a unified perspective for darmon points

A unified perspective for Darmon points2013 CMS Winter Meeting, Ottawa

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Institut fur Experimentelle Mathematik

2Columbia University

3University of Warwick

December 7, 2013

Marc Masdeu A unified perspective for Darmon points December 7, 2013 0 / 17

Birch and Swinnerton-Dyer

Brian Birch Peter Swinnerton-Dyer


The BSD conjecture

Let F be a number field.Let E/F be an elliptic curve of conductor N = NE .Let K/F be a quadratic extension of F .

I Assume (for simplicity) that N is square-free, coprime to disc(K/F ).

Hasse-Weil L-function of the base change of E to K (<(s) >> 0)

L(E/K, s) =∏p|N

(1− ap|p|−s

)−1 ×∏p-N

(1

ap(E) = 1 + |p| −#E(Fp).

− ap|p|−s + |p|1−2s)−1.

Modularity conjecture (which we assume) =⇒I Analytic continuation of L(E/K, s) to C.I Functional equation relating s↔ 2− s.

Coarse version of BSD conjecture

ords=1 L(E/K, s) = rkZE(K).

So L(E/K, 1) = 0BSD=⇒ ∃PK ∈ E(K) of infinite order.


The main tool for BSD: Heegner points

Kurt Heegner

Exist for F totally real and K/F totally complex (CM extension).We recall the definition of Heegner points in the simplest setting:

I F = Q (and K/Q quadratic imaginary), andI Heegner hypothesis: ` | N =⇒ ` split in K.

F This ensures that ords=1 L(E/K, s) is odd (so ≥ 1).


Heegner Points (K/Q imaginary quadratic)

Attach to E a holomorphic 1-form on H = z ∈ C : =(z) > 0.

ΦE = fE(z)dz =∑n≥1

ane2πinzdz ∈ H0(Γ0(N)

Γ0(N) = (a bc d

)∈ SL2(Z) : N | c

,Ω1H).

Given τ ∈ K ∩H set Jτ =

∫ τ

∞ΦE ∈ C.

Consider the lattice ΛE =∫

γ ΦE | γ ∈ H1

(Γ0(N)\H,Z

).

I There exists an isogeny η : C/ΛE → E(C).I Set Pτ = η(Jτ ) ∈ E(C).

Fact: Pτ ∈ E(Hτ ), where Hτ/K is a class field attached to τ .Gross–Zagier: PK = TrHτ/K(Pτ ) is nontorsion iff L′(E/K, 1) 6= 0.

Why did this work?1 The Riemann surface Γ0(N)\H has an algebraic model X0(N)/Q.2 There is a morphism φ defined over Q, mapping Jac(X0(N))→ E.3 The “CM point” (τ)− (∞) gets mapped to φ((τ)− (∞)) ∈ E(Hτ ).


Darmon’s Insight

Drop hypothesis of K/F being CM.

I Simplest case: F = Q, K real quadratic.

However:

I There are no points on the modular curve X0(N) attached to K.

I In general there is no morphism φ : Jac(X0(N))→ E.

I If F not totally real, even the curve X0(N) is missing!

Nevertheless, one can construct local points in such cases. . .I . . . and hope that they are global.


Goals

In this talk I will:1 Explain what is a Darmon point.2 Sketch a general construction.

“ The fun of the subject seems to me to be in the examples.B. Gross, in a letter to B. Birch, 1982”

3 Illustrate with an fun example.


Basic notation

Let v | ∞F be an infinite place of F .I If v is real, then:

1 It may extend to two real places of K (splits), or2 It may extend to one complex place of K (ramifies).

I If v is complex, then it extends to two complex places of K (splits).

n = #v | ∞F : v splits in K.

K/F is CM ⇐⇒ n = 0.I If n = 1 we call K/F quasi-CM.

S(E,K) =v | N∞F : v not split in K

, s = #S(E,K).

Sign of functional equation for L(E/K, s) should be (−1)#S(E,K).I From now on, we assume that s is odd.

Fix a place ν ∈ S(E,K).1 If ν = p is finite we are in the non-archimedean case.2 If ν is infinite we are in the archimedean case.


Non-archimedean History

Henri Darmon Matthew Greenberg Mak Trifkovic


Non-archimedean History

WarningThese constructions are also known as Stark-Heegner points.

H. Darmon (1999): F = Q, quasi-CM, s = 1.I Darmon-Green (2001): special cases, used Riemann products.I Darmon-Pollack (2002): same cases, overconvergent methods.I Guitart-M. (2012): all cases, overconvergent methods.

M. Trifkovic (2006): F imag. quadratic ( =⇒ quasi-CM)), s = 1.I Trifkovic (2006): F euclidean, E of prime conductor.I Guitart-M. (2013): F arbitrary, E arbitrary.

M. Greenberg (2008): F totally real, arbitrary ramification, s ≥ 1.I Guitart-M. (2013): F = Q, quasi-CM case, s ≥ 1.


Archimedean History

Henri DarmonJerome Gartner


Archimedean History

WarningInitially called Almost Totally Real (ATR) points.But this name only makes sense in the original setting of Darmon.

H. Darmon (2000): F totally real, s = 1.I Darmon-Logan (2007): F quadratic norm-euclidean, NE trivial.I Guitart-M. (2011): F quadratic and arbitrary, NE trivial.I Guitart-M. (2012): F quadratic and arbitrary, NE arbitrary.

J. Gartner (2010): F totally real, s ≥ 1.I ?


Our construction

Xavier Guitart M. Mehmet H.Sengun

Available for arbitrary base number fields F (mixed signature).Comes in both archimedean and non-archimedean flavors.All of the previous constructions become particular cases.We can provide genuinely new numerical evidence.


Path integrals: archimedean setting

H = (P1(C) \ P1(R))+ has a complex-analytic structure.SL2(R) acts on Hν through fractional linear transformations:(

a bc d

)· z =

az + b

cz + d, z ∈ H.

Consider a holomorphic 1-form ω ∈ Ω1H.

Given two points P and Q in H, define:∫ Q

Pω = usual path integral.

The resulting pairingDiv0H× Ω1

H → Cis compatible with the action of SL2(R) on H:∫ γQ

γPω =

∫ Q

Pγ∗ω.


Path integrals: non-archimedean setting

Hp = P1(Kp) \ P1(Fp) has a rigid-analytic structure.SL2(Fp) acts on Hp through fractional linear transformations:(

a bc d

)· z =

az + b

cz + d, z ∈ Hp.

Consider a rigid-analytic 1-form ω ∈ Ω1Hp

.Given two points P and Q in Hp(Kp), define:∫ Q

Pω = Coleman integral.

Again, the resulting pairing:

Div0Hp × Ω1Hp→ Kp,

is compatible with the action of SL2(Fp) on Hp:∫ γQ

γPω =

∫ Q

Pγ∗ω.


The group Γ

Let B/F = quaternion algebra with Ram(B) = S(E,K) \ ν.I B = M2(F ) (split case) ⇐⇒ s = 1.I Otherwise, we are in the quaternionic case.

E and K determine a certain ν-arithmetic subgroup Γ ⊂ SL2(Fv):I Let m =

∏l|N, split in K l.

I Let R be an Eichler order of level m inside B.I Fix an embedding ιν : R →M2(ZF,ν).

Γ = ιν(R[1/ν]×1

)⊂ SL2(Fν).


(Co)-homology classes

We attach to E a unique cohomology class

ΦE ∈ Hn(Γ,Ω1

Hν).

We attach to each embedding ψ : K → B a homology class

Θψ ∈ Hn

(Γ,Div0Hν

).

I Well defined up to the image of Hn+1(Γ,Z)δ→ Hn(Γ,Div0Hν).

Cap-product and integration on the coefficients yield an element:

Jψ = 〈Θψ,ΦE〉 ∈ Kν .

Jψ is well-defined up to a lattice

L =〈δ(θ),ΦE〉 : θ ∈ Hn+1(Γ,Z)

.


Conjectures

Jψ = 〈Θψ,ΦE〉 ∈ Kν/L.

Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)There is an isogeny β : Kν/L→ E(Kν).

Proven in some non-arch. cases (Greenberg, Rotger–Longo–Vigni).Completely open in the archimedean case.

Darmon point attached to (E,ψ : K → B)

Pψ = β(Jψ) ∈ E(Kν).

Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)1 The local point Pψ is global, and belongs to E(Kab).2 Pψ is nontorsion if and only if L′(E/K, 1) 6= 0.

Predicts also the exact number field over which Pψ is defined.Includes a Shimura reciprocity law like that of Heegner points.


Examples

Please show themthe example !


Archimedean cubic Darmon point (I)

Let F = Q(r) with r3 − r2 + 1 = 0.F has discriminant −23, and is of signature (1, 1).Consider the elliptic curve E/F given by the equation:

E/F : y2 + (r − 1)xy +(r2 − r

)y = x3 +

(−r2 − 1

)x2 + r2x.

E has prime conductor NE =(r2 + 4

)of norm 89.

K = F (w), with w2 + (r + 1)w + 2r2 − 3r + 3 = 0.I K has class number 1, thus we expect the point to be defined over K.I The computer tells us that rkZE(K) = 1

S(E,K) = σ, where σ : F → R is the real embedding of F .I Therefore the quaternion algebra B is just M2(F ).

The arithmetic group to consider is

Γ = Γ0(NE) ⊂ SL2(OF ).

Γ acts naturally on the symmetric space H

Hyperbolic 3-space

×H3:

H×H3 = (z, x, y) : z ∈ H, x ∈ C, y ∈ R>0.


Archimedean cubic Darmon point (II)

E ; ωE , an automorphic form with Fourier-Bessel expansion:

ωE(z, x, y) =∑

α∈δ−1OFα0>0

a(δα)(E)e−2πi(α0z+α1x+α2x)yH (α1y) ·(−dx∧dzdy∧dzdx∧dz

)

H(t) =

(− i

2eiθK1(4πρ),K0(4πρ),

i

2e−iθK1(4πρ)

)t = ρeiθ.

I K0 and K1 are hyperbolic Bessel functions of the second kind:

K0(x) =

∫ ∞0

e−x cosh(t)dt, K1(x) =

∫ ∞0

e−x cosh(t) cosh(t)dt.

ωE is a 2-form on Γ\H ×H3.The cocycle ΦE is defined as (γ ∈ Γ):

ΦE(γ) =

∫ γ·O

OωE(z, x, y) ∈ Ω1

H with O = (0, 1) ∈ H3.


Archimedean cubic Darmon point (III)

Consider the embedding ψ : K →M2(F ) given by:

w 7→(−2r2 + 3r r − 3

r2 + 4 2r2 − 4r − 1

)Let γψ = ψ(u), where u is a fundamental norm-one unit of OK .γψ fixes τψ = −0.7181328459824 + 0.55312763561813i ∈ H.

I Construct Θ′ψ = [γψ ⊗ τψ] ∈ H1(Γ,DivH).Θ′ψ is equivalent to a cycle

∑γi ⊗ (si − ri) taking values in Div0

H.

Jψ =∑i

∫ si

ri

ΦE(γi) =∑i

∫ γi·O

O

∫ si

ri

ωE(z, x, y).

We obtain, summing over all ideals (α) of norm up to 400, 000:

Jψ = 0.0005281284234 + 0.0013607546066i; Pψ ∈ E(C).

Numerically (up to 32 decimal digits) we obtain:

Pψ?= −10×

(r − 1, w − r2 + 2r

)∈ E(K).


Thank you !Bibliography, code and slides at:http://www.math.columbia.edu/∼masdeu/


What to do next?

1 Available code2 Non-archimedean example3 Details for Cohomology4 Details for Homology5 Bibliography


Available Code

SAGE code for non-archimedean Darmon points when n = 1.

https://github.com/mmasdeu/darmonpoints

I Compute with “quaternionic modular symbols”.F Need presentation for units of orders in B (J. Voight, A. Page).

I Overconvergent method for arbitrary B.I Obtain a method to find algebraic points.

SAGE code for archimedean Darmon points (in restricted cases).

https://github.com/mmasdeu/atrpoints

I Only for the split (B = M2(F )) cases, and:1 F real quadratic, and K/F ATR (Hilbert modular forms)2 F cubic (1, 1), and K/F totally complex (cubic automorphic forms).


https://github.com/mmasdeu/darmonpoints

https://github.com/mmasdeu/atrpoints

Non-archimedean cubic example (I)

F = Q(r), with r3 − r2 − r + 2 = 0 and discriminant −59.Consider the elliptic curve E/F given by the equation:

E/F : y2 + (−r − 1)xy + (−r − 1) y = x3 − rx2 + (−r − 1)x.

E has conductor NE =(r2 + 2

)= p17q2, where

p17 =(−r2 + 2r + 1

), q2 = (r) .

Consider K = F (α), where α =√−3r2 + 9r − 6.

The quaternion algebra B/F has discriminant D = q2:

B = F 〈i, j, k〉, i2 = −1, j2 = r, ij = −ji = k.


Non-archimedean cubic example (II)

The maximal order of K is generated by wK , a root of the polynomial

x2 + (r + 1)x+7r2 − r + 10

16.

One can embed OK in the Eichler order of level p17 by:

wK 7→ (−r2 + r)i+ (−r + 2)j + rk.

We obtain γψ = 6r2−72 + 2r+3

2 i+ 2r2+3r2 j + 5r2−7

2 k and:

τψ = (12g+8)+(7g+13)17+(12g+10)172+(2g+9)173+(4g+2)174+· · ·

After integrating we obtain:

Jψ = 16+9·17+15·172+16·173+12·174+2·175+· · ·+5·1720+O(1721),

which corresponds to:

Pψ = −3

2× 72×

(r − 1,

α+ r2 + r

2

)∈ E(K).


Cohomology

Recall that S(E,K) and ν determine:

Γ = ιν(R[1/ν]×1

)⊂ SL2(Fν).

I e.g. S(E,K) = p and ν = p give Γ = SL2

(OF [ 1p ]

).

Choose “signs at infinity” s1, . . . , sn ∈ ±1.

Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.)There exists a unique (up to sign) class

ΦE ∈ Hn(Γ,Ω1

Hν)

such that:

1 TlΦE = alΦE for all l - N.2 UqΦE = aqΦE for all q | N.3 WσiΦE = siΦE for all embeddings σi : F → R which split in K.


Homology

Let ψ : O → R be an embedding of an order O of K.I Which is optimal: ψ(O) = R ∩ ψ(K).

Consider the group O×1 = u ∈ O× : NmK/F (u) = 1.I rank(O×1 ) = rank(O×)− rank(OF ) = n.

Let u1, . . . , un ∈ O×1 be a basis for the non-torsion units.I ; ∆ψ = ψ(u1) · · ·ψ(un) ∈ Hn(Γ,Z).

K acts on Hv through ιν ψ.I Let τψ be the (unique) fixed point of K× on Hv.

Have the exact sequence

Hn+1(Γ,Z)δ // Hn(Γ,Div0Hν) // Hn(Γ,DivHν)

deg // Hn(Γ,Z)

Θψ ? // [∆ψ ⊗ τψ] // [∆ψ]

Fact: [∆ψ] is torsion.I Can pull back a multiple of [∆ψ ⊗ τψ] to Θψ ∈ Hn(Γ,Div0Hν).I Well defined up to δ(Hn+1(Γ,Z)).


Bibliography

H. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves.Int. Math. Res. Not. (2003), no. 40, 2153–2180.

H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: Algorithms and evidence.Exp. Math., 11, No. 1, 37-55, 2002.

H. Darmon and R. Pollack. Efficient calculation of Stark-Heegner points via overconvergent modular symbols.Israel J. Math., 153:319–354, 2006.

J. Gartner. Darmon points and quaternionic Shimura varieties.Canad. J. Math. 64 (2012), no. 6.

X. Guitart and M. Masdeu. Elementary matrix Decomposition and the computation of Darmon points with higher conductor.Math. Comp. (arXiv.org, 1209.4614), 2013.

X. Guitart and M. Masdeu. Computation of ATR Darmon points on non-geometrically modular elliptic curves.Exp. Math., 2012.

X. Guitart and M. Masdeu. Computation of quaternionic p-adic Darmon points.(arXiv.org, 1307.2556), 2013.

M. Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties.Duke Math. J., 147(3):541–575, 2009.

D. Pollack and R. Pollack. A construction of rigid analytic cohomology classes for congruence subgroups of SL3(Z).Canad. J. Math., 61(3):674–690, 2009.

M. Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields.Duke Math. J., 135, No. 3, 415-453, 2006.


a unified perspective for darmon points

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