s) = r =⇒ rankZE(Q) = r,#X(E/Q) <∞ =⇒ corankZpSelp∞(E/Q) = r.
However, the analytic rank ords=1L(E, s) is often hard to compute in practice, and so a naturalquestion is whether one can prove anything in the “converse” direction: for r = 0, 1,
(3) rcoankZpSelp∞ = r?
=⇒ ords=1L(E, s) = r.
To emphasize the dependence on the choice of auxiliary prime p, such theorems have come tobe referred to as “p-converse theorems”. Recently, there has been great progress on p-conversetheorems. The natural approach is via Iwasawa theory, which Skinner [51] used in the “vertical(i.e. p) aspect” and Zhang [62] in the “horizontal (i.e. tame) aspect” in order to prove the firstp-converse theorems. The assumptions of both loc. cit., however, rule out interesting cases such assupersingular primes p and complex multiplication (CM) elliptic curves. More recently, progresshas been made in generalizing to supersingular primes by Castella-Wan [8] and to the CM case byBurungale-Tian [6].
However, the case of CM at (potentially) supersingular primes p, i.e. those primes inert orramified in the CM field K, entails certain technical difficulties in the Iwasawa-theoretic approach,and so relatively little progress was hitherto made in this case. The main result of this paper isthe following p-converse theorem addressing the potentially supersingular case when p is ramifiedin K.
Theorem 1.1 (Theorem 7.10 and Theorem 6.16). Let E/Q be an elliptic curve with CM by OKand p a prime ramified in K. Then the p-converse implication (3) is true for E.
Our approach introduces objects and constructions from previous work of the author [30] datingback to the author’s thesis [28] coming from relative p-adic Hodge theory, and relating them toclassical Iwasawa-theoretic objects. Roughly speaking, we use a suitable substitute qdR of theordinary Serre-Tate coordinae which is defined on the infinite-level modular curve, and consider qdR-expansions of Coleman power series, which are power series in qdR−1 giving analytic continuationsof Serre-Tate expansions to the overconvergent locus. Using such qdR-expansions, we are able to“geometrically” realize universal norms of local units infinite towers of modular curves, and attachanticyclotomic measures to them. For universal norms of elliptic units, these measures are newanticyclotomic p-adic L-functions in similar flavor to that of Katz [24], interpolating L-values of
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twists of a fixed Hecke character by finite-order anticyclotomic Hecke characters.1The coordinate qdR
was also used in [30] and [28] in order to construct new Katz-type p-adic L-functions interpolatinganticyclotomic characters of varying Hodge-Tate weight and fixed finite type, which give functionscontinuous in the Hodge-Tate weight and not necessarily measures.
In the next section, we give a detailed outline of the main steps and novelties of our approach.
1.2. Outline of the argument. Henceforth fix an imaginary quadratic field K and a prime pramified in K. Our proof of Theorem 1.1 is broadly divided into three steps:
(1) Establishing a 2-variable Rubin-type main conjecture “without p-adic L-function” using theEuler system of elliptic units. This relates the structure of Selp∞(E/Q) to elliptic units.
(2) Constructing a Coleman map from universal norms of units to distributions on Zp. Thisinvolves taking the qdR-expansion of the Coleman power series map, using a canonical deRham coordinate qdR coming from relative p-adic Hodge theory. This Coleman map satisfiesan “explicit reciprocity law”, sending elliptic units to a measure supported on Z×p , whichfor the remainder of the Introduction we will denote by LE .
(3) Using (1) and (2), the rank 0 p-converse theorem immediately follows. For rank 1, we use anadditional argument relating LE to a p-adic L-function Lg,χ0 attached to a Rankin-Selbergpair (g, χ0) that is a “good twist” of the modular form attached to E. This Lg,χ0 is ameasure on Z×p is equal the logarithm of a Heegner point at the trivial character. Using theinterpolation properties of LE and Lg,χ0 , one shows that the Heegner point is non-torsionif and only if a value of LE outside the range of interpolation is nonzero. Using (1) and (2),this latter value is shown to be nonzero under the assumption corankZpSelp∞(E/Q) = 1.
1.2.1. Step 1. Step (1) involves predominantly classical methods of Iwasawa theory, such as thosedeveloped Rubin [39], with new inputs from work on equivariant main conjectures of Kings-Johnson-Leung [23] and a new argument involving Gross’ method for “factorizing” logarithms of elliptic units(cf. [20] and [36]) for the vanishing of the algebraic cyclotomic µ-invariant.2 From class field theory,one gets an exact sequence
0→ E → U→ X → Y → 0
attached to the tower K∞/K = K(E[p∞])/K, where E is the module of universal norms of globalunits, U the module of universal norms of local units, X is the Galois group of the maximalpro-p abelian extension of K∞ unramified outside p, and Y is the Galois group of the maximalpro-p abelian extension of K∞ unramified everywhere. We have a module of elliptic units C ⊂E . (See Section 5 for precise definitions.) Dividing out by C, we get an exact sequence of Λ =ZpJGal(K∞/K)K-modules.
0→ E/C → U/C → X → Y → 0,
1The author wishes to emphasize that our p-adic L-functions satisfy a new type of interpolation property, relatingtwisted Fourier transforms of finite-order anticyclotomic characters χ to χ-twisted central L-values (see (50), (70)and (79)). Thus, our measures should not be viewed as anticyclotomic measures in the traditional sense, eventhough they interpolate anticyclotomic twists. In fact, this type of interpolation for (untwisted) Fourier transformsof anticyclotomic characters is also satisfied by Katz’s ordinary measures, see Remark 3.10.
2The author notes that this step, proving both divisibilities for the Euler system of elliptic units, seems to bethe only place where cyclotomic Iwasawa theory intervenes, as the rest of our methods primarily involve Iwasawatheory on the Z⊕p -extension and anticyclotomic Iwasawa theory. The cyclotomic Zp-extension is more amenable tocomputations of Iwasawa invariants, following the methods of Coates-Wiles [10]. By a reduction in Johnson-Leung-Kings, this is enough to prove the 2-variable equivariant main conjecture for elliptic units, which is itself enough toprove the “rational” (i.e. with p inverted) 2-variable classical elliptic units main conjecture. We remark that thislatter reduction of Johnson-Leung-Kings is reminiscent of the “Easy Lemma” of Skinner-Urban’s proof of the mainconjecture for GL2 ([53]). From a philosophical viewpoint, one should perhaps not expect a ”purely anticyclotomic”proof of our results, as both variables of the 2-variable Euler system of elliptic units should contribute.
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where the Λ-ranks are 0, 1, 1 and 0 respectively. By the aforementioned µ-invariant result andequivariant main conjecture, one gets
(4) charΛ(E/C) = charΛ(Y)
(see Theorem 5.3). Now one chooses a Λ-rank 1 submodule3 U ′ ⊂ U such that U ′ ∩ E = 0. Thechoice of U ′ will depend on whether we are in the rp = corankZpSelp∞(E/Q) = 0 or 1 case. In therp = 0 case, U ′ is the kernel of the reciprocity map (see Section 6), and in the rp = 1 case, U ′ isany subspace with U ′∩E = 0 (which is necessarily distinct from the kernel of the reciprocity map).Then (4) implies a Rubin-type main conjecture
charΛ(U/(C, U ′)) = charΛ(X/rec(U ′))
(see Theorem 5.4), which does not explicitly involve p-adic L-functions.4
1.2.2. Step 2: setup. Step (2) is the most novel part of the argument. Coleman power series(see [14, Chapter I.2]) have long been a starting point in studying the Iwasawa theory of K andrelating elliptic units to p-adic analytic objects. In the height 1 setting (p split in K), it has longbeen known how to obtain measures from Coleman power series (see Chapter I.3 of loc. cit.), andreconstruct the Katz p-adic L-function from Coleman power series of elliptic units (see ChapterII.4 of loc. cit.). Essentially, one uses the fact that all height 1 formal groups are isomorphic to
the formal torus Gm, whose coordinate ring is naturally the Iwasawa algebra on Zp via the Amicetransform. Expanding the Coleman power series into a power series in the natural coordinate onGm, one thus obtains measures; this gives the “Coleman map”
(5) U Coleman power series−−−−−−−−−−−−−→ Γ(ht. 1 formal group) ∼= Γ(Gm) = Λ(Zp).
However, in the case where the underlying formal group of the Coleman power series has height2 (p nonsplit in K), relatively little was known on how to “convert” Coleman power series intomeasures, and so get a Coleman map analogous to (5). This is because the Iwasawa algebra of Z⊕rpcan be viewed via the Amice transform as the coordinate ring of the r-dimensional (component-
wise height 1) formal group G⊕rm , and is not in any obvious way related to a height 2 formal group.Thus it does not seem likely that there is a natural map like (5) into measures, but there is oneinto distributions, as was constructed by Schneider-Teitelbaum in [42], by naturally interpretingColeman power series as the Fourier transform of a distribution. However, for our purposes, wewish to at least relate elliptic units C to a measure, which the construction of loc. cit., and hencesome modification is needed to get a Coleman map with this property.
The starting point of our strategy for constructing an appropriate Coleman map is to follow thestrategy of Tsuji [57, Section I.4] and instead “thicken” Coleman power series to power series in thecoordinate ring of the universal height 2 formal group, and then to pullback via the identity sectionto a function on the underlying deformation space (viewed as a residue disc in an appropriate
3The author notes that in the p inert in K case, Rubin ([40]) has made a great deal of progress in definingcanonical rank-1 subspaces U± ⊂ U, which are defined by ±-type conditions and play the role of U ′ in his theory. Inparticular, contingent on Conjecture 2.2 of loc. cit., one should obtain canonical main conjectures in the p inert case.The p-ramified case seems to have a different flavor as there are no obvious ±-subspaces. However, the p ramifiedassumption ensures the existence of the canonical subgroup on elliptic curves with CM by OK , which will play acrucial role in our constructions.
4The author wishes to point out that the viewpoint to make the distinction between main conjectures “with-out p-adic L-functions” (involving only “generalized Kato classes” coming from Euler systems) and classical mainconjectures “with p-adic L-functions” was perhaps first emphasized by Skinner [52]. In particular, this distinctionis fruitful for the purposes of relating different main conjectures (where the Kato class more transparently relatesto different Selmer groups), and has led to much progress on the BSD formula in recent years. (See, for example,Jetchev-Skinner-Wan [22].)
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modular curve). This latter function is now a power series in one variable (we have “cut” one Zp-rank in the rank-2 automorphism group), and one can study its p-adic variation in this “horizontal”moduli aspect.
Following the methods of the author in [30] and his thesis [28], one can construct a canonicalcoordinate qdR ∈ OB+
dR on the p-adic universal cover of the moduli space, where OB+dR is the
period sheaf from [44]. This coordinate qdR is roughly obtained by exponentiating the fundamentalde Rham period zdR ∈ P1(OB+
dR) introduced in [30], which measures the position of the Hodgefiltration
H1,0 ⊂ H1dR ⊂ H1
et ⊗OB+dR∼= (OB+
dR)⊕2
in universal de Rham cohomology and is the natural counterpart to Scholze’s Hodge-Tate periodz ∈ P1(O) (see [43]), where O is the p-adically completed structure sheaf, measuring the positionof the Hodge-Tate filtration
H0,1 ⊂ H1et ⊗ O ⊂ O⊕2
in universal etale cohomology. Thus, working on the proetale site and in the period sheaf OB+dR,
one can expand functions on the moduli space as a power series in the coordinate qdR − 1, whosecoefficients are themselves functions on the p-adic universal cover. In all, we get a diagram like
U Coleman power series−−−−−−−−−−−−−→ Γ(ht. 2 formal group)
thicken−−−−→ Γ(universal ht. 2 formal group)
identity section−−−−−−−−−→ Γ(moduli space of ht. 2 formal groups)
qdR-expansion−−−−−−−−−→ Γ(p-adic universal cover of moduli space)JqdR − 1K
(6)
where Γ denotes the global sections of the structure sheaf, and Γ the p-adic completion thereof.
1.2.3. Step 2: p-adic boundedness of global qdR-expansions. The key point of working withpower series in the coordinate qdR is this: taking the qdR-expansion of a modular form G definedon a large affinoid U of the p-adic universal cover Y∞ of a modular curve Y (see [43] for a construc-tion), and restricting to the ordinary locus Uord ⊂ U , one recovers recovers Serre-Tate expansions.Moreover, by the results of Coleman [11, Section A.5] on p-adic Banach spaces arising from thecoordinate ring of the overconvergent locus U can ⊂ U , the restriction map from qdR-expansions onthe overconvergent locus to qdR-expansions on the ordinary locus
(7) Γ(U can)JqdR − 1K→ Γ(Uord)JqdR − 1K
is injective. Specializing to any point y ∈ U Ig := Uord ∩ z = 0, the image of GΓ(U can)JqdR − 1Kunder
(8) Γ(U can)JqdR − 1K→ Γ(Uord)JqdR − 1KΓ(Uord)→Γ(y)⊂Cp−−−−−−−−−−−−→ CpJqdR − 1K
is the Serre-Tate expansion of G at the ordinary point y. (Note that y does not have to be a CMpoint, and so the Serre-Tate expansion at y is not necessarily centered at the Serre-Tate canonicallifting.) Thus, one can view qdR-expansions as “analytically continuing” Serre-Tate expansions.Hence one can use this “rigidity” to do computations with qdR-expansions defined on U can, andin particular the coefficients of these qdR-expansions inherit p-adic boundedness properties fromp-adic boundedness of Serre-Tate expansions.
Let us be more precise on the global objects whose qdR-expansions we will consider. In general,the coefficients to qdR-expansions in (6) of Coleman power series can be p-adically unbounded; thefact that Coleman power series are local objects means these coefficients can seemingly behaverather arbitrarily. However, the Coleman power series arises of a global universal norm is a global
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object, in particular a function on the global universal object (universal elliptic curve). The coordi-nate qdR is itself a global coordinate (as it measures the position of the Hodge filtration in universalde Rham cohomology), and so one gets a qdR-expansion map
E → Γ(large affinoid subdomain U of p-adic universal cover of modular curve)JqdR − 1K.
Given an elliptic unit ξE ∈ C, the Coleman power series of ξE is a weight 1 Eisenstein series G5, andso the image of ξE under the above map is the qdR-expansion of a wt. 1 Eisenstein series G (which,by [30], we can view as a function on the p-adic universal cover). The “large affinoid subdomain” Uof the p-adic universal cover contains a canonical proetale cover of the ordinary Katz-Igusa towerYIg (see [24])qdR-expansion of the Eisenstein series G
G(qdR − 1) ∈ Γ(U)JqdR − 1K → Γ(YIg)JqdR − 1K.
Now let U can ⊂ U be the overconvergent locus and as mentioned in the previous paragraph, andby (7) one sees that the G|Ucan(qdR − 1) ∈ Γ(U can)JqdR − 1K has p-adically bounded coefficeints,and is determined by its restriction to Uord (in fact, even to its restriction to U Ig).
Specializing the coefficients of G|Ucan(qdR− 1) to a CM point y ∈ U can (where the CM point liesin the overconvergent locus exactly because of our p ramified assumption), we thus obtain a powerseries
G(y)(qdR − 1) ∈ OCpJqdR − 1K[1/p].It is this power series which gives rise to the measure LE ∈ OCpJZpK[1/p]. In fact, a calculation
on Serre-Tate expansions shows that LE is supported on Z×p ⊂ Zp. See Theorem 3.9 for theconstruction of LE (where it is denoted by LλE , for λE the type (1, 0) Hecke character attached toE).
1.2.4. Step 2: finishing the construction of the Coleman map. As one can view CpJqdR −1K ⊂ Dist(Zp,Cp) (the target being the Cp-algebra of Cp-valued distributions on Zp), by composing(6) with a specialization of coefficients to an appropriate CM point y ∈ U can
Γ(U)→ Γ(y) ⊂ Cp,
we get our desired Coleman map (see Theorem 5.13)
(9) δ : U→ CpJqdR − 1K ⊂ Dist(Zp,Cp).
From the previous section, we see that this map sends the elliptic unit ξE to LE ∈ OCpJZ×p K[1/p](a measure).
We caution the reader that (9) does not satisfy an “obvious” Λ-equivariance property, as unlikein the height 1 case, the ground group Zp on which we are considering distributions is not natu-rally identified with a Zp-extension, rather it can be thought of as the Fourier dual space of theanticyclotomic Zp-extension. (As explained in Remark 3.10, there is a natural way to identify theanticyclotomic tower with its Fourier dual space in the height 1 case.) The equivariance propertywhich (9) satisfies are given by (108) and (118) below, and roughly reads: for any λ ∈ Λ,
δ(λβ)(rχ) = χ−1(λ)δ(β)(rχ)
for any finite-order anticyclotomic character χ, where rχ is the twisted Fourier transform attachedto χ (Definition 3.5), δ(β)(rχ) is the evaluation of the distribution δ(β) ∈ Dist(Zp,Cp) at the locallyconstant function rχ, and χ(λ) is the evaluation of χ on λ ∈ Λ.
5More accurately, the Coleman power series of an elliptic unit is a Siegel modular unit, whose logarithmic derivativeis a weight 1 Eisenstein series, but we are ignoring twists of our Iwasawa-theoretic objects by moment characterswhich account for these logarithmic derivatives in our Introduction.
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As alluded to before, LE also does not satisfy a standard interpolation property, but rather anew type as follows. Given any finite-order anticyclotomic character χ, we have
LE(rχ) = C(χ)L(E,χ−1, 1)
where C(χ) ∈ C×p is some explicit factor (depending on χ). However, evaluating at χ = 1, we haver1 = 1 and so do have a standard equivariance property, see Remark 5.20. It is for this reason thatwe choose not to call LE an anticyclotomic measure (rather it is a measure on the Fourier dual spaceZp of the anticyclotomic tower; again, see Remark 3.10 for an overview of why this distinction doesnot occur in the height 1 case). See (50), (70) and (79) for details on this interpolation property.
1.2.5. Step 3. Step (3) follows the same overall structure as an analogous step in the proof ofprevious results on p-converse theorems, such as that of [39] in the rank 0 case, and such as thatof [6] (see also [2], which has a very similar approach). Namely, we wish to relate elliptic units toHeegner points, and this is done through relating their associated p-adic L-functions. As the rank0 case is relatively standard (see [39, Section 11]), we focus on the rank 1 case in this section. First,we choose a Rankin-Selberg pair (g, χ0) which is a suitable twist of E (in the sense of Section 4.1);in particular, (g, χ0) is chosen in such a way that
(10) ords=1L(E, s) = ords=1/2L((πg)K × χ0, s)
where πg is the GL2-automorphic representation attached to g, (πg)K its base-change to K, andL((πg) × χ0, s) the (unitarily normalized) Rankin-Selberg L-function. In all, we will have threep-adic L-functions in OCpJZ×p K[1/p]: LE (from Theorem 3.9, where it is denoted by LλE ), L′E (fromTheorem 3.28, where it is denoted by d1Lψcχ0 , where ψ = λE/χ0), and Lg,χ0 (from Theorem 4.2,where it is denoted by Lg×χ0). By comparing interpolation properties, one can prove a factorization
(11) LEL′E = C · (κ∗−Lg,χ0)2
(see (88)), where C ∈ C×p is some constant, κ− : Z×p → Z×p is the moment character x 7→ x andκ∗−Lg,χ0 denotes the pullback measure κ∗−Lg,χ0(f) = Lg,χ0(κ−f). We have a p-adic Waldspurgerformula (see (80))
Lg,χ0(1) = logg(Pg,χ0) · C ′
for some Heegner point Pg,χ0 , where C ′ ∈ C×p . Hence specializing (11) to κ−1− , we get
LE(κ−1− )L′E(κ−1
− ) = CC ′2 · log2g(Pg,χ0).
One can show that L′E(κ−1− ) 6= 0 through the interpolation property (Theorem 4.9), and hence
LE(κ−1− ) 6= 0 if and only if Pg,χ0 is non-torsion, which by Yuan-Zhang-Zhang’s Gross-Zagier formula
on Shimura curves ([60]) happens if and only if ords=1L(E, s)(10)= ords=1/2L((πg)K × χ0, s) = 1.
From Steps (1) and (2), one gets the implication that corankZpSelp∞(E/Q) = 1 =⇒ LE(κ−1− ) 6=
0, and hence we establish the rank 1 p-converse theorem.
1.3. Relation to classical problems of arithmetic. Theorem 1.1 has applications to certainclassical problems of arithmetic. First, let Ed : x3 + y3 = d denote the cubic twist family, whichhas CM by Z[
√−3]. It is known that there exists (x, y) ∈ Q⊕2 such that x3 + y3 = d if and only if
rankZEd(Q) > 0. Sylvester ([55]) and later Selmer ([48]) computed the 3-descent of E` for primes` and arrived at the following
corankZ3Sel3∞(Ed/Q) =
0 ` ≡ 2, 5 (mod 9)
1 ` ≡ 4, 7, 8 (mod 9).
“Sylvester’s conjecture” states that8
Conjecture 1.2 (Sylvester’s conjecture).
` ≡ 4, 7, 8 (mod 9) =⇒ rankZEd(Q) = 1,
and in particular there exists (x, y) ∈ Q⊕2 such that x3 + y3 = d.
Applying Theorem 1.1 to K = Q(√−3) and p = 3, we thus resolve Sylvester’s conjecture.
Theorem 1.3 (Theorem 7.11). Sylvester’s conjecture is true.
Another family of elliptic curves of arithmetic interest is the congruent number family Ed : y2 =x3−d2x. It is known that for squarefree d, rankZE
d(Q) > 0 if and only if d is a congruent number.The “congruent number problem” states that
Conjecture 1.4 (Congruent number problem). 100% of squarefree d ≡ 1, 2, 3 (mod 8) are notcongruent numbers, and all d ≡ 5, 6, 7 (mod 8) are congruent numbers.
Smith [54] was able to vastly improve previously known 2-descent methods in order to get precisedistribution results on corankZ2Sel2∞(Ed/Q), showing for 100% of squarefree d:
corankZ2Sel2∞(Ed/Q) =
0 d ≡ 1, 2, 3 (mod 8)
1 d ≡ 5, 6, 7 (mod 8).
In particular, the first part of the congruent number problem immediately follows. By combiningTheorem 1.1 with K = Q(i) and p = 2 with Smith’s results, we obtain the following.
Theorem 1.5 ([54] and Theorem 7.14). 100% of squarefree d ≡ 1, 2, 3 (mod 8) are not congruentnumbers, and 100% of squarefree d ≡ 5, 6, 7 (mod 8) are congruent numbers.
Moreover, this shows that Goldfeld’s conjecture ([18], or see Conjecture 7.13 below) on ranks ofelliptic curves in quadratic twist families is true for the congruent number family Ed, and gives thefirst case for which the full Goldfeld conjecture is established. See Section 7.3 for more details.
1.4. Structure of the paper. In Section 2, we largely recall and develop notions from the localframework of Coleman power series, as well as relevant notions from relative p-adic Hodge theory.This gives the basis for Step (2) above. In Sections 3 and 4, we construct the relevant p-adic L-functions that appear in Steps (2) and (3), and establish the factorization of measures part of Step(3), using much of the material from Section 2. In Section 5, we accomplish Step (1) above contin-gent on the elliptic units main conjecture (the proof of which is relegated to the Appendix, Section8). In Section 6, we establish the rank 0 part of Theorem 1.1, and in Section 7, we establish the rank1 part. In the Appendix (Section 8), we prove the elliptic units main conjecture used in Section5 by invoking the aforementioned work on the equivariant main conjecture of Kings-Johnson-Leung.
Acknowledgements The author would like to thank Christopher Skinner, Ye Tian, Shou-WuZhang and Wei Zhang for helpful discussions, encouragement and advice. The author was supportedby an NSF Mathematical Sciences Postdoctoral Research Fellowship, award number 1803388.
2. Background
Fix an algebraic closure Q, and view all number fields as embedded in Q. Let p be a prime andfix an algebraic closure Qp. Fix an embedding
(12) ip : Q → Qp,
and denote the completion of any number field M ⊂ Q under this embedding by Mp. Let K be animaginary quadratic field, and let p be fixed by this embedding so that Kp is the completion of Kat p.
9
2.1. Coleman power series. Let Kp/Qp be an extension of degree h(= 1, 2), let Lp/Kp anunramified extension with 〈φ〉 = Gal(Lp/Kp), and let F be a relative Lubin-Tate formal groupwith respect to Lp/Kp, which is a formal group defined over OLp with End(F/OLp) = OKp and
with special φ-linear endomorphism f : F → F φ (where F φ denotes the formal group obtained byapplying φ to the coefficients of F ). Let [+] denote the formal group law on F , and let [a] denote
formal multiplication by a ∈ OKp . Let π′ = f ′(0), d := [Lp : Kp], and let fn = fφn−1 fφn−2 · · · f ,
and note that fd = [ξ] for ξ := NmLp/Kp(π′) ∈ OKp . Given a ∈ OKp , let F [a] denote the a-torsion
group scheme. Let F [fn] denote the fn-torsion group scheme. Note that F [fn] = F [πn].
Example 2.1. Suppose A is an elliptic curve with CM by OK defined over OL which has goodreduction at (all places above) p, and suppose Lp/Kp is unramified. Then the formal completion
along the identity, A, is an example of a Lubin-Tate group relative to Lp/Kp. Let p be the prime
of K fixed by the previously chosen embedding Q ⊂ Qp. The special φ-linear endomorphism is the
projection f : A→ A/A[p] ∼= Aφ.
Let π denote a uniformizer of OKp . Let Lp,n = Lp(F [πn]). Then
κ : Gal(Lp,∞/Lp)∼−→ O×Kp , σ(x) = [κ(σ)](x) ∀x ∈ F [p∞]
is the reciprocal of the inverse of the local Artin symbol. (Caution: later we will use κ−1 to denotecompositional inverse of κ.) Let
U = lim←−n
O×Lp,n
with transition maps given by norm, and let U1 denote the principal part. Fix an f -compatible
sequence αn ∈ F φ−n
[πn] of generators, i.e. each αn generates F φ−n
[πn] as a OKp-module, and
fφ−n
(αn) = αn−1. Sometimes α = (αn) is called a generator the Tate module.Note that fixing a formal parameter X on F , we have Γ(F ) ∼= OLpJXK. We have a multiplicative
norm operator Nf : Γ(F )→ Γ(F ), which is uniqely characterized by
Nfg f =∏
$∈F [f ]
g(X[+]$).
Theorem 2.2 (Coleman power series). There is a Gal(Lp,∞/Kp)-equivarant isomorphism
U ∼−→ Γ(F )×,Nf=φ ∼= OLpJXK×,Nf=φ,
characterized by
β = (βn) 7→ gβ(X), gφ−n
β (αn) = βn.
Moreover, gβ satisfies the following properties:
(1) gβ1β2 = gβ1gβ2,(2) for any σ ∈ Gal(Lp,∞/Lp), gβσ = gβ [κ(σ)],
(3) Nfg = gφ, where gφ denotes applying φ to the coefficients of g.
Note that the principal local universal norms U1 is mapped to 1 + (p, X)OLpJXK by β 7→ gβ.
Hence for β ∈ U1,
log(gβ) ∈ LpJXK.
Given a power series G ∈ LpJXK, we define
G(X) := G(X)− 1
p
∑$∈F [π]
G(X[+]$).
10
Following the argument in [14, Lemma I.3.3], given β ∈ U1 we have
loggβ ∈ OLpJXK[1/p].
Hence we get a map
logg : U1 → OLpJXK[1/p].We also get a twisted version
d
ω0logg : U1(κ−1)→ OLpJXK[1/p].
2.2. Semi-local (log-)Coleman map.
Definition 2.3. Let f0 ⊂ OK be an integral ideal prime to p. Let wf0 denote the number of
elements of O×K congruent to 1 (mod f0). Suppose that wf0 = 1. Let L := K(f0), the ray class fieldof conductor f0 over K. Henceforth fix an elliptic curve A such that
(1) A has CM by OK ,(2) A is defined over K(f0),(3) Ators ⊂ A(Kab),(4) A has good reduction at all primes of L not dividing f0.
Definition 2.4. Retain the notation of Defintition 2.3. Fix an ideal f0 ⊂ OK ; we do not imposef0 6= 1 or (f0, p) = 1 unless otherwise specified. Given any ideal c ⊂ OK , let K(c) be the ray classfield over K of conductor c. Henceforth let L = K(f0), and for 0 ≤ n ≤ ∞ let Ln = K(f0)(A[pn]). Asbefore, let A/L be an elliptic curve with complex multiplication by OK , and such that L(Ators)/Kis abelian. Equivalently, we have
ψA/L = ϕ NmL/K
where ψA/L : A×L → C× is the Hecke character associated with A/L by the theory of complex
multiplication and ϕ : A×K → C× is some Hecke character of infinity type (1, 0).
Let
Φn := Ln ⊗K Kp∼=⊕P|p
Ln,P, rp = OLn ⊗OK OKp ∼=⊕P|p
OLn,P
where P|p runs over prime ideals of OLn above p. For simplicity let Φ0 = Φ. Note that we havenorm maps Nmn : Φn → Φn−1 and Nmn : rp → Rn−1. Let
(13) U := lim←−Nmn
r×p .
Definition 2.5. In the notation of (13), let r1p denote the pro-p part of r×p , i.e. the semi-local
principal units. Then let
U1 := lim←−Nmn
r1p.
Let A/OLp denote the formal completion along the identity of A/OLp . Then F = A is a relativeLubin-Tate group of height 2 for the unramified extension Lp/Kp. Then
U1 ⊗Zp Qp =⊕P|p
U1 ⊗Zp Qp
where P|p ranges over all primes of OL above p. Let R = OL⊗OK OK,p, and note that RJT K[1/p] =⊕P|pOL,PJT K[1/p]. We have a semilocal log-Coleman map (cf. [14, II.4.6])
logg : U1 → RJXK[1/p].11
Our fixed embedding Q → Qp specifies exactly one P|p, and so projecting onto the correspondingcomponent we get a map
logg : U1 → OLpJXK[1/p].Given β ∈ U1, we will continue to denote the corresponding Coleman power series by gβ. Note thatwe also have a Gal(L(A[p∞])/K)-equivariant map on the twist
(14) δ :=d
ω0logg : U1(ϕ−1)→ OLpJXK[1/p].
2.3. Functoriality of universal norms. Now assume that K has class number 1, and let E/Qbe a fixed elliptic curve with CM by OK . Let Kn = K(E[pn]). We have universal norms
UE = lim←−n
O×Kn , U1E = lim←−
n
O×,1Kn .
Let λE be the type (1, 0) Hecke character attached to E, and let f be the conductor of λE and let
f(p) be the prime-to-p part of f, and assume wf(p) = 1 (which is true for all but finitely many E).
In particular, if N is the conductor of E then N = NmK/Q(g)|DK | where DK is the fundamental
discriminant of K. Now let L = K(f(p)) as before and let A/L be an elliptic curve with CM by
OK , which is thus a twist of E. Since Ln = K(f(p)pn) ([14, Proposition II.1.6]), we have Kn ⊂ Lnby the theory of complex multiplication (or the Neron-Ogg-Shafarevich criterion). In particular,we have inclusions of principal semilocal units (OKn ⊗ Zp)×,1 → (OLn ⊗ Zp)×,1. Moreover, by themain theorem of complex multiplication, we have
K(f(p)pn) ⊂ K(f(p)pn) ⊂ K(f(p))(E[pn]),
and so K(f(p))Kn = K(f(p))(E[pn]) = K(f(p)pn) = Ln. Hence the norm for the Galois extensionKn/Kn−1 is induced by the restriction of the norm for the Galois extension Ln/Ln−1. Hence, lettingU denote the module of semilocal universal norms associated with A/L from the last section, wehave a natural inclusion
U1E → U1
as well as a natural norm map
U1 Nm−−→ U1E .
Note that we have a natural restriction map res : Gal(L∞/K)→ Gal(K∞/K) and λ res = ϕ, andso we have twisted versions of these maps
U1E(λ−1
E ) → U1(ϕ−1), U1(ϕ−1)Nm−−→ U1
E(λ−1E ).
Let ω0 ∈ Ω1A/OCp
be the normalized invariant differential, i.e. with ω0/dX(0) = 1. Precomposing
(14) with the above inclusion, we get a map
(15) δE :=d
ω0logg : U1
E(λ−1E )→ OLpJXK[1/p].
2.4. Elliptic units and their Coleman power series.
Definition 2.6. Let qz = e2πiz and let L be a lattice Zτ + Z, τ ∈ C. We will now identify L withthe corresponding elliptic curve C/L. Let A(L) = π−1=(τ) (the area of C/L). As in [14, ChapterII.2], we have the Siegel θ-function
θ(z, L) = e6A(L)−1z(z−z) · qτ (q1/2z − qz−1/2) ·
∞∏n=1
((1− qnτ qz)(1− qnτ q−1
z ))12
.
Let a ⊂ OK be an ideal prime to 6fp. Let L be the period lattice of A.
Θ(z;L, a) = θ(z, L)Na/θ(z, a−1L)12
where Na is the positive generator of the ideal NmK/Q(a).
Definition 2.7. Without loss of generality (i.e. replacing A by one of its Galois conjugates ifnecessary), we may assume L = ΩfOK for some Ω ∈ C×.
(16) e(a) = lim←−n
Θ(Ω; pnL, a) ∈ U1
denote the principal universal norm of elliptic units as in [14, Chapter II.4.9] (see Proposition 2.2.4of loc. cit. for a proof of norm-compatibility).
We have the following proposition describing the Coleman power series of e(a).
Proposition 2.8 (Proposition II.4.9 of [14]). Let P (z) be the Taylor series expansion of Θ(Ω −z;L, a) ∈ LJzK, and let logA : A → Cp be the p-adic logarithm. The Coleman power series ge(a) ofe(a) is ge(a)(T ) = P (logA(T )).
For later purposes, we will seek to relate the Coleman power series of elliptic units to Eisensteinnumbers, following Section II.3 of loc. cit. Let
E1(z, L) =1
12
∂
∂zlog θ(z, L)− 1
2zA(L)−1,
E1(z;L, a) = NaE1(z, L)− E1(z, a−1L).
Proposition 2.9 (II.3.1(7) of [14]). We have
− d
λ′A(T )dTge(a)(T ) = − ∂
∂zΘ(z;L, a) = E1(z;L, a).
2.5. Measures. Given a complete Zp-algebra R and a profinite abelian group G, let
RJGK = lim←−H<G,[G:H]<∞
R[G/H],
so that RJGK can naturally be viewed as the R-algebra of measures (i.e. R-linear maps on thespace of continuous functions) on G. For general RJGK-module M and any Zp-algebra R, we letMR = M⊗R.
2.6. Review of construction of measures in the h = 1 case: idea. It is easy to constructmeasures in the h = 1 (OKp = Zp) case, since then base-changing to W = W (Fp), there exists an
isomorphism Gm/W∼−→ F/W , and the coordinate ring of Gm can natural be viewed as measures
on Zp via Amice’s transform (see [14, Chapter I.3]). Thus for β ∈ U1, we can view
µβ := log(gβ) ∈W JZ×p K
where
logg = log g − 1
p
∑$∈F [π]
log(X[+]$) ∈W JXK ∼= W JZpK.
(one checks that the µβ is supported on Z×p ⊂ Zp ∼= G). Recall that G is finite-index in G, and soµβ extends to a define a measure
µβ ∈W JGKas follows. For any open subset U ⊂ G with σU ⊂ G, we define µβ(U) := µβσ(σU).
Note that we can also vary the unramified extension Lp/Kp, and hence can get a two-variablesystem of universal norms lim←−Lp/Kp unramified
ULp , where the transition maps are the natural norm
maps. However, the formal group F only encodes the local units, and so we need a global (or atleast semi-local) model of F in order to get this two-variable system of universal norms.
13
Let A be an elliptic curve with CM by OK , K/Q an imaginary quadratic field, such that p = ppsplits in K. Suppose A has good reduction at p and p, so that A can be defined over L withLp/Kp unramified. So we can view the formal completion A as a global model of F , and as before
A/W ∼= Gm/W .
Now let Lm = L(A[pm]), and Lp,n = Lmp (A[pn]) = Lmp (A[pn]), and we can form ULmp as before.Now suppose we have
β∞ := (βm) ∈ lim←−m
ULmp .
Note that each βm is an element (βmn ) ∈ ULmp . Then for each m we get a Coleman power series
gβm , which by the above gives a measure µβm ∈ W JGmK on Gm = Gal(Lmp,∞/Kp), which one cancheck ([14, Theorem II.4.12]) actually form an inverse system
µβ∞ := lim←−m
µβm ∈ lim←−m
W JGm,W K = W Jlim←−m
GmK = W JG∞K.
Suppose we want to specialize this to the anticyclotomic quotient O×Kp/Z×p (or G/Z×p ). Then we
should get a 1-variable measure, but what is the correct coordinate for this anticyclotomic variable?The correct coordinate comes from geometry.
2.7. The Serre-Tate coordinate. We move our setting to a global modular curve for convenience(previously, we were in a local modular curve, the Lubin-Tate deformation space for F ).
Let E → Y denote the universal elliptic curve (with some tame level structure) over a modularcurve Y of level Γ(N), p - N , so that we can naturally view ordinary/supersingular residue discs asdeformation spaces of p-divisible groups of ordinary/supersingular elliptic curves (so formal heighth = 1, 2 respectively). From now on, we will take the adic generic fiber of E, Y , etc., unlessotherwise indicated. (For h = 1, one can get by with just the rigid analytic fiber.) At times(mainly in the h = 1 case), our discussion will hold on the level of formal schemes, and we willpoint out when this is the case. Given any sheaf F , let Γ(F) denote its global sections (the toposand site will be clear from context).
Definition 2.10 ([24]). The Igusa tower is the pro-formal scheme Y Ig → Yord over W classifyingdeformations of p-divisible groups A[p∞] of ordinary elliptic curves A together with isomorphism
µp∞ ∼= A[p∞], or equivalently Zp∼−→ TpA
et.
Then let D be an ordinary residue disc for F , and let D∞ = Y Ig ×YordD, which is a trivial
Z×p -cover D∞ → D (looking at the action of Z×p on the trivialization Zp∼−→ TpA
et).
Note that our fixed isomorphism θ : Gm/W∼−→ A/W is equivalent to a choice of trivialization
θ : Zp∼−→ TpA
et.For an ordinary residue disc D, Serre-Tate constructed a canonical isomorphism (on the level of
formal schemes)
D/W∼−→ HomZp(TpA
et ⊗ TpAet, Gm/W ),
which then induces an identification of pro-formal schemes
(17) D∞/W ∼=⊔
Zp∼−→TpAet
Gm/W.
(In particular, we see how D∞ → D is a trivial Z×p -cover.) The coordinate T on Gm/W pulls backto a coordinate on D∞/W , called the Serre-Tate coordinate, and which we continue to denote byT . Given an element of the coordinate ring f ∈ Γ(OD∞/W ), we denote its image under the inducedidentification
Γ(D∞/W ) ∼=∏
Zp∼−→TpAet
Γ(Gm/W ) =∏
Zp∼−→TpAet
W JT K
14
by f(T ), and call f(T ) the Serre-Tate (or T -)expansion of f .
2.8. Anticyclotomic measures via Serre-Tate coordinate in the h = 1 case. Note thatG∞ ∼= (OK ⊗Zp)××∆ for some finite abelian group ∆. We can specialize the measure µβ∞ to theanticyclotomic tower G∞/Z×p , where Z×p ⊂ (OK ⊗Zp)× ⊂ G∞ is the canonical inclusion. However,we want to view this specialization naturally as a power series in T .
Recall that µβ∞ = lim←−m µβm , and that each µβm came from a Coleman power series gβm : µβm
was defined from the power series log(gβm) ∈ W JXK. Recall our universal elliptic curve E → Y ,so that for an ordinary residue disc D ⊂ Y , E → D is a universal deformation of an ordinaryp-divisible group. For example, if D contains the CM point corresponding to A, then this is theuniversal deformation of A[p∞] mod p. Base-change everything to W for simplicity. Let E be theformal completion of the universal elliptic curve E along the identity section. We have a canonicalinclusion
(18) Γ(F ) = Γ(A) = W JXK → Γ(D)JXunivK = Γ(E|D),
from the canonical inclusion W ⊂ Γ(D) and sending X 7→ Xuniv. Loosely speaking, choosing an
isomorphism (or equivalently, via (17), a trivialization Zp∼−→ TpA
et) Γ(D) ∼= W JT K, the aboveinclusion is induced by the specialization E → A (“setting T = 0”). Now let gβm(Xuniv) denotethe image of gβm(X) under (18). Then via the identity section
e0 : D → E|D,
we get a section e∗0Xuniv ∈ Γ(D). Using our previously fixed θ : Zp
∼−→ TpAet, (17), we then get a
map Γ(D) → Γ(D∞) (a section of (17)), and then we can take a T -expansion of e∗0Xuniv|D∞ , which
we denote by e∗0Xuniv(T ).
Thus, for each m we get a power series under the composition
gβm(X) 7→ gβm(Xuniv)θe∗07−−−→ gβm(T ) := gβm(e∗0X
univ(T )) ∈W JT K× = Γ(D∞)×.
One can actually package together the gβm via a “half-Coleman map construction” to get powerseries
g−β ∈W JT K×, logg−β := log g−β −1
p
p−1∑j=0
log g−β (ζjp(T + 1)− 1) ∈W JT K.
The latter power series thus gives us an element of W JG/Z×p K, and one can check that the induced
map U1⊗W → W JG/Z×p K is JG/Z×p K-equivariant. The construction of the half-Coleman mapinvolves mimicking Coleman’s original argument and constructing a limit of the gβm interpolating
the βmn . Crucially, we use sections D∞ → E|D∞ induced from our fixed basis of the Tate-module
α to check that this limit converges, instead of just using the identity section 0 : D → E and θ.
2.9. The h = 2 case: idea. However, in the h = 2 case there are no nontrivial maps of formalgroups Gm → F . We will instead proceed by viewing Coleman power series as elements of thecoordinate ring of the universal formal group E → Y for a modular curve Y . But the point isthat pulling back our series to the “base” moduli space Y , height 1 and height 2 deformationspaces are on equal footing (both loci in Y ). On the ordinary locus, the anticyclotomic tower Γis contained in the Galois group of the Igusa tower Y Ig → Y ord (recalled in Definition 2.10). Ifour formal group F , viewed as a point in Y , is “not too far” from Y ord, then we can hope to usethe geometry of a neighborhood of Y Ig to construct measures. More precisely, “not too far” meanson the overconvergent locus, and in order to guarantee that the CM point corresponding to our Fbelongs to this locus, we make the following fundamental assumption henceforth.
Assumption 2.11. Henceforth, assume p is ramified in K.15
This can be done by embedding O×Kp → GL2(Zp) = Gal(Y∞/Y ) where
Y∞ = Y (Γ(pn)) ∈ Yproet
is the Γ(p∞)-level modular curve, which is the inverse limit of the etale covers Y (Γ(pn)) obtainedby adding Γ(pn)-level structure to the moduli data of Y . The quotient O×Kp/Z
×p will be, in some
sense, realized inside the Galois group of the tower Y∞ → Y .For each β ∈ U , we still get a Coleman power series gβ(X), but now it is not at all obvious
how to get a measure from this, since the Galois-action on X is not the same as the group ringaction (in h = 1, we used the fact that F/W is isomorphic to Gm/W , and in this latter group theGalois action and group ring actions coincide). So we want to find some natural coordinate to tryto “expand” gβ(X) in, and it turns out that there are h = 2 analogues of the Serre-Tate coordinateT .
2.10. Period sheaves. For this section, we refer to [44, Section 4, Section 6].Let Y be a local or global modular curve. Let Yproet denote the proetale site as in loc. cit. Let
OY denote the proetale structure sheaf of Y , O+Y denote the integral subsheaf, O+
Y := lim←−nO+Y /p
n
denote the p-adic completion, and OY = O+Y [1/p].
Let B+dR denote the relative de Rham period sheaf on Y , which is equipped with a projection
ϑ : B+dR OY and a filtration Fili = ker(ϑ)iB+
dR. Let OB+dR denote the relative de Rham structure
sheaf, which is equipped with inclusions OY ⊂ OB+dR and B+
dR ⊂ OB+dR and a projection ϑ :
OB+dR OY such that B+
dR ⊂ OB+dR
ϑ−→ OY is ϑ : B+dR OY , and OY ⊂ OB+
dRϑ−→ OY is the
natural p-adic completion. We also have a filtration FiliOB+dR = ker(ϑ)iOB+
dR and a connection
∇ : OB+dR → OB
+dR ⊗ Ω1
Y
(where Ω1Y is the proetale sheaf of differentials) such that the inclusion OY ⊂ OB+
dR∇−→ OB+
dR⊗OYΩ1Y factors through the natural derivation d : OY → Ω1
Y .
2.11. The fundamental de Rham period zdR and the coordinate qdR. In the h = 1 case, thenatural Serre-Tate coordinate on Y Ig gave the right anticyclotomic variable. In h = 2, we will userelative p-adic Hodge theory to introduce a coordinate qdR
6 which extends Serre-Tate coordinatesin some sense (but not the literal sense, since Serre-Tate coordinates can only be defined on localdeformation towers D∞ and are not global functions on Y Ig). We now start working entirely onthe proetale site Yproet of the adic space Y , and in the proetale topos on this site.
The first observation is that, by work of Dwork, Messing and Katz, one can view T + 1 as theexponential of some divided power period measuring the position on D∞ of the Hodge filtration inde Rham cohomology (with respect to a basis of de Rham cohomology provided by θ and θ). Thisperiod is not really a function on D∞ (it is a divided power function), but its exponential T + 1 is.
We seek to generalize this latter viewpoint. On Y∞, we have a Zp-basis (e1, e2) : Zp,Y∞∼−→
TpE|Y∞ . Let ρ : E → Y denote the universal morphism, and let ω := ρ∗Ω1E/Y . Using relative
comparison theorem7
(19) idR : H1dR(E) ⊂ TpE(−1)⊗Zp,Y OB
+dR,
we get an inclusion of the Hodge filtration
idR : ω|Y∞ ⊂ H1dR(E)|Y∞ ⊂ TpE ⊗Zp OB+
dR,Y∞t−1
6Actually, this was first introduced in my thesis.7This is due to Scholze, and is compatible with respect to natural filtrations and connections. We will blackbox the
construction of OB+dR, B+
dR, etc. and just mention their key properties when needed, e.g. filtrations and connections.
16
which by [44, Proposition 7.9] with i = −1 and E = H1dR(E) factors through
(20) idR : ω|Y∞ ⊂ TpE ⊗Zp OB+dR,Y∞
.
Here,
(21) t := log[(〈e1,1, e2,1〉, 〈e1,2, e2,2〉, . . .)] ∈ B+dR(Y∞).
Definition 2.12. Let E → Y denote the universal object, and (e1, e2) : Z⊕2p,Y∞
∼−→ TpE|Y∞ denotethe universal level structure over Y∞.
Using the universal Weil pairing, view e2 ∈ TpE|Y∞ = Hom(E[p∞]|Y∞ , µp∞,Y∞), E[p∞]|Y∞ and
µp∞,Y∞ denoting p-divisible groups over the strong completion Y∞ of Y∞ (i.e. Y∞ ∼ Y∞; see [46] or
[47] for the definition of strong completion and “∼”). Note that OY∞ = OY∞ . Let πHT : Y∞ → P1
denote the Hodge-Tate period map ([43], [9]), and let Vx := x 6= 0 ⊂ P1 denote the standard
affinoid chart, and let Ux := π−1HT(Ux) ⊂ Yx which is an affinoid open. Since Y∞ ∼ Y∞, we have a
homeomorphism |Y∞| ∼= |Y∞|, and so the open Ux ⊂ Y∞ defines a proetale object Ux ∈ Yproet withUx ⊂ Y∞.
Then the fake Hasse invariant ([43], [9]) is
(22) s := e∗2dT
T∈ (ω ⊗OY OY∞)(Ux).
From (20), we get
xdR,ydR ∈ OBdR ∈ (OB+dR ⊗OY OY∞)(Ux)
defined by
(23) idR(s) = xdRe1 + ydRe2.
Note that ϑ : OB+dR → OY extends to OB+
dR ⊗OY OY → OY by ϑ(x⊗ y) = ϑ(x)y.
Definition 2.13. Define
(24) U ′x := Ux ∩ ϑ(ydR) 6= 0 ⊂ Y∞and note that ydR is invertible on U ′x. Define
zdR := −xdR
ydR∈ (OB+
dR ⊗OY OY )(U ′x).
In fact, z+dR lives in OB+
dR.
Proposition 2.14. We have zdR ∈ OB+dR(U ′x).
Proof. Let w be a local generator, say on U ∈ Yproet/U′x (where the latter denotes localization
of the site Yproet to U ′x ∈ Yproet), of the locally free rank-1 sheaf ω. Then under (20) we have
idR(w) = xe1 + ye2 for x, y ∈ OB+dR(U). Then
−xy
= zdR|U ,
and so zdR(U) ∈ OB+dR(U). Now since OB+
dR is a proetale sheaf, by the sheaf gluing property we
have z+dR ∈ OB
+dR(U ′x).
From this, one gets
zdR := zdR (mod t) ∈ OB+dR,Y∞
/(t)(U ′x).17
We call zdR the de Rham fundamental period.8 Then one can define
qdR = exp(zdR − zdR) =∞∑n=1
(zdR − zdR)n
n!
where zdR is the image of zdR under the composition of the relative Fontaine projection (recallt ∈ kerϑ)
ϑ : OB+dR,Y∞
/(t)→ Owith the inclusion
O = B+dR/(t) ⊂ OB
+dR,Y∞
/(t).
The above expression for qdR makes sense since OB+dR/(t) is (kerϑ)-adically complete (note that
zdR − zdR ∈ kerϑ).
2.12. qdR-expansions. Recall there is a natural connection
∇ : OB+dR,Y∞
/(t)→ Ω1Y∞ ⊗OY∞ OB
+dR,Y∞
/(t),
extending the usual exterior derivative d : OY∞ → Ω1Y∞
under the inclusion OY∞ ⊂ OB+dR,Y∞
/(t).
For simplicity, write d = ∇, so that dqdR = ∇(qdR), etc. Importantly, B+dR,Y∞
/(t) are the horizontal
sections of d, and so dzdR = 0. It is easily checked that dqdR/qdR = d(zdR − zdR) = dzdR.As we will make use of the following “intermediate period ring”, we give it a special notation
(which is compatible with the notation of [30, Chatper 5]).
Definition 2.15. Henceforth, letO∆ := OB+
dR,U ′x/(t).
Definition 2.16. Define the qdR-expansion map(25)
Γ(OB+dR,U ′x
/(t)) → Γ(OU ′x)JqdR − 1K, f 7→ f(qdR − 1) :=
∞∑n=0
1
n!ϑ
((d
dqdR
)nf
)· (qdR − 1)n.
The map is indeed injective since the composition
OY∞ ⊂ O∆ϑ−→ OY∞
is the identity. Note that the coefficients of f(qdR − 1) are themselves functions on U ′x, the strongcompletion of U ′x. The qdR-expansion map is compatible with the connection d, i.e. for any sectionw ∈ Ω1
Y∞, letting
∇w = ∇/w : O∆ → O∆
be the induced derivation, we have
∇w(f(qdR)) = (∇w(f))(qdR).
Also note that the coefficients of f(qdR − 1) are horizontal with respect to d, i.e. annihilated by d.
This is because d annihilates the p-adically complete (perfectoid) ring Γ(OU ′x) = Γ(OU∞).
Given any y ∈ U ′x, we define
f(y)(qdR − 1) ∈ Γ(OU ′x)(y)JqdR − 1K
to be the power series obtained by specializing all the coefficients of f(qdR − 1) to y. Note that
(26) ϑ(f(y)(qdR − 1)) = ϑ(f(qdR))(y)
since ϑ(qdR − 1) = 0.
8zdR measures the position of the Hodge filtration (H1,0 ⊂ H1dR), and is the counterpart to Scholze’s Hodge-Tate
period measuring the position of the Hodge-Tate filtration (H0,1 ⊂ H1dR)
18
2.13. Relation of qdR-expansions to classical Serre-Tate expansions. The key property ofqdR-expansions is that they give a way to interpolate Serre-Tate expansions on the “big” Igusa tower(denoted YIg below), as well as analytically continue Serre-Tate expansions to certain proetale coversof the overconvergent locus. It is an important point that qdR is a global object (defined on mostof the infinite-level modular curve Y∞), whereas the Serre-Tate coordinate is only a local object(defined only on the Igusa tower localized to an ordinary residue disc). Thus while qdR-expansionsdo not recover Serre-Tate coordinates directly, they do recover Serre-Tate expansions.
The key to proving this latter fact is first recalling the construction of Katz’s canonical differentialωKatz
can from [26], which is a global (defined on all of Y Ig) differential whose square under the Kodaira-Spencer map is locally d log(1 + T ) where T is the Serre-Tate coordinate. In particular, the squareof ωKatz
can gives a global interpolation of the local differential d log(1 +T ). On the other hand, in [30]and [28], the author constructed a proetale section ωcan of ω ⊗ OB+
dR, defined on a large affinoid
subdomain of Y∞, that specializes to ωKatzcan under restriction to the big Igusa tower YIg.
Recall the universal object π : E → Y , and define the Hodge bundle by
ω := π∗Ω1E/Y .
Definition 2.17. Recall the universal etale section eet : Y Ig → TpEuniv,et|Y Ig , where Euniv → Y ord
denotes the universal (ordinary) object (we will adopt this slight abuse of notation for the remainder
of this definition). Let E denote the formal completion of E along its identity section, and letµp∞,Y ord denote the multiplicative group over Y ord. From the universal Weil pairing
TpEuniv,et × Euniv[p∞]→ µp∞,Y ord ,
we get an identification HomZp(E[p∞], µp∞,Y ord) (see [26, Introduction]). Thus letting dqq be the
canonical differential on µp∞,Y ord , we get a differential
ωKatzcan := e∗et
dT
T∈ ω(Y Ig).
Recall the universal de Rham cohomology H1dR(E), equipped with its Poincare pairing
〈·, ·〉Poin : H1dR(E)×H1
dR(E)→ OY .
Recall the Gauss-Manin connection ∇ : H1dR(E) → H1
dR(E) ⊗OY Ω1Y . Then the Kodaira-Spencer
map is the map of line bundles on Y
KS : ω⊗2 → Ω1Y , KS(w⊗2) = 〈w,∇w〉Poin.
By [25, Appendix], KS is an isomorphism of line bundles on Y .
Theorem 2.18 (Main Theorem of [26]). Let D ⊂ Y ord be any ordinary residue disc, and letDIg = D ×Y ord Y Ig be the localized Igusa tower. Let T be the Serre-Tate coordinate on DIg. Wehave
(27) KS(ωKatz,⊗2can )|DIg = d log(1 + T ).
We now recall the aforementioned canonical differential of [30], as well as a few facts provedabout it in loc. cit.
Definition 2.19 (Chapter 4.9 of [30]). Recall the affinoid subdomain U ′x ⊂ Y∞ from (24), so thatϑ(ydR) is invertible on U ′x. Define
(28) ωcan :=s
ydR∈ (ω ⊗OY OB
+dR ⊗OY OY∞)(U ′x).
19
Proposition 2.20. Recall the map idR of (20). We have
(29) idR(ωcan) = −zdRe1 + e2.
In fact,
ωcan ∈ (ω ⊗OY OB+dR,Y∞
)(U ′x).
Proof. The first identity follows from (23) and (28). For the second, we note that by the firstidentity and zdR ∈ OB+
dR(U ′x), we have
ωcan ∈ (H1dR(E)⊗OY OB
+dR)(U ′x) ∩ (ω ⊗OY OB
+dR ⊗OY OY∞)(U ′x) = (ω ⊗OY OB
+dR,Y∞
)(U ′x).
Definition 2.21 (Big Igusa tower). Let B ⊂ GL2(Zp) denote the usual Borel group (of upper-triangular matrices):
B :=
(Z×p Zp0 Z×p
), B1 :=
(Z×p Zp0 1
).
One can define a B1-cover YIg → Y Ig with YIg ⊂ U ′x ⊂ Y∞. Here YIg ⊂ Y∞ is defined as the locus
of triples (E, e1, e2) (suppressing Γ(N)-level structure for brevity) where e1 : Zp∼−→ TpE. The map
YIg → Y Ig is defined as follows: given (E, e1, e2), we define a map e2 : µp∞ → E by ζ 7→ e, where
e ∈ E[p∞] is the unique section with 〈e, e2〉 = ζ. By the theorem of Tate, this extends to a map
e : Gm → E.
Recall the Hodge-Tate period z (cf. [43] and [9, Section 2], though in our convention, z is thereciprocal of the Hodge-Tate period z in loc. cit.). Then
1/z ∈ OY∞(Ux).
Proposition 2.22. We have
(30) YIg = 1/z = 0.
Proof. Recall that z measures the position of the Hodge-Tate filtration (see loc. cit.). In otherwords, given any local section s−1 of ω−1, say defined on U ⊂ Ux, we have under (19)
idR(s−1) ∈ TpE(−1)⊗Zp,Y OY∞(U) ∼= OY∞(U)⊕2,
say idR(s−1) = x′e1 + y′e2, and 1/z = −y′/x′. On YIg, e1 trivializies the Hodge-Tate filtration, and
so we have y′ = 0 locally on YIg, i.e. 1/z = 0. Using the sheaf property for OY∞ , we are done.
Theorem 2.23 ((4.50) and Proposition 4.38 of [30]). We have
(31) KS(ω⊗2can) = dzdR.
as well as the “p-adic Legendre relation”
(32)xdR
z+ ydR = 1,
and the comparison with Katz’s canonical differential
(33) ωcan|YIg = ωKatzcan |YIg .
20
Proof. Let 〈·, ·〉Weil : TpE×TpE → Zp,Y denote the universal Weil pairing. For the map idR of (19),
by [30, Theorem 4.31] we have for our previously defined t ∈ B+dR(Y∞) from (21)
〈·, ·〉Poin,Y∞ = 〈idR(·), idR(·)〉Weil,Y∞ · t−1.
Now by Proposition 2.20, recalling that ∇(e1) = ∇(e2) = 0, we have
KS(ω⊗2can) = 〈ωcan,∇(ωcan)〉Poin = 〈idR(ωcan),∇(idR(ωcan))〉Weil · t−1
(29)= 〈−zdRe1 + e2, e1〉 · (−dzdR)t−1 = 〈e1, e2〉Weilt
−1 · dzdR = dzdR.
This proves (31).For (32), let s−1 denote the dual of the fake Hasse invariant s under the Poincare pairing 〈·, ·〉Poin.
Then
idR(s−1) = x′e1 + y′e2
for some x′, y′ ∈ OY∞(Ux), with −x′/y′ = z. By functoriality of the Weil pairing (see [30, Proposi-tion 4.8] for details), we have
1 = 〈s, s−1〉Poin = 〈idR(s), idR(s−1)〉Weil · t−1 = 〈idR(e∗2dq
q), x′e1 + y′e2〉Weil · t−1
= 〈e2, x′e1 + y′e2〉Weil · t−1 = −x′.
Hence idR(s−1) = −e1 + 1ze2. Now we have
1 = 〈s, s−1〉Poin = 〈idR(s), idR(s−1)〉Weil · t−1 = 〈xdRe1 + ydRe2,−e1 +1
ze2〉Weil · t−1 =
xdR
z+ ydR,
which gives (32).For (33), we note that by (30) and (32), we have ydR|YIg = 1. Moreover, recall the map YIg → Y Ig
is defined by sending e2 7→ eet. Then
ωcan|YIg =s
ydR|YIg = s|YIg = e∗2
dq
q|YIg = e∗et
dq
q|YIg = ωKatz
can |YIg ,
which gives (33).
Convention 2.24. Note that reducing ωcan modulo our previously defined t ∈ B+dR(Y∞) from (21),
we get
ωcan (mod t) ∈ (ω ⊗OY OB+dR,U ′x
/(t))(U ′x).
Henceforth, we make a slight abuse of notation and denote this by ωcan (we will no longer refer tothe original ωcan ∈ (ω ⊗OY OB
+dR,Y∞
)(U ′x)). We have
ωKatzcan (mod t) = ωKatz
can .
Hence from (31) we get
(34) KS(ω⊗2can) = dzdR.
Since
OY∞ ⊂ O∆ϑ−→ OY∞
is the natural p-adic completion map, from (33) we get
(35) ωcan|YIg = ωKatzcan |YIg .
21
For a fixed ordinary residue disc D ⊂ Y ord, then letting D∞ = D ×Y ord YIg, we have thatD∞ → D∞ is a trivial B1-cover. Then as in [30, Chapter 4.6], the above shows that
(36) d log qdR|D∞ =dqdR
qdR|D∞ = dzdR|D∞
(34)= KS(ω⊗2
can)|D∞(35)= KS(ωKatz,⊗2
can )(27)= d log(1 + T )|D∞ ,
where T |D∞ denotes the pullback of the Serre-Tate coordinate T along D∞ → D∞. We willsee shortly that for any y ∈ YIg, f(y)(qdR − 1) (the latter denoting specialization of coefficientsto y ∈ YIg) is the Serre-Tate expansion centered at y (and not necessarily Serre-Tate expansioncentered at the CM point origin of D∞).
Let us briefly explain what we mean by Serre-Tate expansion centered at y ∈ YIg. Let D ⊂ D∞containing y of the trivial profinite etale cover D∞ → D for an ordinary residue disc D, and let Tdenote the Serre-Tate coordinate on D so that |T (y)| < 1. Then given f ∈ Γ(D∞), its Serre-Tateexpansion f(T ) satisfies
f(T ) = g((1 + T )/(1 + T (y))− 1) = g(T [−]GmT (y))
for a unique power series g(t) ∈ O+YIg(y)JtK[1/p], where [+]Gm denotes the usual multiplicative
formal group law. For simplicity, let
Ty = (1 + T )/(1 + T (y))− 1 = T [−]GmT (y).
Then g(t) is the Serre-Tate expansion of f centered at y, and explicitly we have
(37) g(t) =
∞∑n=0
bntn, bn =
(1
n!
(d
dTy
)nf
)(y).
Proposition 2.25. Suppose f ∈ Γ(OYIg). Then for any y ∈ YIg, the power series f(y)(qdR− 1) isequal to the Ty-expansion f(Ty) of f centered at y.
Proof. By (26), we have
1
n!ϑ
((d
dqdR
)nf(y)(qdR − 1)
)(26)=
1
n!ϑ
((d
dqdR
)nf(qdR − 1)
)(y) =
1
n!ϑ
((d
dqdR
)nf
)(y).
Hence, by (25) and (37), to prove the assertion it suffices to show that
(38)1
n!
((d
dTy
)nf
)(y) =
1
n!ϑ
((d
dqdR
)nf
)(y)
for all n. By (36), we have
(39) d log qdR = d log(1 + T ) = d(log(1 + T )− log(1 + T (y))) = d log(1 + Ty).
By the elementary identity
1
n!qn(d
dq
)n=
( qddq
n
)=
( dd log q
n
),
we have
1
n!qndR
(d
dqdR
)n=
( dd log qdR
n
)(39)=
( dd log(1+Ty)
n
)=
1
n!(1 + Ty)
n
(d
dTy
)n.
Now applying this differential operator to f , applying ϑ : O∆ → OY∞ , specializing to y (notingTy(y) = 0), and noting that
(40) ϑ
(1
n!(1 + Ty)
n
(d
dTy
)nf
)=
1
n!(1 + Ty)
n
(d
dTy
)nf
22
since OY∞ ⊂ O∆ϑ−→ OY∞ is just the natural inclusion, we get
ϑ
(1
n!qndR
(d
dqdR
)nf
)(y) = ϑ
(1
n!(1 + Ty)
n
(d
dTy
)nf
)(y)
(40)=
(1
n!(1 + Ty)
n
(d
dTy
)nf
)(y)
=1
n!
((d
dTy
)nf
)(y),
which (recalling that ϑ(qdR − 1) = 0) gives (38).
As a consequence, we have the following integrality result on YIg. We let
(41) O+(U) := f ∈ O(U) : |f(y)| ≤ 1 ∀y ∈ U,
which by results of Scholze actually defines a proetale sheaf. We let O+ = lim←−nO+/pn denote the
p-adic completion, and O = O+[1/p]. It is not known (and is probably not true) that O+(U) isalways the p-adic completion of O+(U), but if U is perfectoid then this is true.
Corollary 2.26. Given f ∈ Γ(O+YIg), we have
f(qdR − 1) ∈ Γ(O+YIg)JqdR − 1K.
Proof. This follows immediately from the integrality of the Ty-expansion at all y ∈ YIg and using(41).
Lemma 2.27. Let f ∈ Γ(OYIg) and let f(Ty) be the Ty-expansion at y ∈ D ⊂ D∞ for a section
D ∼= D of the trivial profinite etale cover D∞ → D. Then
(42) f(ζjpn(1 + Ty)1/pm − 1) =
(1 j/pn
0 1/pm
)∗f(Ty).
Proof. The first equality of (42) follows from the computations on Serre-Tate coordinates of [4,Section 7], which easily generalize to Ty in place of T .
2.14. Overconvergent locus. Recall that the classical overconvergent locus Y can ⊂ Y is simplythe rigid-analytic locus where there canonical subgroup exists (i.e. the locus cut out by |Ha| >p−p/(p+1), where Ha is any characteristic lift of the Hasse invariant). Let Y (r) denote the rigid-analytic subdomain of Y where |Ha| ≥ p−r. Then for any 0 ≤ r < p/(p+ 1),
Y ord = Y (0) ⊂ Y (r) ⊂ Y can,
where the former inclusion is actually an inclusion of rigid analytic spaces (more precisely, Y (r)is a “strict wide open neighborhood” of Y ord). By results of Coleman on p-adic Banach spacesattached to affinoids, [11, Lemma A5.1, Proposition A5.2], using the fact that Y ord ⊂ Y (r) is aninner (even overconvergent) map of affinoids in sense of the Definition on p. 34 of loc. cit. (asproven in Section B.2 of loc. cit.), this means that
(43) Γ(Y (r)) → Γ(Y ord)
is a completely continuous map of p-adic Banach spaces, where the norm is the Gauss (i.e. supre-mum) norm on both affinoid algebras. Moreover, Coleman (Proposition A5.4 of loc. cit.) showedthat the Gauss norm behaves well with respect to etale maps: if X → Y is an etale morphismof rigid spaces, then Γ(OY ) → Γ(OX) and the Gauss norm on the target extends the one on thesource. In particular, one can consider the proetale covers
Y (r)∞ = Y∞ ×Y Y (r)→ Y (r), Y ord∞ = Y∞ ×Y Y ord,
23
and apply these norm-extension theorems to these proetale objects (i.e. to limits of their etalecoordinate rings). We have universal sections e1, e2 of TpE on Y (r)∞, and we define
YIg(r) := e−11 (TpE|Y (r)∞),
so that YIg(0) = YIg, and YIg(r)→ Y (r) is hence a proetale cover, and
(44) YIg = YIg ×Y (r) Yord.
The above discussion gives the following boundedness result of overconvergent qdR-expansions.
Proposition 2.28. Suppose 0 ≤ r < p/(p + 1). Given f ∈ Γ(OB+dR,YIg(r)
) with f |YIg ∈ Γ(OYIg),
we have
f(qdR − 1) ∈ Γ(O+YIg(r)
)JqdR − 1K[1/p].
Moreover, the map induced by restriction
(45) Γ(O+YIg(r)
)JqdR − 1K[1/p] → Γ(O+YIg)JqdR − 1K[1/p]
is an injection, and so f(qdR − 1) is determined by f |YIg(qdR − 1).
Proof. This follows from Corollary 2.26, (43) being completely continuous (and so the supremumnorms are equivalent norms), the above stated fact that etale maps preserve Gauss norm, and (44).
Corollary 2.29. Let f ∈ OB+dR,Y (YIg(r)). Then
(46) f(ζjpnq1/pm
dR − 1) =
(1 j/pn
0 1/pm
)∗f(qdR − 1).
Proof. This follows immediately from (42) and (45).
2.15. The period Ωp. We conclude by defining one more period that will appear in the interpo-lation properties of our p-adic L-functions.
Definition 2.30. Recall our fixed CM elliptic curve A over L = K(f(p)) from Definition 2.3.Henceforth, fix a generator
ω0 ∈ Ω1A/OL .
A choice of level structure (e1, e2);Z⊕2p
∼−→ TpA, and Γ(N)-level structure on A, we have a pointy = (A, e1, e2) ∈ Y∞ (again suppressing the Γ(N)-level structure from the notation). Recall ωcan ∈(ω ⊗OY OB
+dR,U ′x
/(t))(U ′x), so that we get ϑ(ωcan) ∈ (ω ⊗OY OB+dR,U ′x
/(t))(U ′x). Then if y ∈ U ′x, we
define
(47) Ωp(y) ∈ Cp, ϑ(ωcan)(y) = Ωp · ω0.
3. p-adic L-functions for GL1/K
3.1. Notation. Recall our fixed Q, fixed Qp, and fixed embedding Q ⊂ Qp. For any subfield
M ⊂ Q, we denote by Mp the induced p-adic completion in Qp. Throughout, let λ : A×K/K× → C×
denote the type (1, 0) over an imaginary quadratic field K. Let f denote the conductor of λ, and
let f(p) denote the prime-to-p part of f. In our main application, λ will be the Hecke character λEassociated to en elliptic curve E/Q with CM by OK , or a twist thereof, and K will be of classnumber 1. Assume that p is ramified in K. Thus, in the class number 1 situation, K is of the formK = Q(
√−p) or Q(i). Let p denote the prime ideal of OK above p. We will often conflate notation
and also let λ denote the p-adic avatar of λ.24
Let
ε :=
⌊1
ordp(p)(p− 1)
⌋+ 1 =
1 p > 3
2 p = 3
3 p = 2.
.
Hence letting Γ := 1+pεOKp , we have a natural decomposition O×Kp = µ(OKp)×Γ, where µ(OKp) =
(OKp/pεOKp)× denotes the roots of unity in OKp . Letting q = p if p > 2 and q = 4 if p = 2, we let
Γ− := Γ/(1 + qZp) ∼= Zp.
Let Γ−,n = Γ−/(Γ−)pn ∼= Z/pn. Let G− := O×Kp/(Z
×p O×K). Note that G− = Γ− if p > 3, or if p = 3
and K = Q(√−3), or if p = 2 and K = Q(i).
Choice 3.1. We fix an element of γ ∈ Γ such that the image of γ under the projection Γ→ Γ− isa topological generator. Thus we get an induced identification
Γ− = Zp.
Given χ ∈ Hom(Γ−, µp∞), we will sometimes write χ(x) = χ(γx) for x ∈ Zp.
Henceforth, given any Hecke character ρ : A×K/K× → C×, let f(ρ) ⊂ OK denote its conductor.
Henceforth, let C`(f) denote the ray class group modulo f, so that C`(1) is the ideal class group.
Assumption 3.2. Henceforth, assume that
(1) ordp(f) ≤ ε and f = f (since p is assumed to be ramified, this is equivalent to the equality
on prime-to-p parts f(p)
= f(p)). When λ is the Hecke character attached to an ellipticcurve E/Q with CM by OK , it is known (see [49]) that λ|O×Kp (O×Kp) ⊂ O
×K , and so the first
assumption holds, and moreover λE(x) = λE(x), so that the second assumption holds.(2) Letting η := λ|A×QN
−1Q denote the central character of λ, assume that p|f(η) where f(η)
is the conductor of η. When λ is attached to a CM elliptic curve, η = ηK , the quadraticcharacter attached to K/Q, and by our p ramified assumption we have p|f(η).
(3) p - #C`(1), that is p does not divide the class number of K. Clearly this holds if K hasclass number 1 (such as when λ = λE for the Hecke character λE associated with E/Q).9
3.2. Parametrizing anticyclotomic characters. Choose a nontrivial additive character Ψ :Kp → C× and let π be a uniformizer of OKp (both Ψ and π to be fixed later in Choice 3.12). Thenfor any character χ ∈ Hom(Γ−, µp∞), we have for all x ∈ OKp ,
χ(γx) = Ψ (aχπx)
for a unique aχ ∈ Qp/Zp. Given any a ∈ Qp/Zp, let χa : Γ− → µp∞ denote the character
χa(γx) = Ψ (aπx) .
This shows χ 7→ aχ is a one-to-one correspondence between Hom(Γ−, µp∞) and Qp/Zp.For any subset U ⊂ Zp, let 1U denote the characteristic function of U . Given any a ∈ Qp/Zp, such
that pna ∈ Zp, we have following identity equating the “Fourier transform on Z/pn” of 1pna+pnZpwith χa:
1pna+pnZp(x) =1
pn
pn−1∑j=0
1pna+pnZp(j)Ψ (πjx/pn) = Ψ (πpnax/pn) = χa(x).
9This second assumption is not strictly necessary. It will be easy to see through our proof that we get a p-adicL-function Lλ ∈ OCpJptZpK[1/p], where pt is the p-part of #C`(1).
25
It is an easy calculation that we have the Fourier inversion identity
χa(−x) =1
pn
pn−1∑j=0
χa(j)Ψ (−πjx/pn) =1
pn
pn−1∑j=0
Ψ(aπx)Ψ(−πjx/pn) = 1pna+pnZp(x).
Definition 3.3. Many of our constructions will depend on a fixed compatible sequence of pth-powerroots of unity (1, ζp, ζp2 , . . .). We henceforth fix the choice10
ζpn = Ψ(π/pn).
3.3. A transformation of Zp and twisted Fourier transforms of anticyclotomic charac-ters.
Definition 3.4. Recall our fixed γ ∈ Γ whose image under Γ → Γ− is a topological generator ofΓ−. Given x ∈ Zp, let rp(x) denote the unique solution to the equation
1 + (γ − 1)x = γrp(x).
Explicitly, we can write
rp(x) =log(1 + (γ − 1)x)
log(γ)∈ OKpJxK.
In particular, we get a continuous map rp : Zp → OKp . Note that
(48) rp(x+ pna) ≡ rp(x) (mod pn)
for all n ≥ 0.
We now define certain “twisted Fourier transforms”
rχ : Zp → Qp(µp∞)
of finite-order anticyclotomic characters χ, which are locally constant functions whose Cp-spanforms a uniformly dense subspace of Cp-valued continuous functions on Zp.
Definition 3.5. Hence, given χ : Γ−,n → µp∞ , we define a continuous map
r∗pχ := χ γrp : Γ−,n → µp∞ , (r∗pχ)(x) = χ(γrp(x)).
In keeping with our previous conventions and fixed identification Γ− = Zp, we sometimes write thisas (r∗pχ)(x). If χ : Γ−,n → µp∞ , we then define
(49) rχ : Zp → Qp(µp∞), rχ(x) := (r∗pχ)(x).
Lemma 3.6. If χ : Γ−,n → µpn is imprimitive (i.e. factors through χ : Γ−,n−1 → µpn−1), thenrχ(x) is supported on pZp (i.e. rχ(x) = 0 if x 6∈ pZp).
10Strictly speaking, fixing a choice compatible with Ψ is not necessary, but will simplify the construction andpresentation of the p-adic L-function.
26
Proof. Suppose χ : Γ−,n → µpn is imprimitive. Then p|pnaχ and
rχ(x) =1
pn
pn−1∑j=0
Ψ
(π
pn(pnaχrp(j) + jx)
)
=1
pn
pn−1−1∑b=0
p−1∑a=0
Ψ
(π
pn(pnaχrp(p
n−1a+ b) + (pn−1a+ b)x))
=1
pn
pn−1−1∑b=0
p−1∑a=0
Ψ
(π
pn(pnaχrp(b) + (pn−1a+ b)x
))
=1
pn
pn−1−1∑b=0
Ψ
(π
pn(pnaχrp(b) + bx)
) p−1∑a=0
Ψ
(π
pnpn−1ax
),
where for the penultimate equality, we used (48) and p|pnaχ so that
pnaχrp(pn−1a+ b) ≡ pnaχrp(b) (mod pn).
Now it is clear that the inner sum of the last line is a nonzero function if x ∈ Zp \ pZp.
Let Qp(µp∞) denote the p-adic completion (with respect to (12)) of Qp(µp∞). For a p-adicallycomplete ring R, let Cont(Zp, R) denote the R-algebra of continuous functions Zp → R.
Lemma 3.7. (1) The Qp(µp∞)-span of the set rχχ∈Hom(Γ−,µp∞ ) is uniformly dense in Cont(Zp, Qp(µp∞)).
Moreover, the same statement is true for rχχ∈S where S ⊂ Hom(Γ−, µp∞) is a subset withfinite complement.
(2) The Qp(µp∞)-span of the set rχχ∈Hom(Γ−,µp∞ ) imprimitive is uniformly dense in Cont(pZp, Qp(µp∞)).
Moreover, the same statement is true for rχχ∈S where S ⊂ Hom(Γ−, µp∞)imprimitive is asubset with finite complement.
Proof. (1): Let χ1,n : Γ−,n∼−→ µpn be such that χ1,n(γx) = Ψ
(πpn
), so that χj1,n(x) = Ψ
(jπpn
).
Since the Fourier transform of rχj1,n
is r∗pχj1,n, we have that r∗pχ
j1,n is in the Qp(µp∞)-span of rχ.
We have
1a+pnZp(x) =1
pn
pn−1∑j=0
χj1,n(γrp(x))Ψ
(−πjrp(a)
pn
)=
1
pn
pn−1∑j=0
(r∗pχj1,n)(x)Ψ
(−πjrp(a)
pn
).
The right-hand side of the above identity is in the Qp(µp∞)-span of rχ. Hence all characteristic
functions are contained in the Qp(µp∞)-span of rχ, and so the Qp(µp∞)-span of rχ is uniformly
dense in the Cont(Zp, Qp(µp∞)).For the second statement, note that for any such subset S, χ1,n ∈ S for all n 0. Hence 1a+pnZp
is in the span for all n 0 and all a ∈ Zp, which proves the claim.(2): The first statement follows immediately from (1) and Lemma 3.6. For the second statement,
note that the imprimitive character γx 7→ χ1,n(γpx) = Ψ(
πpn−1
)is in S for all n 0. Hence the
same computation as in (1) shows that 1a+pnZp is in the span for all n 0 and all a ∈ pZp, whichproves the claim.
Remark 3.8. Note that the Qp(µp∞)-valued locally constant functions rχ are not necessarilyp-adically integrally valued (and in fact rarely are).
27
3.4. The p-adic L-function: statement of the theorem. Let
δ =
0 p > 3
1 p = 2, 3.
Then ∆− ∼= Z/pδ. Let OCp = Z + cOK . We will primarily be concerned with c = pn.The purpose of this section is to prove the following theorem.
Theorem 3.9. There is an element Lλ ∈ OCpJZ×p K[1/p] ⊂ OCpJZpK[1/p] satisfying the following:
(1) For any χ ∈ Γ− factoring through χ : Γ−,n∼−→ µpn, we have
(50) Lλ(rχ) =1
pnLalg,(p)(λ−1χ, 0) · Ωp(y)Cλ
for some constants Ωp ∈ C×p , Cλ ∈ Q× that depends only on λ, and Lalg,(p) denotes thestandard algebraically-noramlized L-function for an algebraic Hecke character ε of infinitytype (k, j) multiplied with inverses of Euler factors at p
Lalg(ε−1, 0) =
(√DK
π
)j(j − 1)! (1− ε(p)/p)
(1− ε−1(p)
) L(ε−1, 0)
Ωk−j ∈ Q×
where DK is the (signed) fundamental discriminant of K, Ω is the Neron period of the fixed
CM elliptic curve A/K(f(p)) from Definition 2.3 (note λχ−1 has infinity type (1, 0), andsince p is ramified in K we have p|cond(λχ−1) and so (λχ−1)(p) = 0).
The proof-construction of µλ will be given over the next several sections.
Remark 3.10. Note that this interpolation property differs quite significantly from the classicalsetting, in which we have Lλ ∈W JZ×p K with interpolation formulas like
Lλ(χ) =1
png(χ)L(p)(λ−1χ, 0)Cλ
where g(χ) is the usual Gauss sum: for an anticyclotomic character χ : (Z/pn)× → Q×p , extendedby 0 to Z/pn,
g(χ) =
pn−1∑j=0
χ(j)ζ−jpn .
(See, for example, Brakocevic [4].) However, even this class interpolation implicitly involves aFourier transform, which shows up in the Gauss sum g(χ). Given a character χ : (Z/pn)× → C×p ,extended by zero to a function χ : Z/pn → Cp, we can again define a Fourier transform
χ :=1
pn
pn−1∑j=0
χ(j)exp
(−2πijx
pn
)= χ−1(−x)
g(χ)
pn= χ−1(−x)
χ(−1)
g(χ−1)=χ−1(x)
g(χ−1).
As can be seen in Section 8 of loc. cit., the interpolation in the classical case roughly reads
Lλ(χ) =1
g(χ−1)Lλ(χ−1) =
1
pnL(p)(λ−1χ−1, 0)C = g(χ)L(λχ, 1)C.
The twisted Fourier transforms rχ are thus analogues of the Fourier transform in the ordinarysetting:
χ(x) = χ(x)1
g(χ−1)
in our setting. However, since the locally constant functions rχ are not necessarily p-adicallyintegrally valued (just as the χ(x) above), the right-hand side of the interpolation (50) is not
28
necessarily p-adically integral, and in fact may (and likely does) grow p-adically as n → ∞ eventhough Lλ is a measure.
Remark 3.11. Here,
Ωp := Ωp(y) ∈ C×pis the period (47) associated to the point y ∈ U ′x chosen in Choice 3.12 (see also Proposition 3.14for y ∈ U ′x):
ωcan(y) = Ωp(y)ω0 = Ωpω0.
See (56).
3.5. Construction of the measure: setup. We give a global construction of our GL1/K-typeand GL2/Q-type p-adic L-functions. Later we will use these constructions in order to constructColeman maps from universal norms to measures, which send elliptic units to the GL1/K-typep-adic L-function. The essential idea in both constructions is to introduce analytic continuationsof Serre-Tate expansions obtained from relative p-adic Hodge theory, namely “qdR-expansions”,which “extend” into the overconvergent locus. These qdR-expansions inherit p-adic boundednessproperties of Serre-Tate expansions and have natural actions under the geometric Galois group ofa tower of modular curves, in which we can find the anticyclotomic tower. Thus our measures willbe constructed as bounded power series in qdR − 1.
For an integral ideal f′ ⊂ OK prime to p, let K(f′)[p∞]/K denote the compositum of thering class extension of K of conductor p∞ and the ray class field K(f′)/K modulo f′. Then
Gal(K(f(p))[p∞]/K(f(p))) ∼= G− and
Gal(K(f(p))[p∞]/K) =⊔a
aG−,
where C`(f(p)) is the ray class group modulo f(p) and a runs through some full set of representatives
of C`(f(p)). Fix these a to be integral and prime-to-f(p)p.Recall our quotient G− → Γ−, where Γ− ∼= Zp is the anticyclotomic tower. Recall our splitting
O×Kp ∼= ∆×ΓHence we have a splitting G− = O×Kp/Z×p∼= ∆−×Γ− where ∆− = ∆/(Z/q)× (q = p if
p > 2, q = 4 if p = 2) and Γ− = Γ/(1 + qZp). Note that ∆− = 1 if p > 3, and that ∆ = O×K/±1if p = 3 and K = Q(
√−3), or if p = 2 and K = Q(i). Henceforth fix such a splitting. We note
that the composition Γ− ⊂ G− with the quotient G− → Γ− is x 7→ xpt, where pt is the p-part of
the class number #C`(1). Hence under (2) of Assumption (3.2), we have that this composition isthe identity on Γ−.
Recall f is the conductor of λ. Henceforth, fix a decomposition
Gal(K(f(p))[p∞]/K) =⊔b
bΓ−
where b runs through a fixed set of integral, prime-to-f representatives of C`(f(p))∆− We will firstdefine a measure µλ ∈ OCpJG−K[1/p] as a sum
(51) χ(b)µλ =∑b
µb
for some approriate character χ, where µb ∈ OCpJΓ−K[1/p] = OCpJZpK[1/p]. From µλ, the p-adic
L-function Lλ will be obtained via pushforward along the natural quotient Gal(K(f(p))[p∞]/K)→Γ− = Zp.
29
3.6. Construction of the local measures. Given a congruence subgroup Γ ⊂ GL2(Z), let Y (Γ)be the (geometrically disconnected) GL2-modular curve of level Γ. For N as before (the smallest
positive integer in f(p)), then for Y := Y (Npδ), letting K(Npδ) ⊂ GL2(A(∞)Q ) denote the usual com-
pact open subgroup corresponding to the congruence subgroup Γ(Npδ), there exists an embedding
(52) C`(f(p))∆− = K×\A×,(∞)K /((1+ f(p)OK)(Z+pδOK)×) → GL2(Q)\GL2(A(∞)
Q )×H±/K(Npδ)
where K(N) ⊂ GL2(A(∞)Q ) is the standard open compact subgroup attached to Γ(N). (The ex-
istence of such Y and embeddings is well-known, see [4] for a nice exposition.) Let Y (pn) =Y (Γ(pn)∩Γ(N)), Y1(pn) = Y (Γ1(pn)∩Γ(N)). In essence, we will realize Γ− in the tower Y1(p∞)/Y ,analogously to how Γ− is realized in the Igusa tower in the ordinary case. (We will base-change toOCp so that with respect to our fixed choice of pth-power roots of unity (1, ζp, ζp2 , . . .), the ordinaryIgusa tower may be identified with a rigid analytic subdomain of Y1(p∞)/Y .)
Now choose any CM point y0 ∈ Y corresponding to the trivial ideal class under (52). Then by
the Shimura reciprocity law, there are elements aj ∈ A×,(∞)K giving a fullset of representatives of
C`(f(p))∆−, such that yj := y0 · aj (where “·” denotes the right Hecke action) is the image of theideal class corresponding to aj under (52). Note that since p is ramified in K, y0 ∈ Y (1/2), andhence is in the overconvergent locus. Henceforth fix 1/2 ≤ r < p/(p+ 1), so that y0 ∈ Y (r).
Choice 3.12. Now fix an element π ∈ K such that OK = Z+Zπ and which is a local uniformizerat p (note this exists since p is ramified in K). We now choose an infinite-level CM point y ∈ YIg(r)
above y0 ∈ Y (r), such that (e1, e2) : Z⊕2p
∼−→ TpA with
(53) e1 = [π](e2).
Then
y = (A, e1, e2) ∈ Y∞.We now fix γ ∈ Γ with
π =
γ − 1 p > 3γ−1p p = 2, 3
,
and
〈e1,n, e2,n〉 = ζpn
for our previously fixed choice of sequence (1, ζp, ζp2 , . . .). We henceforth fix this π as our choice of
uniformizer of OKp and fix a nontrivial additive character Ψ : Kp → C× such that
Ψ
(π
pn
)= ζpn
for all n ≥ 0.
Proposition 3.13. View y as a geometric Spa(Qp,Zp)-point. The composition
(54) Qp = OY∞(y) ⊂(OB+
dR,U ′x/(t)
)(y)
ϑ−→ O(y) = Cp
is the canonical inclusion.
Proof. First, observe that O∆(y) is a local ring with maximal ideal (kerϑ)(y). It is clear that the
inclusion Qp ⊂ OY∞(y) ⊂ O∆(y)ϑ−→ O(y) = Cp is the canonical inclusion. Now the statement
follows by applying Hensel’s lemma to the separable extension Qp.
30
Proposition 3.14. For y as in Choice 3.12, we have y ∈ U ′x and letting x 7→ x denote the nontrivalelement of Gal(Kp/Qp), we have
(55) zdR(y) = ϑ(zdR)(y), z(y) = ϑ(z)(y), ydR(y) = ϑ(ydR)(y) ∈ Kp.
Proof. By (53) and the fact that [a] corresponds to multiplication by a on H1,0, we see thatzdR(y) = π. Applying conjugation on H1,0, we get H0,1, and so z(y) ∈ Kp. From (32), we thenget ydR(y) ∈ Kp. Now the properties in (55) follow from this discussion and (54). We have y ∈ U ′xsince zdR(y) and z(y) are defined.
Definition 3.15. Let
(56) Ωp := Ωp(y)
be the period (47) attached to the point y ∈ U ′x from Choice 3.12.
We briefly recall the classical “Lubin-Tate period” (see [42, Appendix]).
Definition 3.16. Taking the limit of the Weil pairing A[pn]×A[pn]→ µpn , we get an identificationof Zp-modules
TpA ∼= Homp-div/ OLp(A[p∞])
(A[p∞], µp∞)
(where the hat denotes p-adic completion, and the Hom is in the category of p-divisible groups
over OLp(A[p∞])). Recall our choice of level structure (e1, e2) : Z⊕2p
∼−→ TpA from Choice 3.12. Inparticular, e2 was an OKp-generator of TpA. Then by the above, e2 induces a map of p-divisible
groups over OLp(A[p∞])
e2 : A[p∞]→ µp∞ ,
which by [56, Proposition 1] (whose proof in fact applies to morphisms of p-divisible groups whosesource and target have noetherian rings of definition), gives us a map of formal groups
e2 : A→ Gm
over OLp(A[p∞]). In particular, we get a map
e∗2 : Ω1Gm→ Ω1
A.
Let dq/q be the canonical differential on Gm, and recall our normalized invariant differential ω0 ∈Ω1A/OLp
. We then define the Lubin-Tate period by
ΩLT :=e∗2dqq
ω0∈ OLp(A[p∞]).
As is shown in [42, Theorem, Appendix], on the normalized p-adic absolute value we have
|ΩLT| = p−r, r :=1
p− 1− 1
e(t− 1)
where e is the ramification index of Kp/Qp and t is the order of the residue field of Kp. (Under ourp-ramfied assumption, e = 2 and t = p, so that r = 1/(2(p− 1)).)
Proposition 3.17.
(57) ΩLT ∈ OLp(A[p∞]) \ OLp(A[p∞]).31
Proof. By general results on formal groups over characteristic zero domains (see [14, Chapter I.1],for example), we have that for maps of formal groups f : F → F ′ over a characteristic zero domainR, the map Hom(F, F ′)→ R, f 7→ f ′(0) is a group morphism injection. One also sees that this map
is equivariant with respect to Galois actions. For the map e2 : A→ Gm, since ω0 is the normalizedinvariant differential on A, we have ΩLT = e′2(0). If ΩLT ∈ OLp(A[p∞]), it would belong to some
finite extension L′p of Lp, and hence would be preserved by Gal(Lp/L′p). However, one can check
that Gal(Lp/Lp) acts on e2 through σ 7→ [κ(σ)], and so acts nontrivially on e2. This violates theequivariance and injectivity of e2 7→ e′2(0), a contradiction.
Using the previous proposition, one can show that Ωp is transcendent over Qp, and is highlyrelated to the Lubin-Tate period ΩLT.
Proposition 3.18.
(58) Ωp ∈ OLp(A[p∞]) \ OLp(A[p∞]).
Proof. Note that for our CM point y = (A, e1, e2) ∈ Y∞, recalling s from (22), we have
s(y) = e∗2dq
q.
By (28), we have have
Ωpω0 = ϑ(ωcan)(y) =ϑ(s)
ϑ(ydR)(y)=
s
ϑ(ydR)(y)=
ΩLT
ϑ(ydR)(y)ω0.
Since ϑ(ydR)(y) ∈ Kp by (55), we have
Ωp =ΩLT
ϑ(ydR)(y)∈ OLp(A[p∞]) \ OLp(A[p∞]),
as desired.
Definition 3.19. Recall η = λ|A×QN−1Q is the central character of λ. Recall Y = Y1(N) as above,
and let
E1,η = L(η, 0) + 2
∞∑n=1
σ0,χ(n)qn, σk,η(n) :=∑
0<d|n
η(d)dk,
which is a weight-1 eigenform on Y . Letting Vp denote the Hecke operator induced by division bythe canonical subgroup, which is defined on Y (r) (on which the canonical subgroup exists), so thatwe have a weight 1 eigenform on Y (r),
E[1,η(q) := E1,η|1−V = E1,η(q)− E1,η(qp) = 2
∞∑p-n,n=1
σ0,η(n)qn,
where in the last equality, we use our assumption p|f(η) from Assumption 3.2. Let ω denote theusual Hodge bundle of weight 1 modular forms on Y , and let ωU ′x denote the pullback along U ′x → Y .Then we can view
E[1,η ∈ Γ(ωYIg(r))
is the associated differential on Y (r). Let ωcan ∈ Γ(ωU ′x) be the canonical differential as in [30,
Chapter 5], which as explained in loc. cit. extends Katz’s differential ωKatzcan ∈ Γ(ω|YIg) from [26]
from YIg to YIg(r). Then let
G[1,η :=E[1,ηωcan
∈ Γ(OB+dR,YIg(r)
/(t)),
32
which is a weight 1 generalized p-adic modular form in the sense of [30, Chapter 5.7].
Definition 3.20. We let
G[1,η(qdR − 1) ∈ Γ(OYIg(r))JqdR − 1K
denote the qdR-expansion. Since ωcan extends Katz’s canonical differential from [26], we have
G[1,η|YIg ∈ Γ(OYIg).
Hence by Proposition 2.28, we have
G[1,η(qdR − 1) ∈ Γ(O+YIg(r)
)JqdR − 1K[1/p].
Given any point y′ ∈ YIg(r), let G[1,η(y)JqdR − 1K denote the image of G[1,η(qdR − 1) under thespecialization map on coefficients
Γ(O+YIg(r)
)→ Γ(O+YIg(r)
)(y) ⊂ OCp .
Hence,
G[1,η(y′)(qdR − 1) ∈ OCpJqdR − 1K[1/p].
We have an identification
OCpJqdR − 1K[1/p] = OCpJZpK[1/p]
using the Amice transform (see [14, Chapter I.3]).
3.7. The global measure.
Definition 3.21. Let yb correspond to the ideal class b ∈ C`(f(p))∆− under (52), and let y =(A, e1, e2) correspond to the trivial ideal class. We have a corresponding isogeny πb : A→ A/A[b].Since b is integral and prime to pN , this induces a point πb(A, e1, e2) = yb. Fix a differentialω0 ∈ Ω1
A/Z. Then π∗bω0 = λ(b)ω0. Define
(59) µb := (π∗bG[1,η)(qdR − 1) = λ−1
A (b)G[1,η(yb)(qdR − 1) ∈ OCpJqdR − 1K[1/p].
Let λA denote the type (1, 0) Hecke character attached to A (i.e. λA NmK(f(p))/K = ΨA where
ΨA is the reciprocity character given by the main theorem of complex multiplication for A). Thenλ/λA is a finite-order Hecke character over K. Hence, following (51) we define
(60) µλ :=∑b
(λA/λ)(b)µb ∈ OCpJGal(K(f(p))[p∞]/K)K[1/p].
Recall our identification Γ− = Zp induced by our choice of γ ∈ Γ. Now we let r : Gal(K(f(p))[p∞]/K)→Γ− = Zp denote the natural quotient. Via pushforward, we get a map
r∗ : OCpJGal(K(f(p))[p∞]/K)K[1/p]→ OCpJZpK[1/p], r∗µ(f) = µ(f r).
We define
Lλ := r∗µλ.
3.8. Interpolation and proof of Theorem 3.9. Recall the GL2(Qp)-action on Y∞ (see [30,Chapter 3.3] or [9, Section 2.1]), acting on the right. Henceforth, let
γj/pn,m =
(1 j/pn
0 1/pm
).
33
Proposition 3.22. Write yb = (Ab, e1, e2), where Ab has CM by Z + pδOK . We have
(61) yb · γj/pn,n+δ = (Ab,n, e′1, e′2) · γrp(j),
where Ab,n = Ab/〈e2 (mod pn)〉 has CM by Z + pn+δOK , (e′1, e′2) : Z⊕2
p∼−→ TpAb,n is the p∞-level
structure determined by the definition of the GL2(Qp)-action, and we view γrp(j) ∈ Γ ⊂ O×Kp ⊂GL2(Zp). The points yb · γj/pn,n+δ for 0 ≤ j ≤ pn − 1 represent C`(f(p))∆−Γ−,n under (52).
Proof. For g ∈ GL2(Qp), let g∨ = g−1det(g). Note that the kernel of the underlying isogeny forγj/pn,n+δ is the kernel of(
pn+δ
(1 j/pn
0 1/pn+δ
))∨=
(pn+δ jpδ
0 1
)∨=
(1 jpδ
0 pn+δ
)(mod pn)
which is 〈jpδe1 +e2 (mod pn)〉. Now if we let j = 0, we see that this kernel is exactly 〈e2 (mod pn)〉and so Ab · γ0/pn,n+δ = Ab,n. Finally, since
jpδe1 + e2 = [jπpδ + 1](e2) = [γrp(j)](e2),
we get (61) (for a review of Shimura’s reciprocity law, see [4, Proposition 6.4]). For the second
assertion, we note that since Ab,n · γrp(j) for 0 ≤ j ≤ pn − 1 are non-isogenous elliptic curves with
CM by Z + pn+δOK , and so give a full set of representatives of the isomorphism classes of suchcurves.
Lemma 3.23. We have
G[1,η(yb)(q1/pn+δ
dR ζjpn − 1) = (γ∗j/pn,n+δG[1,η)(yb)(qdR − 1) = λ−1
A (γrp(j))G[1,η(yb·γrp(j))(qdR − 1).(62)
Proof. The first equality of (62) follows from (46) and the second from (61) and the fact that for
any differential ω0 on Ab, (γrp(j))∗ω0 = λA(γrp(j))ω0 = λ(γrp(j)) since λA|Γ = λ|Γ.
Note for any function f : Z/pn → Cp, we can write
f(x) =∑
a∈Z/pnf(a) · 1a+pnZp(x) =
1
pn
∑a∈Z/pn
f(a)
pn−1∑j=0
ζ−japn ζjxpn =1
pn
pn−1∑j=0
f(j)ζjxpn .
Applied to f(x) = rχ(x), this gives
(63) rχ(x) =1
pn
pn−1∑j=0
rχ(j)ζjxpn =1
pn
pn−1∑j=0
χ(γrp(j))ζjxpn .
Proof of Theorem 3.9. Note that on p-adic avatars, we have λ|Γ = λA|Γ because the algebraic Heckecharacter underlying λ has conductor at p dividing pε by (1) of Assumption 3.2. Hence on p-adicavatars
(64) λ(γ) = λA(γ).34
For χ : Γ−,n → µp∞ ,we have
Lλ(rχ) =∑b
(λA/λ)(b)µb(rχ)(63)=∑b
(λA/λ)(b)µb
1
pn
pn−1∑j=0
χ(γrp(j))ζjxpn
(59)=
1
pn
∑b
λ−1(b)
pn−1∑j=0
χ(γrp(j))G[1,η(yb)(ζjpnq
1/pn+δ
dR − 1)|qdR−1=0
(62),(64)=
1
pn
∑b
pn−1∑j=0
λ−1(b)χ(γrp(j))λ−1(γrp(j))G[1,η(ybγrp(j))(qdR − 1)|qdR−1=0
=1
pn
pn−1∑j=0
∑b
(λ−1χ)(bγrp(j))G[1,η(ybγrp(j)) =1
pn
∑b′
(λ−1χ)(b′)G[1,η(yb′),
(65)
where the sum is over b′ ∈ Gal(K(f(p))/K)∆−Γ−n , yb′ represent the b′ in the tower Y∞ → Y , and in
the penultimate equality we used χ(b) = 1 since χ ∈ Γ−. Recalling our fixed generator ω0 ∈ Ω1A/Z,
which by transport of structure via the Hecke action πb′ determines generators ω0(yb′) ∈ Ωπb(A)/Zp ,
which thus all satisfy
(66) ωcan = Ωp · ω0(yb′)
for some (common) Ωp ∈ O∆(y) with (recalling the projection ϑ : O∆ OY∞)
ϑ(Ωp) = Ωp(y) =: Ωp
(from (56)). Then
Ω−1p
∑b′
(λ−1χ)(b′)G[1,η(yb′) ∈ Q×.
Hence
Ω−1p
∑b′
(λ−1χ)(b′)G[1,η(yb′)(54)= ϑ
(Ω−1p
∑b′
(λ−1χ)(b′)G[1,η(yb′)
)
= Ω−1p ϑ
(∑b′
(λ−1χ)(b′)G[1,η(yb′)
).
(67)
We thus have
Ω−1p ϑ
(∑b′
(λ−1χ)(b′)G[1,η(yb′)
)(67)= Ω−1
p
∑b′
(λ−1χ)(b′)G[1,η(yb′)(66)= Lalg,(p)(λ−1χ, 0)Cλ,
where the last equality follows from the calculations of [24] (cf. also [14, Chapter II.4]) relating CMtoric period sums of algebraic Eisenstein series with algebraically normalized Hecke L-values for χis primitive. We finally note that on the level of qdR-expansions, and so for the power series definingthe measure Lλ in (60), the application of ϑ on the left-hand side of the above equation correspondsto plugging in qdR− 1 = 0. Hence this gives (50). Hence (1) of Theorem 3.9 is established. For (2),we first have the following.
Lemma 3.24. If χ is not primitive, then∑b′
(λ−1χ)(b′)G[1,η(yb′) = 0.
35
Admitting this Lemma, then the above shows that µλ(rχ) = 0 if χ is primitive. By Lemmas 3.6and 3.7, this implies µλ is supported on Zp \ pZp = Z×p . Now (2) of Theorem 3.9 is proved.
Proof of Lemma 3.24. Assume χ : Γ−,n → µpn is imprimitive, and so factors through χ : Γ−,n−1 →µpn−1 . Then we can break up the b into cosets for Gal(K(f(p))/K)∆−Γ−,n−1. We have∑
b′∈Gal(K(f(p))/K)∆−Γ−n
(λ−1χ)(b′)G[1,η(yb′)
=∑
a∈Gal(K(f(p))/K)∆−Γn−1,−
(λ−1χ)(a)∑
σ∈Γn,−/Γn−1,−
G[1,η(yaσ)
=∑
a∈Gal(K(f(p))/K)∆−Γn−1,−
(λ−1χ)(a)Up(G[1,η)(ya)
where Up is the Atkin Up-operator (see [25, Chapter 3] for the definition). By definition of [,
Up(G[1,η) = 0, and so the above expression is 0, as desired.
This finishes the proof of Theorem 3.9.
3.9. An interpolation property of a twist of Lλ. For later applications, we also want tocompute values of the twist of our measure by λ/λc, where ρc = ρ c for any Hecke character ρ
over K, and c : A×K∼−→ A×K is complex conjugation.
Definition 3.25. For k ≥ 0, we define a differential operator on measures:
dk : OCpJqdR − 1K→ OCpJqdR − 1K[
1
zdR(y)− z(y)
], dk =
qdRd
dqdR+
k
zdR(y)− z(y).
Here z ∈ Γ(OY∞) is Scholze’s Hodge-Tate period (see [44] or [30] for a discussion), and “(y)” denotesthe image after tensoring with ⊗OY∞OY∞(y). We note that
d0 =qdRd
dqdR, dk = d0 +
k
zdR − z.
Recall the relative Fontaine projection ϑ : OB+dR,Y∞
/(t) → OY∞ ([44, Section 4]), which is a
morphism of proetale sheaves. Let
(68) dk := ϑ dk : OCpJqdR − 1K→ OCpJqdR − 1K[
1
ϑ(zdR − z)(y)
]⊂ OCpJqdR − 1K[1/p].
Note that d0 = d0 and
dk = d0 +k
ϑ(zdR − z)(y).
Lemma 3.26. We have ϑ(zdR)(y), ϑ(z)(y) ∈ OKp, and letting x denote the complex conjugate ofx, we have
(69) ϑ(zdR)(y) = −ϑ(zdR)(y), ϑ(z)(y) = −ϑ(z)(y).
Proof. This follows from the fact that zdR(y) and z(y) parametrize the H1,0 and H0,1 eigenspacesunder the CM action of the de Rham cohomology of a CM elliptic curve, and these spaces areinterchanged by complex conjugation. See [30, Chapter 7] for example.
Definition 3.27. Let κ− : Zp → Zp denote the moment map x 7→ x.
The next theorem computes the moment map36
Theorem 3.28. Suppose we are in the situation of Theorem 3.9. Then for any χ : Γ−,n∼−→ µpn,
(1)
(70) (d0Lλ)(rχ) = µλ(κ−rχ),
(2)
(71) (d1Lλ)(rχ) = pn+δ(Lλ(κ−rχ) +1
ϑ(zdR − z)(y)Lλ(rχ)) = pδLalg,(p)(λ−1χλcA/λA, 0) · Ω3
pC′λ
for some C ′λ ∈ Q×, Ωp ∈ C×p as in (56).
Proof. For (1), recall that d0 = qdRddqdR
. Now the statement follows from (59), (60) and Kummer’s
logarithmic derivative ([14, Chapter I.3]).
Now we prove (2). First note that by the chain rule, pn+δd1g(qdR − 1)|qdR−1=0 = d1g(q1/pn+δ
dR −1)|
q1/pn+δ
dR −1=0. By [30, Chapter 5], d1G
[1,η is a generalized p-adic modular form of weight 3. Hence
by the same calculations as in the first equality of (59) and the proof of (62), we have
d1G[1,η(yb)(ζ
jpnq
1/pn+δ
dR − 1)|q1/pn+δ−1=0
= λ−3(γrp(j))pn+δd1G[1,η(ybγrp(j))(qdR − 1)|qdR−1=0
= λ−3(γrp(j))pn+δd1G[1,η(ybγrp(j))(qdR − 1),
(72)
and
(73) d1G[1,η(yb) = λ−3(b)(π∗bd1G
[1,η)(y).
Now we have
d1Lλ(rχ) =∑b
(λA/λ)(b)d1µb(rχ)(63)=∑b
(λA/λ)(b)d1µb
1
pn
pn−1∑j=0
χ(γrp(j))ζjxpn
(59)=
1
pn
∑b
(λλ2A)−1(b)
pn−1∑j=0
χ(γrp(j))pn+δd1G[1,η(yb)(ζ
jpnq
1/pn+δ
dR − 1)|qdR−1=0
(62)(64)= pδ
∑b
pn−1∑j=0
(λλ2A)−1(b)χ(γrp(j))(λλA)−1(γrp(j))d1G
[1,η(ybγrp(j))(qdR − 1)|qdR−1=0
= pδpn−1∑j=0
∑b
((λλ2A)−1χ)(bγrp(j))d1G
[1,η(ybγrp(j)) = pδ
∑b′
((λλ2A)−1χ)(b′)d1G
[1,η(yb′),
(74)
where the sum is over b′ ∈ Gal(K(f(p))/K)∆−Γ−n .Write y = (A, e1, e2). Let ω0 ∈ Ω1
Ab′/Zbe a generator. Moreover, by (66) and [30, Theorem 5.48],
there exist Ωp ∈ O∆(y)×, ϑ(Ωp) = Ωp, Ω∞ ∈ C×, such that for all b′,
(75) Ω−3p ϑ
(d1(π∗b′G
[1,η)(y)
)(54)= ϑ
(Ω−3p d1(π∗b′G
[1,η)(y)
)= Ω−3
∞ δ1(π∗bE[1,η)(A,ω0),
where δ1 is the real analytic Maass-Shimura operator (see [24, Chapter 1-2]).Finally, we have
Ω−3p ϑ
(∑b′
((λλ2A)−1χ)(b′)d1G
[1,η(yb′)
)(75)= ϑ
(Ω−3p
∑b′
((λλ2A)−1χ)(b′)d1G
[1,η(yb′)
)= Lalg,(p)(λ−1χλcA/λA, 0)C ′λ.
37
where the last equality follows by the calculations of [24] (see also [14, Chapter II.4]). We finallynote, as in the last step of the proof of (50), that the application of ϑ on the left-hand side of theabove equation corresponds to plugging in qdR − 1 on qdR-expansions, and so on the level of thepower series defining the measure d0Lλ (see (60) and (68)). Together with (74), this finishes theproof of (2).
4. Anticyclotomic p-adic L-functions for GL2/Q
4.1. A “good” twist of λ. Given any Hecke character ρ over K, let f(ρ) denote its conductor.
Consider the local character λ|O×Kp : O×Kp → Q×.11 We now choose any Dirichlet character χ0 with
χ0|O×Kp = λ|O×Kp , and then consider the twist
ψ := λ/χ0 : A×K/K× → C×.
Again, this is an algebraic Hecke character of infinity type (1, 0), and we will use ψ to also denoteits p-adic avatar. By definition of ψ, we have that f(ψ) is prime-to-p.
Let g = θψ, so that (g, χ0) be a Rankin-Selberg pair with central character compatibility ωg =
χ−10 |A×Q . Let πg denote the automorphic representation of D×(AQ) associated with g, and let (πg)K
denote the base change to K. Let ε((πg)K , χ0, 1/2) denote the root number of the Rankin-SelbergL-function L((πg)K × χ0, s) (attached to the Jacquet-Langlands lifting of (πg)K × χ0). Note thatsince χ is ramified at p (because λ is), but (πg)p is unramified at p and hence a principal series, wehave
(76) εp((πg)K , χ0, 1/2) = +1.
Recall ηK is the quadratic character attached to K/Q. Now let D be a quaternion algebra over Qwhich satisfies for every prime ` of Q,
(77) ε`((πg)K , χ0, 1/2) = χ`(−1)ηK,`(−1)ε`(D),
where ε`((πg)K , χ0, 1/2) =∏v|` εv((πg)K , χ0, 1/2) and χ` =
∏v|` χv where v|` runs over primes of
K above `. Note that by our choice of χ0, we have χp(−1) = λp(−1) = ηK,p(−1). Hence by (76)and (77), we have
εp(D) = +1,
and so D is split at p, i.e. D ⊗Q Qp∼= M2(Qp).
We henceforth also impose the assumption that ε((πg), χ0, 1/2) = −1. Note that since the Galoisrepresentation attached to (πg)K is ρg|Gal(K/K)
∼= ψ ⊕ ψc, we have
L((πg)K × χ0, s) = L(ψχ0, s+ 1/2)L(ψcχ0, s+ 1/2) = L(λ, s+ 1/2)L(λcχ0/χc0, s+ 1/2).
Now choose χ0 to also satisfy ε(λcχ0/χc, 1) = +1 (which occurs infinitely often, even half the time,
see [61]), and using the results of Rohrlich [35], we can choose χ0 so that
L(λcχ0/χc0, 1) 6= 0.
Since the base change (πg,p)Kp of the local representation πg,p (of GL2(Qp)) to Kp is a principalseries, by choosing χ0 of sufficiently high order, so that χ0/χ
c0 is sufficiently ramified at p, we have
that εp(g, χ0) = +1. Hence, by the above we have
(78) ords=1/2L((πg)K × χ0, s) = ords=1L(λ, s) = ords=1L(E, s).
In summary, we have made the following choice of χ0.
11In the class number 1 situation, by the theory of complex multiplication we further know that λ|O×Kp
(O×Kp) ⊂ O×K .
38
Choice 4.1. χ0 : A×K/K× → Q× is a finite-order Hecke character with
(1) χ0|O×Kp = λ|O×Kp ,
(2) ε(λcχ0/χc0, 1) = +1.
(3) L(λcχ0/χc0, 1) 6= 0.
Henceforth, let
Y∞ := lim←−U
YU ∈ YU,proet
and fix a point as in [60, Section 1.3 of Introduction]
P ∈ YK×∞ ,
i.e. a CM point for K. Viewing πg ∈ Hom(Y∞, Ag)⊗Q, we get a Heegner point
Pg,χ0χ :=
∫Gal(Kab/K)
πg(Pσ)⊗ (χ0χ)(σ)dσ ∈ (Ag ⊗ χ0χ)Gal(K/K) ⊗Q.
4.2. Anticyclotomic Rankin-Selberg p-adic L-functions. We henceforth assume that λ isassociated with an elliptic curve E/Q with CM by K. We have a Rankin-Selberg (GL2/Q-type)p-adic L-function interpolating finite-order twists.
Theorem 4.2. There is an element Lg×χ0 ∈ OCpJZ×p K[1/p] ⊂ OCpJZpK[1/p] satisfying the following
interpolation property. For any χ ∈ Γ− with χ : Γ−,n∼−→ µpn, we have
(1)
(79) (d0Lg×χ0)2(rχ) =pn+δ
p2nLalg,(p)(g, χ0χ, 1) · Ω4
pCg,
(2)
(80) Lg×χ0(rχ) =1
pnG[(Pg,χ0χ) · cg =
1
pnlogωg(Pg,χ0χ) · c′g,
for some constants Cg, cg, c′g ∈ Q× that depends only on f , Ωp ∈ C×p as in (47), and where
L(g, χ, s) = L((πg)K × χ, s − 12) is the Rankin-Selberg L-function centered at 1, and L(p) denotes
the L-function with the Euler factor at p removed, “alg” denotes the algebraic normalization as in[1].
4.3. Form of the measure. Following (51), we will construct a measure µ ∈ OCpJGal(K(f)[p∞]/K)K[1/p]as a measure
(81) µg×χ0 =∑b
χ0(b)µg×χ0,b
for some measures µg×χ0,b ∈ OCpJΓ−K[1/p].
4.4. Construction of the measure.
Definition 4.3. Let ωg be the 1-form associated with g, and let ωg[ = ωg|1−UpVp be the p-depletion
of g as before, using the fact that the quaternion algebra D is split at p (Dp∼= M2(Qp)) so
that the Atkin Up and Vp operators are defined. For a sufficiently small open compact subgroup
U ⊂ D×(A(∞)Q ), we can view g as a weight 2 eigenform on a Shimura curve YU associated with U ,
and let YU (r) be the locus cut out by |Ha| ≥ p−r as before (with 0 ≤ r < p/(p + 1)). Again, we39
have g[ is also a well-defined eigenform on YU (r) (since the canonical subgroup exists on YU (r)),and moreover letting ωg[ be its associated 1-form on YU , we have
G[2 :=ωg[
ω⊗2can∈ Γ(OYIg(r))
as well as its p-adic logarithm
G[ := logωg[∈ Γ(OYU (r)),
which is indeed a rigid-analytic function satisfying (see [30, Chapter 9])
(82)qdRd
dqdRG[
(68)= d0G
[ = G[2.
(This last equality follows exactly because d0 is the p-adic Maass-Shimura weight-raising operatoron weight 0 forms.)
For simplicity, we slightly abuse notation and let Y∞ = YU,∞ denote the Γ(p∞)-level Shimura
curve over YU (recall D is split at p). Let also let YIg,YIg(r) be the corresponding Igusa towers asbefore. Then we take the qdR-expansion to get
G[(qdR − 1) ∈ Γ(O+YIg(r)
)JqdR − 1K[1/p] = Γ(O+YIg(r)
)JZpK[1/p].
As before, we letG[(y)(qdR−1) denote the image under the specialization of coefficients Γ(O+YIg(r)
)→Γ(O+
YIg(r))(y) ⊂ OCp , and we get
G[(y)(qdR − 1) ∈ OCpJqdR − 1K[1/p] ⊂ OCpJZpK[1/p].
We now fix an embedding
(83) C`(f)∆− = K×\A×,(∞)K /((1 + fOK)(Z + pnOK)×) → D×\D×(A(∞)
Q )×H±/U.
Definition 4.4. Let γb ∈ D×(AQ) correspond to the ideal class b under (83), and let yb = P · γb.Then we define
(84) µg×χ0,b := G[(yb)(qdR − 1) ∈ OCpJqdR − 1K[1/p].
Hence, from (81) we get
(85) µg×χ0 :=∑b
χ0(b)µg×χ0,b ∈ OCpJGal(K(f)[p∞]/K)K[1/p].
Now we let r : Gal(K(f)[p∞]/K)→ Γ− = Zp denote the natural quotient. Via pushforward, we geta map
r∗ : OCpJGal(K(f)[p∞]/K)K[1/p]→ OCpJZpK[1/p], r∗µ(f) = µ(f r).
We define
Lg×χ0 := r∗µg×χ0 .40
4.5. Proof of Theorem 4.2.
Proof of Theorem 4.2. Analogously to (74), we have
d0Lg×χ0(rχ) =∑b
χ0(b)d0µg×χ0,b(rχ)(63)=∑b
χ0(b)d0µg×χ0,b
1
pn
pn−1∑j=0
χ(γrp(j))ζjxpn
(59)=
1
pn
∑b
χ0(b)
pn−1∑j=0
λ−2(b)χ(γrp(j))d0G[(yb)(ζ
jpnq
1/pn+δ
dR − 1)|q1/pn+δ
dR −1=0
(62)=
pn+δ
pn
∑b
χ0(b)
pn−1∑j=0
λ−2(b)χ(γrp(j))λ−2(γrp(j))d0G[(yb)(qdR − 1)|qdR−1=0
(62)=
pn+δ
pn
pn−1∑j=0
∑b
χ0(b)χ(γrp(j))λ−2(bγrp(j))d0G[(yb)
=pn+δ
pn
pn−1∑j=0
∑b
(λ−2χ0χ)(bγrp(j))d0G[(ybγrp(j)) =
pn+δ
pn
∑b′
(λ−2χ0χ)(b′)d0G[(yb′).
(86)
By (66) and [30, Theorem 5.48], there exist Ωp ∈ O∆(y)×, ϑ(Ωp) = Ωp, Ω∞ ∈ C×, such that forall b′,
(87) Ω−2p ϑ
(d0π
∗b′G
[(y))
(54)= ϑ
(Ω−2p d0(π∗b′G
[)(y))
= Ω−2∞ δ0(π∗bω
[g)(A,ω0),
where δ0 = ddz is the real analytic Maass-Shimura operator (see [24, Chapter 1-2]).
Finally, we have
Ω−4p ϑ
(pn+δ
∑b′
(λ−2χ0χ)(b′)d0G[(yb′)
)2
=
(Ω−2p
∑b′
(λ−2χ0χ)(b′)d0G[(yb′)
)2
=
(pn+δ
∑b′
(λ−2χ0χ)(b′)δ0(π∗bG[)(A,ω0)
)2
= pn+δLalg,(p)(g, χ0χ, 1)C ′g
where the last equality follows from the explicit Waldspurger formula ([60, Theorem 1.4] and [7]).Together with (86), this finishes the proof of (1).
The proof of (80) is entirely analogous, using G[ = logω[g . For the final equality, note that
logωg(Pg,χ0χ) = logω[g(Pg,χ0χ) · c for some constant c, since χ0 is ramified at p (cf. [30, Proposition
8.3]).
4.6. A factorization of measures and special values. In this section, we relate the GL1/Kand GL2/Q measures in the case when g = θψ.
Theorem 4.5. We have the following identity in OCpJZ×p K[1/p]:
(88) Lλ ·(d0Lψχc0 +
1
ϑ(zdR − z)(y)Lψχc0
)= Lλ · d1Lψχc0 = (d0Lg×χ0)2 · Cg×χ0 ,
where Cg×χ0 ∈ C×p is some constant depending only on the pair (g, χ0).41
Proof. From the three interpolation identities (50), (71) and (79), this holds on the specializationsof (88) to rχ for all χ ∈ Hom(Γ−, µp∞). Now (88) follows from Lemma 3.7.
Recall our moment function κ− : Zp → Zp, κ−(x) = x. Restricted to Z×p ⊂ Zp, we get the
moment character κ− : Z×p → Z×p as well as its inverse κ−1− : Z×p → Z×p . Since all our measures are
elements of OCpJZ×p K[1/p] ⊂ OCpJZpK[1/p] (i.e. supported on Z×p ⊂ Zp), we can evaluate them at
the character κ−1− .
Proposition 4.6. We have Lψχc0(1) 6= 0.
Proof. By (2) of Choice 4.1, we have that L(ψcχ0, 1) 6= 0. Hence by (71) we have Lψχc0(1) 6= 0.
Definition 4.7. Recall that F univ → M is the universal height-2 formal group, where M is thedeformation of a fixed height 2 formal group over the residue field of Lp. Choose M to correspond
to the height-2 group A (mod pOLp). Let ωuniv0 ∈ Ω1
Funiv/Mdenote the normalized differential.
Recall that the rigid-analytification of M is the residue disc of the reduction of y0 ∈ Y (Fp), and so
letting M∞ = Mad ×Y Y∞, we may consider ωcan|M∞ . Then define Ωcan ∈ O∆(M∞) by
ωcan|M∞ = Ωcanωuniv0 |M∞ , ϑ(ωcan|M∞) = ϑ(Ωcan)ϑ(ω0|M∞) = ϑ(Ωcan)ω0|M∞ .
In particular,
(89) ϑ(Ωcan)(y) = Ωp
(see (47)). Then by (31), using the fact that KS is OY∞ (and hence O∆)-linear, we have
KS(ωuniv,⊗2
0 |M∞)
= KS
((ωcan|M∞
Ωcan
)⊗2)
=1
Ω2can
dzdR|M∞ .
Hence,
(90)qdRd
dqdR=
d
dzdR= Ω−2
can
d
KS(ωuniv,⊗2
0 |M∞) .
Proposition 4.8. We have
(91)Lψχc0(κ−1
− )
Ωp∈(Lp,∞ \ Lp,∞
)∪ 0.
Proof. We have
Lψχc0(κ−1− )
Ωp=
1
Ωpϑ
((qdRd
dqdR
)−1
Lψχc0
)(1)
(90),(60),(82)=
1
Ωpϑ(Ωcan)−2(y)ϑ
((d
KS(ω⊗20 )
)−1
Lψχc0
)(1)
(89)= Ω−3
p ϑ
((d
KS(ω⊗20 )
)−1
Lψχc0
)(1).
(92)
We claim that1
Ωpϑ
((d
KS(ω⊗20 )
)−1
Lψχc0
)(1) ∈ Qp.
to see this, note
1
Ωpϑ
((d
KS(ω⊗20 )
)−1
Lψχc0
)(1) = ϑ
((d
KS(ω⊗20 )
)−1 Lψχc0Ωcan
)(y).
42
Noting that we have Ωcan(y) = Ωp by (66), then from (60) we see that in the stalk at y(Lψχc0Ωcan
)y
∈ OY∞,y
and so ((d
KS(ω⊗20 )
)−1 Lψχc0Ωcan
)y
∈ OY∞,y,
which, specializing to y and using OY∞,y(y) ⊂ Qp, gives the claim. To finish the proof of theProposition, invoke (58) with (92).
The following non-vanishing statement on a special value at κ−1− (outside the range of inter-
polation) of the second factor on the left-hand side of (88) is crucial. In particular, it uses thenon-vanishing L-value assumption of Choice 4.1 on the pair (g, χ0); the existence of such a χ0 usedthe deep (and ineffective) theorem of Rohrlich [35].
Theorem 4.9. We have
(93)
(d0Lψχc0 +
Lψχc0ϑ(zdR − z)(y)
)(κ−1− ) 6= 0.
Proof. We have(94)(
d0Lψχc0 +Lψχc0
ϑ(zdR − z)(y)
)(κ−1− ) = d0Lψχc0(κ−1
− ) +Lψχc0(κ−1
− )
ϑ(zdR − z)(y)
(82)= Lψχc0(1) +
Lψχc0(κ−1− )
ϑ(zdR − z)(y).
We now divide into two cases.
Case 1: Lψχc0(κ−1− ) = 0. In this case, the right-hand side of (94) is Lψχc0(1), which is nonzero by
Proposition 4.6. This gives (93).
Case 2: Lψχc0(κ−1− ) 6= 0. This case requires more work, and relies on showing establishing the
p-adic transcencence of Lψχc0(κ−1− ) established by (91). We will argue by contradiction. Assume
(94) is zero. Then
ϑ(zdR − z)(y) ·Lψχc0(1)
Ωp= −Lψχc0(κ−1
− )
Ωp.
By (55) and (70), the left-hand side of the above expression is in Qp, whereas the right-hand side
is in OLp(A[p∞]) \OLp(A[p∞]) by (91). Since every element of Qp belongs to a finite extension of Qp,
and no element OLp(A[p∞]) \ OLp(A[p∞]) belongs to such a finite extension, we have a contradiction.
The following Corollary will be key in linking the non-vanishing of a Heegner point to the corank1 assumption on the Selmer group.
Corollary 4.10. We have
(95) Lλ(κ−1− ) = (d0Lg×χ)2 (κ−1
− ) · C ′ = log2ωg(Pg,χ0) · C ′′
for some C ′, C ′′ ∈ C×p . Hence if the left-hand side is not zero, we have that the Heegner point
Pg,χ0 ∈ (Ag ⊗ χ0)Gal(K/K) ⊗Q is non-trivial.43
Proof. By (88) we have the identity in OCpJZ×p K[1/p]
Lλ ·(d0Lψχc0 +
1
ϑ(zdR − z)(y)Lψχc0
)= (d0Lg×χ)2 · Cg×χ0
for some Cg×χ0 ∈ C×p , and by (93), we see that the specialization to κ−1−
Lλ(κ−1− ) · C ′′′ = (d0Lg×χ)2 (κ−1
− ) · Cg×χ0
(82),(80)= log2
ωg(Pg,χ0) · C ′g×χ0
for C ′′′, Cg×χ0 ∈ C×p . Now (95) follows.
Under the assumption corankZpSelp∞(E/Q) = 1, we will show that the left-hand side of (95) isnonzero using an “anticyclotomic Rubin-type main conjecture” in the height 2 setting, which weformulate and prove in the next sections.
5. Rubin-type main conjectures and explicit reciprocity laws
For this remainder of the paper, assume E/Q is an elliptic curve with CM by OK and with p aprime ramified in K. In particular, K has class number 1.
5.1. Review of the relevant Iwasawa modules. Briefly, let A/L denote an arbitrary ellipticcurve with CM by OK , and as before let Ln = L(A[pn]). Let
E = lim←−n
O×Ln , E1 = lim←−n
O×,1Ln
denote the module of global universal norms. Let E1denote the p-adic closure of E1 inside U1. Let
M∞/L∞ be the maximal pro-p abelian extension of L∞ unramified at all places of M∞ not abovep, let N∞/L∞ denote the maximal pro-p abelian extension of L∞ unramified everywhere, and letX = Gal(M∞/L∞) and Y = Gal(N∞/L∞). Recall our fundamental exact sequence
0→ E1 → U1 → X → Y → 0.
Now let f0 ⊂ OK be an integral ideal that is prime to p, and with wf0 = #x ∈ O×K : x ≡ 1(mod f0), and let L = K(f0). As explained in [14, Chapter II.1.4], there exists an elliptic curveA/L with CM by OK , and a Hecke character λA of type (1, 0) such that λA NmL/K = ΨA whereΨA is the Hecke character attached to A by the theory of complex multiplication. Let C ⊂ E be the
module generated by elliptic units as in [14, Chapter II.1.2 and Chapter III.1], and let C1denote
the p-adic closure in U1. We have a norm map U1 → U1E induced by the norm of the extension
L/K (unramified at p). Define E1E and C1
E as the images of C1and E1
, respectively.
In this case, E1E has (ZpJGal(K∞/K)K)-rank 1, C1
E has rank 1, U1 has rank 2, X has rank 1, and
Y has rank 0. Dividing by C1E , we get an exact sequence of ZpJGal(K∞/K)K-modules
(96) 0→ E1E/C
1E → U1
E/C1E → XE → YE → 0.
The nonzero terms of this exact sequence have ranks 0, 1, 1 and 0, respectively. Choose and fix adecomposition
(97) Gal(K∞/K) = ∆K × ΓK
where ΓK ⊂ Gal(K∞/K) is the maximal Zp-free subgroup. Note that since K has class number 1,
we have a natural identification through restriction resK∞/K∞ΓK∼−→ Gal(K∞/K), where K∞/K is
the Z⊕2p -extension.
44
5.2. Rubin-type main conjectures (without p-adic L-functions).
Definition 5.1. Given a group G, let G denote the group of Q×p -valued charaters on it. Given a
ZpJGal(K∞/K)K-module M and a character χ ∈ ∆K , let Zp[χ] denote the extension of Zp generatedby values of χ, let Qp[χ] = Zp[χ][1/p] and let
Mχ := x ∈M ⊗Qp[χ] : δ(x) = χ(x) ∀δ ∈ ∆K
denote the χ-isotypic component viewed as a Zp[χ]JΓKK[1/p]-module. (Note that Mχ is a directsummand of M ⊗Zp Qp[χ] and can be viewed as a subspace or quotient.) Recall the type (1, 0)
Hecke character λE over K attached to E, and viewing λE : Gal(K∞/K) ∼= ΓK ×∆K → Q×p as ap-adic Galois character, let χE be the restriction to ∆K of λE .
Given a OKpJ∆K × ΓKK[1/p]-module M , let
M(λ−1E ) := M ⊗Kp Kp(λ
−1E )
where Kp(λ−1E ) is the one-dimensional vector space with Gal(K/K) acting by λ−1
E . Similarly, for a
ZpJ∆A × ΓAK[1/p]-module define M(λ−1A ).
Proposition 5.2. E1χE
(λ−1E ), C1
χE(λ−1E ) are free OKpJΓKK[1/p]-modules of rank 1, and U1
E,χE(λ−1E )
is a free OKpJΓKK[1/p]-module of rank 2.
Proof. See [39, Lemma 11.8]. Note that in our setting, we have inverted p.
Recall that χE = λE |∆K. Taking the χE-isotypic component of (96) and twisting by λ−1
E , we get
(98) 0→ E1E,χE
(λ−1E )/C1
E,χE(λ−1E )→ U1
E,χE(λ−1E )/C1
E,χE(λ−1E )→ XE,χE (λ−1
E )→ YE,χE (λ−1E )→ 0.
In the Appendix, we use work of Kings-Johnson-Leung ([23]) on equivariant main conjectures toprove:
Theorem 5.3 (Rational elliptic units main conjecture).
charOKpJΓKK[1/p]
((E1
/C1)χE (λ−1
E ))
= charOKpJΓKK[1/p](YχE (λ−1E )).
Admitting Theorem 5.3, we would then expect to get a main conjecture relating U1E/C
1E and
X 1χE
(λ−1E ). However, these latter OKpJΓKK[1/p]-modules have rank 1, and so vanishing character-
istic ideals, and thus do not admit an obvious main conjecture. To this end, one can formulate an“artificial” main conjecture in the following way, suggested by the latter half of [39, Section 11].Choose a decomposition
(99) U1E,χE
(λ−1E ) ∼= U1 ⊕ U2
where Ui ∼= OKpJΓKK[1/p] such that U2 ∩ E1χE
(λ−1E ) = 0. Then U1
E,χE(λ−1E )/(C1
E,χE(λ−1E ), U2) and
XE,χE (λ−1E )/rec(U2) are torsion OKpJΓKK[1/p]-modules, and Theorem 5.3 implies the following:
Corollary 5.4 (Rubin-Type main conjecture).
charOKpJΓKK[1/p]
(U1E,χE
(λ−1E )/(C1
E,χE(λ−1E ), U2)
)= charOKpJΓKK[1/p]
(XE,χE (λ−1
E )/rec(U2)).
Proof. This follows from (98) and additivity of characteristic ideals.
45
5.3. Relating Rubin-type main conjectures to L-values and p-adic L-functions: outline.The flexibility to vary the choice of U2 as in (99) is crucial in our proofs of rank 0 and rank 1converse theorems. For the rank 0 case, we will take U2 to be the kernel of the map δ|qdR−1=0 for δin (105). This is essentially the same choice of U2 taken in [39, Section 11], where U2 is the kernel ofa certain logarithmic derivative map (denoted by δ in loc. cit.). In fact, our δ|qdR−1=0 is essentiallythe logarithmic derivative in loc. cit. One can show, using Wiles’ explicit reciprocity law, that
δ|qdR−1=0(C1E,χE
(λ−1E )) = C · L(λ−1, 0) = C ′ · L(E/Q, 1)
The assumption corankZpSelp∞(E/Q) = 0 implies, via Wiles’ explicit reciprocity law, that U2 ∩C1χE
(λ−1E ) = 0 and hence U2∩E1
χE(λ−1E ) = 0, so that U2 satisfies the assumptions of (99). Moreover,
Theorem 5.4 implies
δ|qdR−1=0(C1E,χE
(λ−1E )) 6= 0,
and as a result we obtain the rank 0 p-converse theorem (cf. Theorem 11.18 of loc. cit.).
For the rank 1 case, we choose U2 with U2 ∩ E1χE
(λ−1E ) = 0. Then from (117) below, we have a
mapδ′ : U1 → OCpJqdR − 1K[1/p]
satisfying the “explicit reciprocity law”
δ′(C1E,χE
(λ−1E )) = (κ−1
− )∗LλE .The assumption corankZpSelp∞(E/Q) = 1 implies
LλE (κ−1− ) = δ′(C1
E,χE(λ−1E ))/(ΓK−1)
Theorem 5.4= charOKpJΓKK[1/p]
(XE,χE (λ−1
E )/rec(U2))/(ΓK−1) 6= 0
where (ΓK−1) is the augmentation ideal of OKpJΓKK[1/p], and (κ−1− )∗LλE ∈ OCpJZ×p K[1/p] denotes
the pullback measure(κ−1− )∗LλE (f) = LλE (κ−1
− f).
Now (95) gives the p-converse theorem.
5.4. Choice of decomposition of module of semi-local universal norms: rank 1 case. Wesee to find an appropriate submodule U ′ ⊂ U1(λ−1
E )χE to quotient out by to formulate our mainconjecture. As mentioned above, the choice of U ′ depends on whether corankZpSelp∞(E/Q) is 0 or1. In the rank 0 case, U ′ will simply encode the Kummer local condition, whereas in the rank 1case it must be something else in order to relate to a smaller torsion Selmer group.
Definition 5.5. Let
(100) U0 = x ∈ U1E(λ−1)χE : ∃α ∈ OKpJΓK[1/p], αx ∈ ξEOKpJΓK[1/p] = CE(λ−1)χE.
Proposition 5.6. U0 is a free rank-1 OKpJΓKK[1/p]-module.
Proof. By definition, U1E,χE
(λ−1E )/U0 is OKpJΓKK[1/p]-torsion free, and by freeness of U1
E,χE(λ−1E )
we see that U0 has rank 1.We now proceed along the same lines as in the proof of [40, Lemma 4.1]. Let β1, β2 be a
OKpJΓKK[1/p]-basis of U1E,χE
(λ−1E ). Since ΓK ∼= Z⊕2
p , we have OKpJΓKK[1/p] ∼= OKpJT1, T2K[1/p],which is a UFD. Let Z ⊂ U1
E,χE(λ−1E ) be the maximal free submodule of U0. Then sinceOKpJΓKK[1/p]
is a UFD, U1E,χE
(λ−1E )/Z is torsion-free by maximality of Z, and hence so is the submodule U0/Z.
However, U0/Z has rank 0 and so is torsion and torsion-free, meaning U0 = Z.
Let (ΓK − 1) ⊂ OKpJΓKK[1/p] denote the augmentation ideal. Given a OKpJΓKK[1/p]-moduleM , let
M/(ΓK − 1) = M ⊗OKpJΓKK[1/p] OKpJΓKK[1/p]/(ΓK − 1).
46
Choice 5.7. Choose any decomposition of OKpJΓKK[1/p]-modules U1E,χE
(λ−1E ) = U1⊕U2 such that
each Ui is OKpJΓKK[1/p]-free of rank 1 and
(101) U2 ∩ E1E,χE
(λ−1E ) = 0.
(Such a decomposition with U2 as above will be fixed later in Choice 7.5.) Note that this exists
since E1E,χE
(λ−1E ) ∼= OKpJΓKK[1/p] and U1
E,χE(λ−1E ) ∼= OKpJΓKK[1/p]⊕2.
Proposition 5.8.
rec(U2) ⊂ XE,χE (λ−1E )
is OKpJΓKK[1/p]-free of rank 1.
Proof. This follows from the identity ker(rec) = E1E,χE
(λ−1E ) and (101).
Let βi be a OKpJΓKK[1/p]-basis of Ui. Then we can write
β0 = λ1β1 + λ2β2.
Let
pr1 : U1E,χE
(λ−1E ) = U1 ⊕ U2 → U1
denote projection onto the first factor. Then
(102) pr1(β0) = λ1β1.
Now let ξ be any OKpJΓKK[1/p]-generator of C1E,χE
(λ−1E ) and write
(103) ξ = λ0β0
for λ0 ∈ OKpJΓKK[1/p]. Then
pr1(ξ) = λ0pr1(β0).
Corollary 5.9.
λ0λ1 = charOKpJΓKK[1/p](XE,χE (λ−1
E )/rec(U2)).
Proof. We have
λ0λ1(102),(112)
= charOKpJΓKK[1/p] (U1/(pr1(ξ))) = charOKpJΓKK[1/p]
(U1E,χE
(λ−1E )/(C1
E,χE(λ−1E ), U2)
)Theorem 5.4
= charOKpJΓKK[1/p](XE,χE (λ−1
E )/rec(U2)).
5.5. Explicit reciprocity laws. Recall our previously chosen modular curve Y = Y (N), where
N is the smallest positive integer in f(p). Since p - N , the supersingular locus Y ss is a finite etalecover of a finite union of height-2 Lubin-Tate deformation spaces M . Let F univ → M denote theuniversal deformation. As explained in [57, Section 1.1], F univ has a O×Kp-action, and M also has
a O×Kp-action that factors through O×Kp/Z×p and which is compatible with the structure morphism
F →M . We let
M(r) := M ∩ Y (r),
and then pick 1/2 ≤ r < p/(p + 1) as before, so that M(r) is an affinoid subdomain of theoverconvergent locus. Let F univ(r) denote
F univ ×M M(r).47
Note that the universal canonical subgroup on Y (r) restricts to a canonical subgroup for F univ,which gives a O×Kp-equivariant morphism funiv : F univ → F univ,φ. Thus, we have an associated
multiplicative norm operator
Nfuniv : Γ(F univ(r))→ Γ(F univ(r)).
Proposition 5.10 (cf. Section 4 of [57]). Given a Coleman power series gβ ∈ Γ(F )×,Nf=φ, there
is a unique element gunivβ ∈ Γ(F univ(r))×,Nfuniv=φ.
Proof. Observe that one has the operator (φ−1Nfuniv)m : Γ(F univ(r))→ Γ(F univ(r)) for all m ≥ 0.
As in [57, Section 4], one chooses an arbitrary lift gβ of gβ to Γ(F univ(r)) and shows that
(φ−1Nfuniv)∞(gβ) = limm→∞
(φ−1Nfuniv)m(gβ)
is a well-defined element of Γ(F univ(r))×,Nfuniv=φ independent of the choice of gβ.
Hence we get a map
(104) thicken : Γ(F )×,Nf=φ → Γ(F univ)Nfuniv=φ.
Lete : M(r)→ F univ(r)
denote the identity section. We will write the superscript “ad” when we want to emphasize thatwe are working on the adic generic fiber (rather than on the formal scheme). Let M∞ denote theadic generic fiber of infinite-level Lubin-Tate space, so that M∞ ∈Mad
proet. Then let
M∞(r) := M∞ ×Mad M(r)ad.
Definition 5.11. Let U1 denote the module of principal local universal norms as in Section 2.1 forthe height 2 Lubin-Tate OKp-module F = A for the unramified extension Lp/Kp (where A denotesthe formal completion along the identity of A/OLp). Recall Gal(Lp,∞/Kp) ∼= Γ×∆0 for some finite
abelian group ∆0, and where Γ ⊂ O×Kp ∼= Gal(Lp,∞/Lp) is the maximal Zp-free submodule as before
(so that Γ ∼= Z⊕2p ). Since K has class number 1, we can identify Γ = ΓK and ∆0 = Gal(Lp,∞/Kp,∞),
where Kp,∞/Kp is the maximal Z⊕2p subextension of Lp,∞/Kp. Let
χF := κ|∆0 .
In keeping with previous notation, given a ZpJΓ×∆0K-module M and a character χ ∈ ∆0, let Zp[χ]denote the extension of Zp generated by values of χ, let Qp[χ] = Zp[χ][1/p] and let
Mχ := x ∈M ⊗Zp Qp[χ] : σ(x) = χ(σ)x ∀σ ∈ ∆0denote the χ-isotypic component of M ⊗Zp Qp (viewed as both a subspace and a subquotient ofM ⊗Zp Qp and as a OKpJΓK[1/p]-module).
Definition 5.12. Let ωFuniv denote the normalized invariant differential on F univ (i.e. the uniqueinvariant differential with ωFuniv/dX|X=0 = 1). Recall our previously fixed CM point (A, e1, e2) ∈Y∞(r) (since p is ramified in K, |Ha(A)| = p−1/2 and we chose 1/2 ≤ r < p/(p + 1)). Then wedefine a composition of maps
δ0 : U1χF
(κ−1)β 7→gβ−−−−→ Γ(F )×,Nf=φ(κ−1)⊗Zp Qp
thicken−−−−→ Γ(F univ(r))×,Nfuniv=φ(κ−1)⊗Zp Qp
dωFuniv
log
−−−−−−→ Γ(F univ(r))⊗Zp Qpe∗−→ Γ(M(r))⊗Zp Qp
qdR-exp−−−−−→ Γ(M∞(r))JqdR − 1K
Γ(M∞(r))→Γ(M∞(r))(y)⊂Cp−−−−−−−−−−−−−−−−−−→ CpJqdR − 1K ⊂ Dist(Zp,Cp).
(105)
48
Given β ∈ U1χF
(κ−1), one thus gets δ0(β) ∈ CpJqdR − 1K ⊂ Dist(Zp,Cp). One can then extend δ(β)to a distribution on ∆0 × Zp as follows. Given U ⊂ ∆0 × Zp and σ ∈ ∆0 such that σU ⊂ 1× Zp ⊂∆0 × Zp, we define
δ0(β)(U) = δ0(σ(β))(σU).
Then letting r : ∆0 × Zp → Zp denote the projection, we get a pushforward distribution r∗δ0 ∈Dist(Zp,Cp); in fact, by construction r∗δ0 ∈ CpJqdR − 1K. This defines a map
(106) δ := r∗δ0 : U1χF
(κ−1)→ CpJqdR − 1K.
Recall our fixed elliptic curve A/L with CM by OK , where L = K(f(p)), whose associated Heckecharacter ψA satisfies ψA = λANmL/K for some type (1, 0) Hecke character λA overK. Recall Ln =L(A[pn]). Fix a decomposition Gal(L∞/K) ∼= ΓA × ∆A with ΓA the maximal Zp-free subgroup,and compatible with the previous decomposition Gal(K∞/K) ∼= ΓK × ∆K under the restrictionmap Gal(L∞/K)→ Gal(K∞/K). (Recall K∞ = K(E[p∞]).) Let χA denote the restriction to ∆A
of λA. Recall our inclusion U1E ⊂ U1, and our decomposition U1⊗Zp Qp =
∏U1⊗Zp Qp. From this,
we get maps U1χA
(λ−1A )→ U1
χF(κ−1). Denote the composition
(107) locp : U1E,χE
(λ−1E ) → U1
χA(λ−1A )→ U1
χF(κ−1).
Theorem 5.13 (Coleman map). The map (105) precomposed with locp defines a map (which wewill also denote by δ)
δ : U1E,χE
(λ−1E )⊗Qp Cp → Dist(Zp,Cp)
satisfying:
(1) For all α ∈ OCpJΓKK[1/p] all β ∈ U1E,χE
(λ−1E )⊗Qp Cp and all χ ∈ Hom(Γ−, µp∞),
(108) δ(αβ)(rχ) = χ−1(α)δ(β)(rχ).
(2) The free OKpJΓKK[1/p]-module C1E,χE
(λ−1E ) has a generator ξE such that for some constant
C ∈ C×p ,
(109) δ(ξE) = LλE .
Remark 5.14. (1) can be viewed as a kind of “twisted” OKpJΓKK[1/p]-equivariance. In a standardOKpJΓKK[1/p]-equivariance (as in the height 1 case of [14]), one would instead replace the “rχ” onboth sides of the identity with “χ”. However, in our case rχ is necessary, as the coordinate qdR− 1does not have a natural ΓK-action as in the height 1 case, but rather a natural Z×p -action; it is a
coincidence in the height 1 case that the Z×p -action coincides with the anticyclotomic Galois actioninduced by the ΓK-action.
(2) in the statement is the “explicit reciprocity law” alluded to in the previous section.
Proof of Theorem 5.13. (1) follows from the identify
dlog(gσβ) = dlog(gβ [κ(σ)])
for all σ ∈ Γ = ΓK , which in turn follows from Theorem 2.2.For (2), we first define ξE ∈ U1
E,χE(λ−1E ) as follows. Recall the elliptic unit e(a) ∈ U1 from (16),
and let
e(a)χA :=1
#∆A
∑σ∈∆A
χ−1A (σ)e(a)σ ∈ U1
χA,
and e(a)χA(λ−1A ) ∈ U1
χA(λ−1A ) its twist by λ−1
A . Recalling the map N : U1χA
(λ−1A ) → U1
E,χE(λ−1E )
induced by the norm NmL/K and the identity LKn = Ln, we then define
(110) ξE :=1
(Na− χA(a))[L : K]N(e(a)χA(λ−1
A )) ∈ U1E,χE
(λ−1E ).
49
Hence ξE maps to 1Na−χA(a)e(a)χA(λ−1
A ) under the inclusion U1E,χE
(λ−1E ) → U1
χA(λ−1A ).
Now by Proposition 2.9, we have
d
ω0log gξE (T ) =
∑σ∈∆A
χ−1A (σ)E1(logA(T );L, a)σ =
∑σ∈∆A
χ−1A (σ)E1(logA(T ), L)σ.
(Note we are using additive notation in the above expression, as multiplication by scalars inU1E,χE
(λ−1E ) is through the extension of scalars ⊗ZpCp.) Applying Proposition 5.10 to gξE , and
observing that the norm operator Nfuniv on M(r) corresponds to the Up-operator on Y (r) underthe embedding M(r) ⊂ Y (r), we see that
e∗d
ωFuniv
log(thicken(gξE )) =∑
b∈C`(f(p))∆−
π∗bG[1,η|M∞(r).
Hence
e∗d
ω0log (thicken(gξE )) =
∑b∈C`(f(p))∆−
π∗bG[1,η|M∞(r)
qdR-exp7−−−−−→∑
b∈C`(f(p))∆−
π∗bG[1,η|M∞(r)(qdR − 1)
O(M∞(r))→O(M∞(r))(y)⊂Cp7−−−−−−−−−−−−−−−−−−−→∑
b∈C`(f(p))∆−
π∗bG[1,η(y)(qdR − 1)
(60)= LλE .
(111)
Let U0 ⊂ U1E,χE
(λ−1E ) be as in (100). Recall the basis β0 of U0, and write
(112) ξE = λ0β0
for λ0 ∈ OCpJΓKK[1/p].
Corollary 5.15. δ restricts to a map
(113) δ|U0 : U0 ⊗Qp Cp → Dist(Z×p ,Cp),
that is the restriction to U0 ⊗Qp Cp factors through Dist(Z×p ,Cp) ⊂ Dist(Zp,Cp).
Proof. From (1) and (2) of Theorem 5.13, we have for all characters χ : Γ− → µp∞ ,
LλE (rχ) = δ(ξE)(rχ) = χ(λ0)δ(β0)(rχ).
Recalling LλE ∈ OCpJZ×p K[1/p] (and in particular is supported on Z×p ), we see by Lemma 3.6 andthe previous equation that δ(β0)(rχ) = 0 for all but possibly finitely many imprimitive χ. Thus,by Lemma 3.7 we see that δ(β0) is 0 on any subset U ⊂ pZp, and hence is supported on Z×p .
Proposition 5.16. Suppose β ∈ U0 (see Definition 100). Then for any a ∈ Z/(p− 1)×Zp, lettingji ⊂ Z≥0 be any sequence converging to a in Z/(p− 1)× Zp, we have that
(114) limj0→a
(qdRd
dqdR
)j0δ|U0(β)|qdR−1=0
converges to a value in Cp.
Proof. Suppose β ∈ U0, so that there is α ∈ OKpJΓKK[1/p] with αβ = ξE . By the same reasons asin the proof of (108), we see that for any j ∈ Z≥0,
1(α)
(qdRd
dqdR
)jδ|U0(β)|qdR−1=0 = 1(α)
(qdRd
dqdR
)jδ|U0(β)(1) =
(qdRd
dqdR
)jδ|U0(ξE)(1)
(109)=
(qdRd
dqdR
)jLλE (1).
(115)
50
We need a lemma.
Lemma 5.17.
1(α) 6= 0.
Proof. Suppose 1(α) = 0. Then (115) shows(qdRddqdR
)jLλE (1) = 0 for all j ≥ 0. By Amice’s
transform, this implies LλE = 0 as a measure on Zp. However, from (50), the fact that the rootnumber of L(λ−1χ, 0) is 0 for infinitely many χ ([61]), and the main result of Rohrlich [35], we seethat LλE 6= 0, a contradiction.
Now we finish the proof of Proposition 5.16. From (115), we have
(116) 1(α)
(qdRd
dqdR
)jδ|U0(β)|qdR−1=0 =
(qdRd
dqdR
)jLλE (1)
(82)= LλE (κj−).
Now letting j0 ⊂ Z≥0 converge to j in Z/(p− 1)× Zp so that κj0− → κa− as functions Z×p → Z×p ,and invoking (113) and Lemma 5.17, we see that the right-hand side of (116) converges and so(114) converges.
Definition 5.18. Also define12
(117) δ′ :=
(qdRd
dqdR
)−1
δ|U0 |qdR−1=0 := limm→∞
(qdRd
dqdR
)−1+(p−1)pm
δ|U0 |qdR−1=0 : U0 ⊗Qp Cp → Cp.
(Note that unlike for δ, we have plugged in “qdR − 1 = 0”, i.e. evaluated at the trivial character 1,in the definition of δ′.)This definition makes sense by Proposition 5.16 for a = −1.
We summarize some key properties of δ′.
Corollary 5.19. The map δ′ satisfies
(1) For all α ∈ OCpJΓKK[1/p] all β ∈ U0 ⊗Qp Cp,
(118) δ′(αβ) = χ−1(α)δ′(β).
(2) We also have
(119) δ′(ξE) = LλE (κ−1− ).
Proof. This follows from (1) and (2) of Theorem 5.13, (115) and (109).
12The notation suggests that one view δ′ as the “anticyclotomic derivative” of δ. In fact, δ′ is a limit of derivativesgiven by the formula
δ′ = limm→∞
(qdRd
dqdR
)−1+pm(p−1)
δ|qdR−1=0.
On the other hand,(qdRddqdR
)δ′ = δ, and so δ′ can also be viewed as the formal integral
δ′ =
∫δdqdR
qdR|qdR−1=0.
This difference in philosophical conventions is reflected in the literature in the terminologies “p-adic Gross-Zagierformula” (cf. [1]) and “p-adic Waldspurger formula” (cf. [33], [30]).
51
Remark 5.20. Note that (108) and (118) are not standard OKpJΓKK[1/p]-equivariance propertiesthat one often sees in height 1 Coleman maps (in particular, the “rχ”s would be replaced byessentially χ divided by a Gauss sum, see (3.10)). However, at the (primitive) trivial characterχ = 1, we have r1 = 1, and hence the maps
δ|qdR−1=0 : U1E,χE
(λ−1E )→ Cp, δ′|qdR−1=0 : U1
E,χE(λ−1E )→ Cp
are OKpJΓKK[1/p]-equivariant in the obvious sense.
5.6. Choice of decomposition of module of semi-local universal norms: a preview ofthe rank 0 case. Our choice of U0 in this case will follow that of [39, Section 11]. We will chooseU2 = ker δ|qdR−1=0. As in loc. cit., using Wiles’ explicit reciprocity law describing the local Kummerpairing, one can relate U2 to the Kummer local condition at p describing the global Selmer groupSelp∞(E/Q). In particular, under the assumption corankZpSelp∞(E/Q) = 0 we will show in Section
6 that U2 ∩ E1E,χE
(λ−1E ) = 0 and that there is a map
Hom(X 1E,χE
/rec(U2),Kp/OKp)Gal(K∞/K) → Selp∞(E/Q)
with finite cokernel. By Theorem 5.4 and (109), this implies C · L(E, 1) = LλE (1) 6= 0, for someC ∈ C×p , and hence L(E, 1) 6= 0 which gives the rank 0 p-converse theorem.
6. Rank 0 p-converse theorem
As before, E/Q denotes an elliptic curve with CM by OK , and p a prime that ramifies in K.Descent in the rank 0 case essentially entails replacing the map “δ” in [39, Section 11.4] with themap
δ|U0 |qdR−1 : U0 → Cp,and following largely the same arguments as in loc. cit. In essence, our map δ|U0 is a power serieslifting of δ from loc. cit., which is itself simply the reciprocity law map.
6.1. The Selmer group via local class field theory. For certain reasons of convenience, we willemphasize the GL1/K viewpoint in this section rather than the GL2/Q-viewpoint (for example,viewing E[p∞] as the Gal(K/K)-moduleKp/OKp(λE) rather than the standard Gal(Q/Q)-module).Also, for convenience, we will use λ for λE . Hence we will use the following notations.
Definition 6.1. Let T (λ) := OLp(λ), the free rank-1 OLp-module with Gal(Kp/Kp) acting throughλ, V (λ) := T (λ)⊗Zp Qp, W (λ) := V (λ)/T (λ) = T (λ)⊗Zp Qp/Zp. We define
S(λ) = ker
(∏v
locv : H1(K,W (λ))→∏v
H1(Kv, E)
)where the product runs over all places of Kv. Following the notation of [39], we define the relaxedSelmer group
S′(λ) = ker
∏v-p
locv : H1(K,W (λ))→∏v-p
H1(Kv, E)
,
where the product runs over all places of K not dividing p.
Definition 6.2. Given a group H and an H-module M , let MH denote the invariants by H. Givena OKpJGal(K∞/K)K-module and a character χ ∈ ∆K , let
M |χ := M ⊗OKpJGal(K∞/K)K,χ OKp [χ]JΓKK,
where OKp [χ] is the extension of OKp obtained by adjoining the values of χ, and the tensor productmaps χ : ∆K → OKp [χ]×. In other words, M |χ is the maximal subquotient of M on which ∆K
52
acts through χ. Note that this is different from Mχ in our setting, as the latter is a OKpJΓKK[1/p]-submodule and subquotient of M ⊗Zp Qp. It is easy to see that there is a natural identification
M |χ ⊗Zp Qp∼−→Mχ.
Recall our previously fixed decomposition (97), which identifies Gal(K∞/K) ∼= ΓK ×∆K . Notealso that since p is totally ramified in K∞/K, there is a natural identification Gal(K∞/K) =Gal(Kp(E[p∞])/Kp) with respect to our fixed embedding (12).
For the remainder of this section, all Homs are in the category of OKp-modules unless otherwiseindicated.
Proposition 6.3 (Theorem 11.2 of [39]). The restriction map H1(K,W (λ)) → H1(K∞,W (λ))induces a map with finite kernel and cokernel
S′(λ)→ Hom(XE ,W (λ))Gal(K∞/K) = Hom(XE |χE ,W (λ))Gal(K∞/K).
Proof. This is well-known. See [39, Proof of Theorem 11.2] for detailed references, or [37, Lemma1.2] for a detailed outline.
Lemma 6.4 (cf. Lemma 11.6 of [39]). The natural restriction map
H1(Kp,W (λ))res−−→ H1(Kp(E[p∞]),W (λ))Gal(Kp(E[p∞])/Kp)
induces a map
(120) H1(Kp,W (λ))→ Hom(U1E |χE ,W (λ))Gal(Kp(E[p∞])/Kp)
of OKpJΓKK = OKpJGal(Kp(E[p∞])/Kp(E[p∞]))K-modules with finite kernel and cokernel.
Remark 6.5. The above map is not necessarily an isomorphism (unlike in loc. cit.) because we donot necessarily have E(Kp)[p] = 0 in our setting (whereas the assumptions of loc. cit. force this).
Proof. By inflation-restriction, we have
0→ H1(Kp(E[p∞])/Kp,W (λ)Gal(Kp(E[p∞])/Kp))inf−→ H1(Kp,W (λ))
res−−→ H1(Kp(E[p∞]),W (λ))Gal(Kp(E[p∞])/Kp) → H2(Kp(E[p∞])/Kp,W (λ)Gal(Kp(E[p∞])/Kp)).
From [41, Lemma 2.2], we see that H1(Kp(E[p∞])/Kp,W (λ)) is finite. Since E[p∞](Kp) is finite,we see by local duality that H2(Kp(E[p∞])/Kp,W (λ)) is finite. Hence the map
res : H1(Kp,W (λ))res−−→ H1(Kp(E[p∞]),W (λ))Gal(Kp(E[p∞])/Kp)
has finite kernel and cokernel. We will finish by showing that the target of this map is naturallyisomorphic to Hom(U1
E |χE ,W (λ))Gal(Kp(E[p∞])/Kp) under the Artin reciprocity map.
Since Gal(Kp/Kp) acts on W (λ) = E[p∞] through λ : Gal(Kp(E[p∞])/Kp(E[p∞])) → O×Kp , we
have that Gal(Kp(E[p∞])/Kp(E[p∞])) acts trivially on W (λ), and so
H1(Kp(E[p∞]),W (λ))Gal(Kp(E[p∞])/Kp) = Hom(Gal(Kp(E[p∞])/Kp(E[p∞])),W (λ))Gal(Kp(E[p∞])/Kp)
= Hom(Gal(Kp(E[p∞])ab/Kp(E[p∞]))|χE ,W (λ))Gal(Kp(E[p∞])/Kp).
(121)
By local class field theory, since K has class number 1, we have an isomorphism
(122) U1E × Z ∼−→ Gal(Kp(E[p∞])ab/Kp(E[p∞])).
53
By Lemma [39, 11.5], we have that χE is non-trivial on the decomposition subgroup of ∆ above p,
and so Z|χE = 1. Hence (122) gives an isomorphism
(123) U1E |χE
∼−→ Gal(Kp(E[p∞])ab/Kp(E[p∞]))|χE .This, along with (121), gives (120).
This has the following consequence on maximal divisible subgroups. Given a group G, let Gdiv ⊂G denote the maximal divisible subgroup.
Corollary 6.6. The natural restriction map
H1(Kp,W (λ))res−−→ H1(Kp(E[p∞]),W (λ))Gal(Kp(E[p∞])/Kp)
induces an isomorphism
(124) H1(Kp,W (λ))div∼−→ Hom(U1
E,χE|χE ,W (λ))Gal(Kp(E[p∞])/Kp)
of OKpJΓKK = OKpJGal(Kp(E[p∞])/Kp(E[p∞]))K-modules.
Proof. From (120), we have a map with finite kernel and cokernel
H1(Kp,W (λ))→ Hom(U1E |χE (λ−1),Kp/OKp)Gal(Kp(E[p∞])/Kp)
= Hom((U1E |χE (λ−1)
)Gal(Kp(E[p∞])/Kp),Kp/OKp)
where U1E |χE (λ−1) = U1
E |χE⊗OKpOKp(λ−1). Hence we get a map of finitely-generatedOKp-modules
with finite kernel and cokernel(U1E |χE (λ−1)
)Gal(Kp(E[p∞])/Kp) → Hom(H1(Kp,W (λ)),Kp/OKp).Given a finitely generated OKp-module M , let Mfree denote the OKp-free part. Tensoring with⊗ZpQp, we then get an isomorphism of Kp-vector spaces(
U1E,χE
(λ−1))Gal(Kp(E[p∞])/Kp) ∼−→ Hom(H1(Kp,W (λ)),Kp/OKp)free
= Hom(H1(Kp,W (λ))div,Kp/OKp).Applying Hom(·,Kp/OKp) again, we arrive at (124).
The χE-part U1E,χE
rec−−→ XE,χE of the local reciprocity map U1E
rec−−→ XE induces a map
Hom(XE,χE ,W (λ))→ Hom(U1E,χE
,W (λ)).
We denote this map by f → f |U1E,χE
. We have the following commutative diagram (cf. [39, p. 62]):
(125)
Hom(XE,χE ,W (λ))Gal(Kp(E[p∞])/Kp)
0(E(Kp)⊗OKp Kp/OKp
)div
H1(Kp(E[p∞]),W (λ))div
(H1(Kp(E[p∞]), E)[p∞]
)div
0
Hom(U1E,χE
,W (λ))Gal(Kp(E[p∞])/Kp)
β
φ ∼=
where the middle row is exact by the local descent exact sequence. By (125), and the fact that theimage of the Kummer map
E(Kp)⊗OKp Kp/OKp → H1(Kp(E[p∞]),W (λ))54
is contained in H1(Kp(E[p∞]),W (λ))div (see [3]), we have that it is equal to the image of(E(Kp)⊗OKp Kp/OKp
)div→ H1(Kp(E[p∞]),W (λ))div.
Hence by Proposition 6.3,
(126) S(λ)div = f ∈ Hom(XE,χE ,W (λ))Gal(Kp(E[p∞])/Kp) : f |1UE,χE ∈ im(φ).
6.2. Interlude: relating reciprocity maps to good reduction twists. Recall f is the conduc-tor of λ = λE as well as the fixed CM elliptic curve A/L from Definition 2.3, where L = K(f(p)) so
that wf(p) = 1, (f, p) = 1, and F = A is a relative Lubin-Tate group for the unramified extension
Lp/Kp.This ensures that K(E[pn]) ⊂ K(fpn) = L(A[pn]) (where the first equality follows because
K(E[pn])/K is ramified at primes dividing f(p)p, and the last equality follows from [14, PropositionII.1.6]). Denote the associated tower of local units by U ′, and the associated tower of principal localunits by (U ′)1. Recalling that U1 denotes the tower of local units attached to E, by the previoussentence, we have U1 ⊂ (U ′)1. Moreover, the norm from L(A[pn])→ K(E[pn]) induces a norm mapNm : (U ′)1 → U1. Denote the type (1, 0) Hecke character associated with A by λA, and viewed asa Galois character on Gal(L(A[p∞])/K) let χA = λA|Gal(L(A[p∞])/K∞), where K∞/K denotes the
unique Z⊕2p -extension of K.
Proposition 6.7. We have
(127) λ/χE = λA/χA
as characters Gal(K∞/K)→ O×Kp.
Proof. By the theory of complex multiplication ([38, Theorem 5.11]), we see that λ/χE and λA/χAboth map Gal(K∞/K) into 1 + pεOKp with finite cokernel. Moreover, since λ and λA differ bya finite-order character (both are of infinite type (1, 0)), we have that λ/χE and λA/χA differ bya finite-order character. Since the images of both λ/χE and λA/χA lie in the torsion-free group1 + pεOKp , this finite-order character must be trivial, and so we have λ/χE = λA/χA on all ofGal(K∞/K).
Definition 6.8. Let
δE : U1E,χE
→ Hom(E(Kp,∞)⊗OKp Kp/OKp ,W (λ))⊗Zp Qp,
δA : U1A,χA
→ Hom(A(Lp,∞)⊗OKp Kp/OKp ,W (λA))⊗Zp Qp
be the natural Kummer maps precomposed with the local Artin maps (cf. [14, Chapter I.4]). Inother words, δE is
U1E,χE
local rec−−−−−→ XE,χEKummer−−−−−→ Hom(E(Kp)⊗OKp Kp/OKp ,W (λ))⊗Zp Qp,
and similarly with δA. Note that since the local reciprocity map is nonzero and the Kummer map isinjective, δE and δA are nonzero maps. In particular, since by the p-adic logarithm logp : E(Kp)→Kp, we have as Kp-vector spaces
Hom(E(Kp)⊗OKp Kp/OKp ,W (λ))⊗Zp Qp∼= Hom(Kp/OKp ,Kp/OKp)⊗Zp Qp = Kp,
and similarly with δA. Hence the targets of δE and δA are isomorphic to Kp, and hence these mapsare surjective.
Note that since A(Lp,∞) = E(Lp,∞) (by (127)), there is a trace map TrLp,∞/Kp,∞ : A(Lp,∞) =E(Lp,∞)→ E(Kp,∞).
55
Proposition 6.9. There is commutative diagram of OKpJGal(K∞/K),OKpK[1/p]-equivariant maps
(128)
U1E,χE
Hom(E(Kp,∞)⊗OKp Kp/OKp ,W (λ))⊗Zp Qp
U1A,χA
Hom(A(Lp,∞)⊗OKp Kp/OKp ,W (λA))⊗Zp Qp
U1E,χE
Hom(E(Kp,∞)⊗OKp Kp/OKp ,W (λ))⊗Zp Qp,
δE
Tr∗Lp,∞/Kp,∞
δA
Nm incl∗
δE
where the first right vertical arrow is pullback by TrLp,∞/Kp,∞ : A(Lp,∞) → E(Kp,∞), the secondright vertical arrow is pullback by the inclusion incl : E(Kp,∞) ⊂ E(Lp,∞) = A(Lp,∞), and the leftvertical arrows and right vertical arrows are isomorphisms of OKpJGal(K∞/K),OKpK[1/p]-modules.
Proof. This follows immediately from the restriction/corestriction functorialities of the local reci-procity map.
6.3. The value on elliptic units.
Choice 6.10. Henceforth, fix a Kp-generator u of E(Kp)⊗OKp Kp∼= Kp, so that we can identify
Hom(E(Kp)⊗OKp Kp/OKp ,Kp/OKp) = Kp,
and whence view the target of δE as in Kp.
Let
δE(λ−1) : U1E,χE
(λ−1)→ Kp
denote the twist of δE by λ−1. Recall our map (105)
δ|qdR−1=0 : U1E,χE
(λ−1)→ Cp.
Proposition 6.11. Let ξE ∈ C1E,χE
denote the generator from (110), and let Ω∞ denote the Neronperiod of E/Q. We have
(129) δE(λ−1) = C0δ|qdR−1=0
and
(130) ker(δE(λ−1)) = ker(δ|qdR−1)
for some C0 ∈ C×p , and
(131) δE(λ−1)(ξE) = Lλ(1) · C1 =L(E, 1)
Ω∞· C2
for some constants C1, C2 ∈ C×p .
Proof. By (128) and (107) it suffices to show δA(λ−1A )(locp(e(a))) = Lλ ·C ′1 = L(E, 1) ·C ′2 for some
C ′1, C′2 ∈ C×p . Recall ω0 is the normalized invariant differential on F = A. Note that for any
β ∈ U1E,χE
(λ−1) ⊂ U1χA
(λ−1A ),, we have (see [14, p. 20])
d
ω0loggβ(0) =
(1− f ′(0)
p
)d
ω0log gβ(0)
where f : A→ A/A[π] = Aφ is the natural quotient viewed as an morphism f(X) on formal groups.Thus by Wiles’ explicit reciprocity law ([14, Chapter I.4]), we have δA(λ−1
A ) = C ′0δ|qdR−1=0 for some56
constant C ′0 ∈ C×p , which gives (129). Moreover δA(λ−1A ) = ker(δ|qdR−1=0) and (130) immediately
follows. Moreover, one computes
δA(λ−1A )(e(a)) = C ′3
d
ω0log gloc(e(a))(0) = C ′4
d
ω0logglocp(e(a))(0)
(111)= δ|qdR−1=0C
′′4 δ(e(a))
(109)= C ′′4Lλ(1)
for some constants C ′3, C′4, C
′′4 ∈ C×p . This gives the first equality.
The second equality follows from (50). We are done.
6.4. Description of the Selmer group via Wiles’s explicit reciprocity law. Continue toretain the choice of Choice 6.10. We now choose an appropriate decomposition of U1
E,χE(λ−1), as
in Section 5.6 (recall λ = λE in this section).
Proposition 6.12 (cf. Lemma 11.9 of [39]). There exists a decomposition
U1E,χE
(λ−1) = U1 ⊕ U2
where
(132) U2 = ker(δE(λ−1)).
Proof. First choose an arbitrary such decomposition, and let β1, β2 be the generators of U1, U2,respectively. As mentioned above,
δE(λ−1) : U1E,χE
(λ−1) ∼= OKpJΓKK⊕2 → Kp
is surjective. HenceδE(λ−1)(β1) = αδE(λ−1)(β2)
for some α ∈ Kp. Now replace β2 by β2 − αβ2, and replace U2 by the OKpJΓKK[1/p]-modulegenerated by this new β2. Then the new decomposition satisfies (132).
Proposition 6.13 (cf. Proposition 11.10 of [39]). Let
δW (λ) := maps u 7→ δE(u)v : v ∈W (λ).We have
(133) im(φ) = δW (λ) = Hom(U1E,χE
/ ker(δE),W (λ))Gal(Kp(E[p∞])/Kp).
Proof. From the definitions we have
(134) im(φ) = δW (λ) ⊂ Hom(U1E,χE
/ ker(δE),W (λ))Gal(Kp(E[p∞])/Kp)
As abelian groups, we have
im(φ) ∼=(E(Kp)⊗OKp Kp/OKp
)div
∼= Kp/OKp ,
δW (λ) ∼= W (λ) ∼= Kp/OKp ,and
Hom(U1E,χE
/ ker(δE),W (λ))Gal(Kp(E[p∞])/Kp)
∼= Hom((U1E,χE
/ ker(δE))
(λ−1),Kp/OKp)Gal(Kp(E[p∞])/Kp)
∼= Hom(OKp ,Kp/OKp) = Kp/OKp .So all terms of (134) are isomorphic to Kp/OKp , and hence all the inclusions are actually equalities.
A key consequence is the following.57
Theorem 6.14. There is a natural identification
(135) S(λ)div = Hom((X )χE/rec(ker(δE)),W (λ))Gal(K∞/K).
Hence,
(136) #S(λ)div = #Hom((X )χE/rec(ker(δE)),W (λ))Gal(K∞/K).
Proof. This follows immediately from (133) and (126).
Proposition 6.15. We have(137)
#(Hom((X )χE/rec(ker(δE)),W (λ))Gal(K∞/K)) ∼ #(Hom(U1E,χE
/(C1E,χE
, ker(δE)),W (λ))Gal(K∞/K))
where “∼” means one side is finite if and only the other side is.
Proof. This follows from the same arguments as in [39, Proof of Theorem 11.16], using the Rubin-type Main Conjecture (Theorem 5.4) and the fact that XE has no nonzero pseudo-null submodulesby [19, Theorem 2].
Theorem 6.14 gives us the rank 0 p-converse theorem.
Theorem 6.16 (Rank 0 p-converse). Suppose E/Q is an elliptic curve with CM by OK and p isramified in K. Then corankZpSelp∞(E/Q) = 0 implies L(E, 1) 6= 0, i.e. ords=1L(E, s) = 0.
Proof. We have
#S(λ)div(136)= #(Hom(XE,χE/(rec(δE)),W (λ))Gal(K∞/K))
(137)∼ #(Hom(U1E,χE
/(ker(δE), C1E,χE
),W (λ))Gal(K∞/K)).
(138)
Now we have
#(Hom((U1E,χE
/(ker(δE), C1E,χE
),W (λ))Gal(K∞/K)))
= #φ ∈ Hom(U1E,χE
/ ker(δE),W (λ))G∞ : φ(C1E,χE
) = 0= v ∈W (λ) : δE(C1
E,χE)v = 0
(131)= v ∈W (λ) : C4
L(E, 1)
Ω∞v = 0 = #
(OKp/
(C2L(E, 1)
Ω∞OKp
))where C2 ∈ C×p . Now the statement immediately follows since S0(λ) = Selp∞(E/Q) by Shapiro’slemma.
7. Rank 1 p-converse theorem
7.1. The rank 1 p-converse theorem for CM elliptic curves with p ramified in the CMfield.
Proposition 7.1. Suppose corankZpSelp∞(E/Q) ≤ 1. Then XE,χE (λ−1E )/(ΓK−1) is a 1-dimensional
Kp-vector space.
Proof. See [37, Remark before Section 5], using the fact that rankOKE(K) = rankZE(Q).
We will briefly need two different (non-Bloch Kato) Selmer group.58
Definition 7.2. For each (possibly infinite) prime ` of Q, fix an algebraic closure Q` and anembedding Q ⊂ Q` (for ` = p, we continue to use (12)). Let G` = Gal(Q`/Q`) be the decompositiongroup I` be the inertial subgroup of G`. Define for all 0 ≤ n ≤ ∞,
Selurpn(E/Q) = ker
(∏`
res` : H1(Q, E[pn])→∏`
H1(G`/I`, E[pn])
).
We also let
Selrelpn(E/Q) = ker
∏6=p
res` : H1(Q, E[pn])→∏`6=p
H1(G`/I`, E[pn])
Note that by Shapiro’s lemma, Selrel
pn(E/Q) = S′(λE) in the notation of Definition 6.1. (Recall thatλ = λE in Section 6.) Moreover, by Proposition 6.3, we have
(139) Selrelpn,div(E/Q)div = S′(λ)div = XE,χE (λ−1
E )/(ΓK − 1).
Similarly, by class field theory and essentially the same argument as for Proposition 6.3 mutatismutandis, we have
(140) Selurpn(E/Q)div = YE,χE (λ−1
E )/(ΓK − 1).
Corollary 7.3. Suppose corankZpSelp∞(E/Q) ≤ 1. Then YE,χE (λ−1E )/(ΓK − 1) = 0.
Proof. By Proposition 7.1 and (139), we have corankZpSelrelpn(E/Q) = 1. Note that the local condi-
tion at p of Selrelpn(E/Q) is H1(Qp, E[pn]), whose orthogonal complement under Tate local duality is
0. Moreover, the local condition at ` for Selurpn(E/Q) is, by the inflation-restriction exact sequence,
H1(G`/I`, E[pn]I`) = ker(res` : H1(Q, E[pn])→ H1(G`/I`, E[pn])
).
In particular, at ` = p we have E[pn]Ip = 0 since Kp(E[pn])/Kp is totally ramified. Hence
H1(Gp/Ip, E[pn]Ip) = 0. Hence Selrelp∞,div(E/Q) and Selur
p∞(E/Q) have local conditions that aremutually orthogonal complements at all primes ` (see [12, Theorem 2.17 and ensuing discussion]).Now by Theorem 2.17 and Theorem 2.18 of loc. cit. and Shapiro’s lemma, (using the fact thatE[pn] ∼= E[pn](1) canonically by the Weil pairing E[pn]× E[pn]→ µpn)
#Selrelpn(E/Q)
#Selurpn(E/Q)
=#H0(K, (OK/pn)(λE))
#H0(K, (OK/pn)(λE))
#H1(Kp, (OK/pn)(λE))
#H0(Kp, (OK/pn)(λE))
=#H1(Kp, (OK/pn)(λE))
#H0(Kp, (OK/pn)(λE))= #H0(Kp, E[pn])#(OK/pn).
Since #H0(Kp, E[pn]) is constant as n → ∞, by taking n → ∞ and using (139) and (140) we seethat
dimKp
(YE,χE (λ−1
E )/(ΓK − 1))
= dimKp
(XE,χE (λ−1
E )/(ΓK − 1))− 1 = 0,
which gives the desired statement.
Proposition 7.4. Suppose corankZpSelp∞(E/Q) = 1. Recall U0 from (100), and δE from Definition6.8. We have
(141) U0 = ker(δ|qdR−1=0)(130)= ker(δE(λ−1
E )).
In particular, U0 is a direct summand of U1E,χE
(λ−1E ).
59
Proof. Since corankZpSelp∞(E/Q) = 1, by the p-Selmer rank-analytic rank parity conjecture (knownfor all elliptic curves over Q, see [15]), we have L(E/Q, 1) = 0, and so by (50) we have δ(ξE)(1) = 0.Hence ξE ∈ ker(δ|qdR−1=0), and so by (100) we have ker(δ|qdR−1=0) ⊂ U0. By [39, Lemma 11.9],
ker(δ|qdR−1=0) is a direct summand of U1E,χE
(λ−1E ) (our δ|qdR−1=0 is a multiple of δE(λ−1
E ) by (129),
and the latter is is the twist by λ−1E of the reciprocity map “δ” in loc. cit.). Thus, by Proposition
5.6 we have (141) and the direct summand statement.
Choice 7.5. Henceforth, fix a choice of decomposition
U1E,χE
(λ−1E ) = U0 ⊕ U2,
so that in particular U2 ∩ E1E,χE
(λ−1E ) = 0, and U2 satisfies the condition (101) of Choice 5.7. In
particular, we may and do take U1 = U0, and so λ1 = 1, λ2 = 0 in Section 5.4.
Remark 7.6. It is interesting to remark how ker(δE(λ−1E )) plays different roles in the rank 0
situation, as in [39, Section 11] (where it is denoted by “U2” in the notation of loc. cit., see Lemma11.8 of loc. cit.; note that this differs from our use of “U2”), and in our rank 1 situation. In theformer situation, it plays the role of the (Kummer) local condition defining the usual Selmer group,and hence takes the role of U2 in (5.4). In the latter situation, it plays the role of U1 = U0, thedomain of our “derivative” operator δ′, whereas a different local condition U2 is chosen for (5.4).
Reducing (98) modulo (ΓK − 1), we get
U1E,χE
(λ−1E )/(C1
E,χE(λ−1E ), (ΓK − 1))→ XE,χE (λ−1
E )/(ΓK − 1)→ YE,χE (λ−1E )/(ΓK − 1)→ 0.
Hence by Corollary 7.3 we have a surjection of Kp-vector spaces
(142) U1E,χE
(λ−1E )/(C1
E,χE(λ−1E ), (ΓK − 1)) XE,χE (λ−1
E )/(ΓK − 1)
where the target is 1-dimensional.
Proposition 7.7. We have U2/(ΓK − 1) 6= 0 and
rec(U2)/(ΓK − 1) = XE,χE (λ−1E )/(ΓK − 1).
Proof. We have U1E,χE
(λ−1E ) = U0 ⊕U2. By Proposition 6.14 and the fact that U0 = ker(δE) (since
δE = δ|qdR−1=0), we have rec(U0)/(ΓK − 1) = 0. Hence rec(U2)/(ΓK − 1) 6= 0 by (142), and soU2/(ΓK − 1) 6= 0.
Corollary 7.8. We have(charOKpJΓKK[1/p]XχE (λ−1
E )/rec(U2))/(ΓK − 1) 6= 0.
Proof. This follows from Proposition 7.1 and Proposition 7.7.
Recall the basis β0 of U0. The following Theorem makes crucial use of our corankZpSelp∞(E/Q) =1 assumption. In essence, we use this assumption and (141) to identify U0 as the kernel ker(δE) ofthe reciprocity map from Definition 6.8. Then we proceed by contradiction: Assume δ′(β0) = 0.By the functoriality of the reciprocity map (128), we conclude that if δ′(β0) = 0, then δ′ wouldbe the zero map for every elliptic curve E′/Q with CM by OK . However, by previous formulas(namely (95), this would imply that every such E′/Q has rank 0, which is clearly known to be false(we exhibit a list of examples known to have rank 1).
Theorem 7.9. Suppose corankZpSelp∞(E/Q) = 1. Then
(143) δ′(β0) 6= 0.60
Proof. Note that since K has class number 1, Kn ⊗K Kp = Kn,p, and so (107) induces an identifi-cation
locp : U1E,χE
(λ−1E )
∼−→ U1F,χF
(κ−1)
of OKpJΓKK[1/p]-modules. In particular, for any elliptic curve E′/Q with CM by OK , we have
logp : U1E′,χE′
(λ−1E′ )
∼−→ U1F ′,χF ′
(κ−1)
for F ′ the formal group A′ of an appropriate twist A′ of E′, defined over some unramified exten-sion L′p/Kp. By Lubin-Tate theory, the modules U1
F ′,χF ′(κ−1) form an inverse system indexed by
unramified extensions L′′p ⊃ L′p ⊃ Kp, height 2 formal modules F ′′ for L′′p/Kp and F ′ for L′p/Kp,respectively, with transition maps
NmL′′p/L′p
: U1F ′′ U1
F ′ .
On isotypic components, this induces isomorphisms of OKpJΓKK[1/p]-modules
(144) U1F ′′,χF ′′
∼−→ U1F ′,χF ′
.
Let U ′0 ⊂ U1E′,χE′
(λ−1E′ ) be as in (100) for E′. By (141) and (128) applied to E and A = E′, we then
have under the identifications (144)
U0(141)= ker(δE(λ−1
E ))(128)
ker (δE′(λ−1E′ )
(141)= U ′0.
Hence β0 induces a basis of U ′0. Write λ′0β0 = ξE′ for λ′0 ∈ OKpJΓKK[1/p]. Then by (118) and (119),we see that (recalling r1 = 1)
0 = λ′0(1)δ′(β0) = δ′(ξE′) = LλE′ (κ−1− ).
Now assume that E′ has global root number -1. Hence, by (95), we see that the Heegner pointPθλE′/χ0
,χ0 = 0, and hence by the Gross-Zagier formula for Shimura curves [60, Theorem 1.2] and
(78), we see that ords=1L(E′, s) 6= 1, for any E′/Q with CM by OK (and p ramified in K). However,this is clearly not true, as for any imaginary quadratic field K of class number 1 and p (the uniqueprime) ramified in K, there exists E′/Q such that ords=1L(E′, s) = 1. For example, in Cremona’slabeling:
(1) For K = Q(i), p = 2, take E′ = 256b1 : y2 = x3 − 2x.(2) For K = Q(
√−2), p = 2, take E′ = 256a1 : y2 = x3 + x2 − 3x+ 1.
(3) For K = Q(√−3), p = 3, take E′ = 225a2 : y2 + y = x3 + 1.
(4) For K = Q(√−7), p = 7, take E′ = 441d1 : y2 + xy + y = x3 − x2 − 20x+ 46.
(5) For K = Q(√−11), p = 11, take E′ = 121b1 : y2 + y = x3 − x2 − 7x+ 10.
(6) For K = Q(√−19), p = 19, take E′ = 361a1 : y2 + y = x3 − 38x+ 90.
(7) For K = Q(√−43), p = 43, take E′ = 1849a1 : y2 + y = x3 − 860x+ 9707.
(8) For K = Q(√−67), p = 67, take E′ = 4489a1 : y2 + y = x3 − 7370x+ 243528.
(9) For K = Q(√−163) p = 163, take E′ = 26569a1 : y2 + y = x3 − 2174420x+ 1234136692.
Hence we have a contraction, and (143) is established.
Theorem 7.10 (Rank 1 p-converse theorem). Let E/Q be an elliptic curve with CM by OK andsuppose p is ramified in K. Suppose corankZpSelp∞(E/Q) = 1. Then ran(E/Q) = ralg(E/Q) = 1and #X(E/Q) <∞.
61
Proof. By Corollaries 5.9 and 7.8, we see that (λ0λ1)(1) 6= 0, which implies λ0(1) 6= 0. From (143),we have δ′(β0) 6= 0. Since ξE = λ0β0, we see from (118) and (119), and the fact that r1 = 1, that
LλE (κ−1− ) = (κ−1
− )∗LλE (r1) = δ′(ξE)(r1) = λ0(1)δ′(β0)(r1) = λ0(1)δ′(β0) 6= 0.
Now the statement follows from (95).
7.2. Sylvester’s conjecture. Recall the cubic twist family of elliptic curves
Ed : x3 + y3 = d.
Sylvester conjectured in 1879 ([55]) that for d = p prime, if d ≡ 4, 7, 8 (mod 9) then Ed has arational solution.
Theorem 7.11. Sylvester’s conjecture is true. That is, for any prime p, if p ≡ 4, 7, 8 (mod 9)then there exist x, y ∈ Q such that x3 + y3 = p.
Remark 7.12. Previously, the case p ≡ 4, 7 (mod 9) was announced by Elkies [16], though thefull proof remains unpublished. See also the article of Dasgupta-Voight [13], which gives anotherproof of Elkies’s result under additional assumptions.
Proof of Theorem 7.11. By standard 3-descent (see [13]), we have corankZ3(Sel3∞(Ed)) = 1 ford ≡ 4, 7, 8 (mod 9). Now the Corollary follows immediately from Theorem 7.10 with p = 3 andK = Q(
√−3).
7.3. Goldfeld’s conjecture for the congruent number family. For a general elliptic curveE : y2 = x3 + ax + b, let ran(E/Q) = ords=1L(E/Q, s) denote the analytic rank. Let Ed : y2 =
x3 + ad2x + bd3 be the quadratic twist by Q(√d). The celebrated conjecture of Goldfeld states
that:
Conjecture 7.13 (Goldfeld’s conjecture [18]). For r = 0, 1,
limX→∞
#
0 < |d| < X : d squarefree, ran(Ed/Q) = r
# 0 < |d| < X : d squarefree=
1
2.
The best known unconditional general results towards Goldfeld’s conjecture are [29], [31], [32]and [54] for certain CM elliptic curves including the congruent number family. In particular, theresults of Smith imply the “algebraic” Goldfeld conjecture for the prime 2, in which the analyticranks are replaced with corankZ2Sel2∞ . Using the results of loc. cit., we can establish Goldfeld’sconjecture for certain elliptic curves among those considered in loc. cit. including the congruentnumber family.
Theorem 7.14. Suppose E/Q is an elliptic curve with E(Q)[2] ∼= (Z/2)⊕2 and no cyclic 4-isogenydefined over Q. Suppose further that E has CM by K = Q(i) or Q(
√−2). Then Goldfeld’s
conjecture (Conjecture 7.13) is true for E.In particular, 100% of squarefree d ≡ 1, 2, 3 (mod 8) are not congruent numbers and 100% of
squarefree d ≡ 5, 6, 7 (mod 8) are congruent numbers, and Goldfeld’s conjecture is true for thecongruent number family Ed : y2 = x3 − d2x.
Proof. The corankZ2Sel2∞ results of Smith [54] imply that for r = 0, 1, corankZpSel2∞(Ed/Q) = r
for 50% of squarefree d (ordered by |d|). Note that each Ed has CM by OK . Now applying Theorem6.16 and Theorem 7.10 for p = 2 and K = Q(i) or Q(
√−2) to these Ed, we arrive at the statement.
62
In particular, the above theorem has the following consequence for the congruent number familyEd : y2 = x3 − d2x, which together with the results of [54] resolves the congruent number problemin 100% of cases.
Theorem 7.15. Consider Ed : y2 = x3− d2x. Then ords=1L(Ed/Q, s) = 0 for 100% of squarefreed ≡ 1, 2, 3 (mod 8), ordering by increasing |d|, and ords=1L(Ed, s) = 1 for 100% of squarefreed ≡ 5, 6, 7 (mod 8), ordering by increasing |d|. In particular, 100% of squarefree integers d ≡ 5, 6, 7(mod 8) are congruent numbers.
Proof. This follows from Theorem 7.14 and the work of Gross-Zagier [21] and Kolyvagin [27].
8. Appendix: Elliptic units main conjecture via equivariant main conjecture
In this Appendix, we prove Theorem 5.3 (the elliptic units main conjecture). In [39], Rubin provesthis conjecture in many cases, under relatively mild assumptions which are however not satisfiedin our situation (e.g. p|#O×K in certain applications of interest). To circumvent the technicalassumptions of Rubin’s Euler system argument, we appeal to the results of Johnson-Leung andKings ([23]) on equivariant main conjectures. In particular, the results of Section 5 of loc. cit.imply Theorem admitting the vanishing of the total µ-invariant of appropriate extensions K ′∞/Kwhere Gal(K ′∞/K) ∼= Zp ×∆′, ∆′ finite abelian.
In the p split in K situation, the authors of loc. cit. take the choice K ′∞ = K(p∞) and appealto results of Robert ([34]) on the vanishing of the total µ-invariant.13 In the p nonsplit situation,we take K ′∞ = K(µp∞), and introduce a new argument computing the total µ-invariant as theµ-invariant of a product of two Kubota-Leopoldt p-adic L-functions. This latter argument comesdown to computing the µ-invariant as the p-adic valuation of a sum of logarithms of elliptic units(i.e. proving an “algebraic p-adic Kronecker limit formula”) following the valuation computation of[10], and using Gross’ “facotrizing” method from [20] in order to show these logarithms of ellipticunits are products of speical values of Kubota-Leopoldt p-adic L-functions.14Then the vanishingfollows from the theorem of Ferrero-Washington [17] (see also [50]).
8.1. The cyclotomic algebraic µ-invariant. We continue the notation of the previous sections.
Definition 8.1. Recall U as defined in (13), and recall we let L = K(f(p)) with f(p) the prime-to-p
part of f. Let f be the smallest positive rational integer belonging to f(p) (so that (f) = Nm(f(p))), letL+ = KQ(µf ), and let L+
n = L+(µpn) and L+∞ = L+(µp∞), and so Gal(L+
∞/K) ∼= Gal(L+∞/L
+)×Gal(L+/K) ∼= Zp×∆+ for some finite abelian group ∆+. For 0 ≤ n ≤ ∞, let M+
n be the maximalpro-p abelian extension of L+
∞ which is unramified outside p, and let N+n be the maximal pro-p
abelian extension of L+∞ which is unramified everywhere. The we let
X+ := Gal(M+∞/L
+∞) = lim←−
n
Gal(M+n /L
+n ).
Let Γ+ ⊂ Gal(L+∞/L) be the maximal subgroup isomorphic to Zp, so that
(145) (X+)Γpn
+= Gal(M+
n+s/L+∞)
where s = ordp(q), so that s = 1 if p > 2 and s = 2 if p = 2. By the non-vanishing of the p-adicregulator ([5]), one can show that Gal(M+
n /L+∞) is finite. Note that X+ is a ZpJGal(L+
∞/K)K-module. For any χ ∈ ∆+, we can consider the χ-isotypic component X+
χ .
13In fact, the argument of [34] is based on the method of Sinnott [50] reproving the vanishing of the µ-invariantof the Kubota-Leopoldt p-adic L-function (i.e. the theorem of Ferrero-Washington [17]).
14We note that Rubin previously extended Gross’ argument to the p-ramified case in [36].
63
Moreover, for each 0 ≤ n ≤ ∞, letK ⊂ K+n ⊂ K(µp∞) be the unique subfield with Gal(K+
n /K) ∼=Z/pn if n < ∞, and Gal(K+
∞/K) ∼= Zp if n = ∞. Note that (by standard theory of cyclotomicextensions), K+
n /K is totally ramified. Let
Γ′+ = Gal(K+∞/K).
Note that if we let t ∈ Z≥0 such that K+∞ ∩ K(1) = K+
t , then we have a natural identification
Γ+ = (Γ′+)pt
given by restriction from L+∞ to K+
∞. As L+∞/Q is an abelian extension of Q which is a
finite extension of the cyclotomic Zp-extension of Q, by [19], we X+ is a torsion ZpJGal(L+∞/K),ZpK-
module.
Remark 8.2. Note that X+ is not the cyclotomic specialization of X , as the latter would haverank at least 1. In fact, there is a natural map from the latter into the former.
8.2. Computation of algebraic Iwasawa invariants: a valuation calculation. We followthe strategy of [10] for studying
ordp(#(X+)Γpn
+) = ordp(#Gal(M+
n+s/L+∞)).
We need some lemmas and propositions which are refinements of results from loc. cit. We follow[14, Chapter III.2] throughout this section.
Definition 8.3 (p-adic regulator). Let F/K be any abelian extension of degree r. Let σ1, . . . , σrbe the embeddings F → Cp. (Note that all embeddings induce the same embedding Kp → Cp,since there is only one prime above p.) Let E be a subgroup of finite index in O×F , and choosegenerators e1, . . . , er−1 for E/Etors. Then the p-adic regulator of E is defined to be
Rp(E) = det(log σi(ej))1≤i,j≤r−1.
This is well-defined up to sign because log NmF/K(e) = 0 for e ∈ E. As short-hand, let
Rp(F ) = Rp(O×F ).
Theorem 8.4 ([5]). Assume F/K is abelian (as is the case in Definition 8.3). Then Rp(E) 6= 0.
Definition 8.5. We follow the notation and presentation of [14, Chapter III.2.4]. Fix an abelianextension F/K of degree r as in Definition 8.3. For each prime P of F above p (the unique prime ofK above p), let wP denote the number of p-power roots of unity in FP. Let Φ = F⊗KKp =
∏P|p FP,
and let U be the group of principal units in Φ. Then the p-adic logarithm gives a homomorphism
log : U → Φ
whose kernel has order∏
P|pwP, and whose image is an open subgroup of Φ. Let q = p if p > 2
and q = 4 if p = 2. Let E ⊂ O×F be a subgroup of finite index, and let D = D(E) = E · 〈1 + q〉,and let D be its closure in Φ×, and 〈D〉 the projection of D to U . We write Φ(F ), U(F ), E(F )when we refer to these objects for a specific F . Let ∆p(F/K) denote the sum over P|p of therelative discriminants of FP/Kp. Given any number field F , let h(F ) denote its class number andlet h(F )[p∞] denote the p-part of the class number.
Definition 8.6. Suppose we are in the situation of Definition 8.1, i.e. with F = L+n . Then letting
Un = U(L+n ) and En = E(L+
n ), Artin reciprocity gives
Un/〈En〉 ∼= Gal(M+n /N
+n ).
Recall that N+n (resp. M+
n ) is the maximal pro-p abelian extension unramified everywhere (resp.unramified outside p) of L+
n . Note that N+n ∩ L+
∞ = L+n and L+
∞ ⊂ Mn. Let Yn ⊂ Un be thesubgroup such that
(146) Yn/〈En〉 ∼= Gal(M+n /N
+n L
+∞).
64
Recall that L+n = LK+
n , and K+n ⊂ K(µp∞) is such that [K+
n : K] = pn. In particular, K+n /Q is
an abelian extension. Consider the norm map NmL+n /Q : Un → 1 + pOKp .
Lemma 8.7 (cf. Lemma III.2.6 of [14]). We have Yn = NmL+n /Q|Un.
Proof. If u ∈ Yn, then since Un/Yn ∼= Gal(N+n L
+∞/N
+n ) ∼= Gal(L+
∞/L+n ), we have (u, L+
∞/L+n ) = 1
(as usual, denoting the Artin symbol of an abelian extension E′/E by (·, E′/E)). Then by thefunctoriality of the Artin symbol, we get (u, L+
∞/L+n ) = (NmL+
n /Q(u),K+∞/Q) = 1. Hence the idele
NmL+n /Q(u), which is 1 outside of p, is necessarily 1. The argument in reverse shows the converse.
Let Dn = D(L+n ), and similarly with Dn.
Lemma 8.8. Assume that we are in the situation of Definition 8.1, i.e. with F = L+n . Let
pδ||[L+ : K], so that pn+δ||[L+n : K]. Then we have [log(Un) : log(〈Dn〉)] <∞, and in fact
ordp([log(Un) : log(〈Dn〉)]
)= ordp
qpn+δRp(L+n )√
∆p(L+n /K)
·∏P|p
(wPN(P))−1
,
where P runs over all primes of L+n above p.
Proof. This is [10, Lemma 8], the argument of which goes through using εd = 1 + q in place ofεd = 1 + p.
Corollary 8.9. Retain the situation of Lemma 8.8. Let w(E) be the number of roots of unity inE. Then
ordp([Un : 〈Dn〉]
)= ordp
qpn+δRp(L+n )
w(L+n )√
∆p(L+n /K)
∏P|p
N(P)−1
,
where P runs over all primes of L+n above p.
Proof. This is an immediate consequence of Lemma 8.8 using the snake lemma, see [10, Lemma 9]for details.
Proposition 8.10 (cf. Proposition III.2.17 of [14]). Retain the notation of Lemma 8.9. Recallpt = [(K(1) ∩K+
∞) : K] and let pe = [(L ∩K+∞) : K]. We have
(147) ordp([M+
n : L+∞])
= ordp
qpn+e−th(L+n )Rp(L
+n )
w(L+n )√
∆p(L+n /K)
·∏P|p
(1− N(P)−1)
,
where the product on the right-hand side runs over primes of L+n above p.
Proof. Let Dn = En · 〈1 + q〉. Then since NmL+n /K
(En) = 1, we have NmL+n /Q(〈Dn〉) = 1 +
q2pn+δ−1Zp. Now we have a diagram
(148)
0 〈En〉 〈Dn〉 1 + q2pn+δ−1Zp 0
0 Yn Un 1 + qpn+t−e−1Zp 0
NmL+n /Q
NmL+n /Q
,
where the horizontal rows are exact and the vertical arrows are injective, by the above discussionand Lemma 8.7. Then [L+
n : K] is exactly divisible by pn+δ, and so (147) follows from Lemma 8.9,65
(146), [M+n : L+
∞] = [M+n : N+
n L+∞][N+
n L+∞ : L+
∞], and the fact that [N+n L
+∞ : L+
∞] = [N+n : L+
n ] =h(L+
n )[p∞].
Definition 8.11. Let ∆′+ = Gal(L+∞/K
+∞) and recall Γ′+ = Gal(K+
∞/K) so that Gal(L+∞/K) =
∆′+ × Γ′+. For χ ∈ ∆′+, recall the torsion ZpJΓ′+K-module (X+)χ. We let
fχ := charOKp [χ]JΓ′+(X+χ ).
Denote the µ-invariant by µinv(f+χ ) and the λ-invariant by λinv(f+
χ ) and λinv(g+χ ).
Corollary 8.12. Let K+t = K+
∞ ∩ K(1), and recall that s = 1 if p > 2 and s = 2 if p = 2. Wehave for all n 0,(149)∑χ∈∆′+
µinv(f+χ ) · pt+n−1(p− 1) +
∑χ∈∆′+
λinv(f+χ ) = 1 + ordp
(hRp
w√
∆(L+
n+s/K)/hRp
w√
∆(L+
n+s−1/K)
).
Proof. By (147), the right-hand side of (149) is ordp([M+n+s : L+
∞]/[M+n+s−1 : L+
∞]). Now (149)follows from (145) and Weierstrass preparation (using the standard definitions of µinv and λinv).
8.3. Computation of analytic Iwasawa invariants: application of analytic class numberformula and p-adic Kronecker limit formula. We continue to follow [14, Chapter III.2].
Lemma 8.13. For any finite-order character ν ∈ Γ′+, define mν = r if ν((Γ′+)rs) = 1 but
ν((Γ′+)pr−1
) 6= 1. Retaining the notation of Corollary 8.12, we have for all n 0,
(150)∑χ∈∆′+
µinv(f+χ ) · pt+n−1(p− 1) +
∑χ∈∆′+
λinv(f+χ ) = ord
( ∏ν,mν=t+n
ν(g+χ )
).
Proof. This is just [14, Lemma III.2.9].
Definition 8.14. For any ramified character ε ∈ Gal(L+∞/K), let g = cond(ε), and let g denote
the smallest positive integer in g. Then define
Sp(ε) := − 1
12gwg
∑c∈C`(g)
ε−1(c) log φg(c).
Fix (ζfpn)n a compatible system of fpnth roots of unity, and for a character ε ∈ Gal(L+∞/K) of
conductor dividing fpn define the Gauss sum by
g(ε) :=1
fpn
∑σ∈Gal(L+
n /K)
ε(a)ζ−afpn ,
and letSn := ε ∈ Γ′+ : pn||cond(ε),
where the conductor is computed viewing ε : Gal(K(f(p)p∞)/K)→ Q×p (so that Γ+ is the image of
Γ ⊂ Gal(K(f(p)p∞)/K) under the restriction from K(f(p)p∞) to L+∞ = K(µfp∞), and Γ+ = (Γ′+)p
t).
Proposition 8.15. In the notation of (8.14), for all n 0 we have
(151) ord
(∏ε∈Sn
g(ε)Sp(ε)
)= ordp
(hRp
w√
∆p
(L+n )/
hRp
w√
∆p
(L+n−1)
).
66
Proof. This follows from the same argument as in [14, Chapter III.2.11], which is a standardcalculation using the analytic class number formula.
8.4. Relating to Kubota-Leopoldt p-adic L-functions. We will show that the total µ-invariantof X+ is 0 by showing it is equal to the total µ-invariant of a product of Kubota-Leopoldt p-adicL-functions, which has µ-invariant 0 by the fundamental result of Ferrero-Washington (see also[50]).
The following Proposition relates the quantities Sp(ε) to a special value of a product of Kubota-Leopoldt p-adic L-functions. This is reminiscent of a theorem of Gross [20] on the factorization ofthe Katz p-adic L-function specialized to the cyclotomic line.
Definition 8.16. Recall ε ∈ Gal(L+∞/K). As L+
∞/Q is an abelian extension, ε has two extensions
ε+, ε+εK ∈ Gal(L+∞/Q) restricting to ε, where εK denotes the quadratic character attached to
K/Q. Without loss of generality, let ε+ denote the even extension, i.e. with ε+(−1) = 1 and letε− = ε+εK .
Proposition 8.17. We have
g(ε)Sp(ε) =Lp(ε
−1− ω, 0)
2
Lp(ε+, 1)
2,
where Lp(χ, s) denotes the Kubota-Leopoldt p-adic L-function (see [58, Chapter 5]).
Proof. We note that for n 0, wg = 1. By argument of [20, Proof of Theorem 3.1], we have
g(ε)Sp(ε) = −fpnB1,ε−g(ε+)
4
∑a∈(Z/(fpn))×
ε−1+ (a) logp(C
+(a)),
where C+(a) = (1 − ζafpn)(1 − ζ−afpn) is the cyclotomic unit in loc. cit. The right-hand side of the
above equation is, by the special value formulas recalled in loc. cit., equal toLp(ε−1
− ω,0)
2Lp(ε+,1)
2 .
8.5. Vanishing of the µ-invariant.
Corollary 8.18. Recall the notation of Definition 8.11. Then for any χ ∈ ∆′+, we have
µinv(f+χ ) = 0.
Proof. This follows from Proposition 8.17 and the vanishing of the µ-invariant of (half of) anybranch of half the Kubota-Leopoldt p-adic L-function 1
2Lp, established first by Ferrero-Washington,see for example [50].
8.6. Proof of the Elliptic Units Main Conjecture from vanishing of the cyclotomicalgebraic µ-invariant. In this section, we prove Theorem 5.3 from the result of [23] (when pis ramified and K has class number 1), using as input the vanishing of the cyclotomic algebraicµ-invariant.
Proof of Theorem 5.3. This will follow from the the results of [23], and our results on the vanishingof the algebraic total µ-invariant of X+. By Corollary 8.18, we have that X+ is a finitely generatedZp-module. Recall that N+
n is the maximal unramified pro-p abelian extension of L+n , and Y+ =
lim←−n Gal(N+n /L
+n ). The natural surjection X+ → Y+ implies that Y+ is also a finitely generated
Zp-module. Now let
H := ker(Gal(L∞/K)→ Gal(L+∞/K)) ∼= Zp ×∆′0
67
for some finite abelian group ∆′0. Since Zp[∆′0] is a finitely generated Zp-module, Y+⊗Zp Zp[∆′0] is
also a finitely generated Zp-module. Moreover, it is clear that as L+∞ is the compositum of Q(µp∞)
and a finite abelian extension, all primes are finitely decomposed in L+∞ = L+(µp∞) and all primes
of L+ = Q(µf )K above p are totally ramified in L+∞/L
+.
Now specialize to the case where K has class number 1, f(p) = 1; then L = K, ∆′0 = 1, H ∼= Zp.Hence in the notation of loc. cit,
H2(OK [1/pf ],OKp [∆′0]JGal(L+∞/K)K)(1)
is a finitely generated OK ⊗Z Zp-module, and hence as in Corollary 5.10 of loc. cit.,
H2(OK [1/pf ],ΩOKp (1))
is a finitely generated OKpJHK ∼= OKpJZpK-module by an application of Nakayama’s lemma forcompact modules over completed group rings. Now the argument of Lemma 5.11 of loc. cit. goesthrough replacing the H of loc. cit. with the above H to show that Conjecture 5.5 of loc. cit.holds. By [23, Theorem 5.7, Section 5.4], this gives the Theorem.
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