Proc. Nad. Acad. Sci. USAVol. 82, pp. 2212-2216, April 1985Mathematics

Pade approximations and diophantine geometryD. V. CHUDNOVSKY AND G. V. CHUDNOVSKYDepartment of Mathematics, Columbia University, New York, NY 10027

Communicated by Herbert Robbins, November 2, 1984

ABSTRACT Using methods of Pade approximations weprove a converse to Eisenstein's theorem on the boundednessof denominators of coefficients in the expansion of an algebra-ic function, for classes of functions, parametrized by mero-morphic functions. This result is applied to the Tate conjectureon the effective description of isogenies for elliptic curves.

meromorphic functions of finite order of growth -p. Thismeans that there exist entire functions H(u), Hj(u), ...,HJ(R) of 17 in C9 such that Uj(7) = Hj(77)/H(g) (i = 1, ..., n)and

maxIIH(u9)1, IH,(Ii)l, ..., IHn(u9)11 <c explaTPI [1.2]

We use a Pade approximation technique to prove the g-di-mensional criterion of algebraicity of a function converse toEisenstein's theorem: a function f(X), x = (xl, ..., Xg), havingalgebraic coefficients of the expansion at x = 0 from an alge-braic number field K with "controllable" denominators, is analgebraic function over K(x), provided that f(T) and x aresimultaneously uniformized near x = 0 by meromorphicfunctions in C9 of finite order of growth. We apply our re-sults to Tate conjectures (1, 2) on the bijectivity of a mapHom(A, B) 0 Z1 -- Hom(T,(A), T1(B)) and on the finitenessof isomorphism classes of abelian varieties B isogenous to afixed abelian variety A defined over K and having a polariza-tion of a given degree. For elliptic curves, the conjecture onthe finiteness of classes of isogeny is a corollary of Siegel'stheorem on integral points (3, 4). The first Tate conjecturefor elliptic curves was proved by Serre (3) for all cases butthe one in which A and B have integral invariants and nocomplex multiplication. Faltings (5) recently proved (ineffec-tively) the finiteness of the isogeny classes for arbitrary abe-lian varieties, thus providing solutions to the Tate, Shafare-vich, and Mordell conjectures. We show how our generalresult yields a simple proof of isogeny of two elliptic curvesdefined over Q. having identical traces of Frobenious opera-tors for a bounded number of primes. Here we use Honda'stheorem (6). The technique from ref. 7 allows us to deter-mine effectively all elliptic curves K-isogenous to a givenone. Applications of our methods to Tate conjectures for ar-bitrary abelian varieties and to the Grothendieck conjectureon linear differential equations will be reported elsewhere(8).Section 1. The Main Result

Here we prove the criterion of algebraicity and algebraic in-dependence of functions, uniformized by meromorphic func-tions in C9, and having "bounded" denominators of algebra-ic numbers occurring as coefficients of their power seriesexpansions at x = D. We consider n functions fi(X), ..., MT)analytic in x = (xi, ..., x,) at x = 0. We assume that thereexist n meromorphic functions U1(9), ..., U(ui) of g varia-bles 1 = (ui, ..., ug) in C9 such that there exists a change ofvariables x 2± 17 with the following properties. First, 0 a goand the Jacobian D(M)/D(X) of the transformation is nonsin-gular at;x =X and all elements of the matrix (aui/dxj)Fj=1 at x= 0 are bounded by co. Second, we assume that in the neigh-borhood of 1 = WO we have the uniformization

fj(X) = Uj((7) and fj(U) = Uj((uo) [1.1]

for all 'a in the polydisk D7 = {1 E Cg:Jui - ui,01 s T, i = 1,..., g} and 17o = (ulO. ..., ugO). Finally, the analyticity ofUj(17) at 17 = WO is expressed as

[1.3]

These assumptions on f1(X), ..., fjGx) with n 2 g + 1 willbe invoked everywhere in this section. We put m = (ml, ....mg) for nonnegative integers mi 2 0 (i = 1, ..., g) and wedenote |m| = ml + ... +mg,m! m1! ...mg ! m =x1 '...xg9 and am= (d/dxl)ml ... (a/axg)mgLEMMA 1.1. Under the assumptions above, let P(zl,

Zn) be a polynomial in zl, ..., Zn oftotal degree at most D andof height at most H, such that the function R(X) Le, p(fi(X),. fn(X)) has a zero at X = Xo of order at least M:R(X) = Em

am X, am = O for Iml < M. Then for Iml = M, we have

lam| = lamR(x)l=Ujl s YD (cog)mHCjjnexp{aM} (M/D)-M/PM!/F(M/g + l)g [1.4]

Here M . D > g. deProof: In the notation above, we put F(1) =d H(17)D

P(U(u), ..., Un(17)). Since fj(T) = Uj(u) = Hj( )/H(17) (j = 1, ..., n) near 17 = 7!, F(17) is an entire function andRM = F(7) near x = 0 (or near 77 = 77o). Thus, according tothe assumptions of Lemma 1.1, F(u) has a zero of order atleast M at 1 = 7o. We can now apply the Cauchy integralformula in the polydisk DT to an entire function F(19):

amF(77) JOD1T F(J6)d{+l [1.5]

m! -a=o (2fi)g (dDT - aO)M+1where aODT is a hull of the polydisk DT. It follows from Eq.1.5 that

JamlF(V)6=aO/m!j -c CQ+n*H-expjaDTP} T-m9 [1.6]

where Iml = M. We can now express 9mRG?)ji for Iml = Min terms of aX F(RM)l 1O, using the fact that a F(17)l = 0,whenever Inl < M. To do this, we use the formula for partialdifferentiation, ad5.p = IJ- 1MijJa/p, where M(u) = (MQj=1is the matrix inverse of the Jacobian matrix J(A) = (axA(M)/aui)fj=1. By our assumptions, absolute values of all elementsof M(Qo) are bounded by co. Thus, by applying the differenti-ation rule M times and taking into account the fact that F(M)has a zero at 77 = 'go of order at least M, we obtain

la R(F)Lj=7l . jH(7o)j-D(cog)Mfor] = 1, ..., n. Third, we assume that Uj(u), ..., UI(U) are max{llo F(-a)16=l 0:Inl = M}.

We put now T = (M/D)11" in Eq. 1.6 for M > D. Then the

combination of Eqs. 1.6 and 1.7 implies Eq. 1.4 and the lem-

ma is proved.

2212

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"

in accordance with 18 U.S.C. §1734 solely to indicate this fact.

[1.7]

IH(lio) >- y, 0 < y --- 1.

Dow

nloa

ded

by g

uest

on

June

10,

202

0

Proc. NatL. Acad. Sci USA 82 (1985) 2213

For an algebraic number field K of degree d = [K:Q] weconsider d imbedding a |-+ a(of) ofK into C: o' = 1, ..., d. Thenthe size [-&I of an algebraic number a E K is defined as J=max{Ia(0'l:o = 1, ..., d}. For a E K we denote by d(a) thedenominator of a-i.e., a rational integer d(a) . 1 such thatad(a) is an algebraic integer. Also we denote by den{ao,aN} the common denominator of ao, ..., aN from K.THEOREM 1.1. Let a system of functions f1(X), ..., fU(x)

satisfy all the assumptions above with n .-g + 1. Denotefi(x) = Em a(').Xm (i = 1, ..., n) and f1(X) *- fM(x)Emam;k...,k, Xm for non-negative integers k1, ..., k,. Assumethat am E K (i = 1,..., n) and denote by DM the commondenominator of{am;kj,...,kn : ml . M; k1 + ... + kn < M}. Leta = lim supM gC max{ja(m')j` :|ml S M; i = 1, ..., n} and =lim supM g. logjDMj/M log M. 1. If o- < oo and 6 < [1 -g/n]/(dp), then the functions f1(x), ..., fn(X) are algebraicallydependent over K. 2. Denote ff)(x) = Xm(a4) (U)Xmfor a =1, ..., d. Assume that the system offunctions f~< (x), ...,fn(x) satisfies the uniformization assumptions above for any cr= 1, ..., d. Then conditions o- < o and 6 < [1 - g/n]/p implythat the functions fi(x), ..., fn(X) are algebraically dependentover K. 3. Assume that the functions fi(X), .., fn(X) satisfy asystem of g independent algebraic differential equations ina/axi, ..., a/axg.. Then for any Mo there exists an effectiveconstant M1 depending only on this system of differentialequations, on a, co, a, y, K, and Mo such thatthe assumption in case 1 on 6 can be weakened tosupM0<MC.M1 log DM/(M log M) < [1 - g/n]/(dp) and theassumption in case 2 on ( can be weakened to supM(<M5M1log DM/(M log M) < [1 - g/n]/p. Moreover, the statementsof I and 2 can be improved to "the functions f1(X), ..., fn(X.)are related by an algebraic relation over K of total degree <W1.sProof: Let us assume that a- < w and that for some 8> 0, 6

< 11 - g/(n - 8)]/(dp) in case 1 and 6 < [1 - g/(n - 8)]/p incase 2. Thus, we may assume that log|DMj < [1 - g/(n - 8)]x M log M/(dp) in case 1 and logIDMj < [1 - g/(n - 3)]-Mlog M/p in case 2 for any sufficiently large M (or for 1 <MMM1 in case 3).LEMMA 1.2(4). Let M and N be integers, N > M > 0, and

let uj j (i = 1, ..., M; j = 1, ..., N) be algebraic integers in Kwith sizes at most U (21). Then there exist algebraic inte-gers xj, ..., XN in K, not all zero, satisfying I ljuj xj = 0: i= 1, ..., M and such that Ixjx S c1 (ciNU)M/(NM): j = 1,M. Here cl = cl(K) > 0.

In the proof below D denotes an integer parameter satisfy-ing a finite set of inequalities D 2 Di for effective constantsDi = Di(K, 8, g, n, o-, ...).LEMMA 1.3. Let 1/4 > 8> 0 and n -g > E > 0. Then for

any D - Do(K, 8, g, n, a-, E) there exists a nonzero polynomi-al P(x1, ..., xn) E K[x1, ..., xn] with integral coefficients oftotal degree in x, ..., xn at most D and with the followingproperties. The sizes of the coefficients of P(x1,..., xn) arebounded by DclD' , where c1 = c1(K, n, g, 8, E) > 0. Thefunction R(x1, ..., xg) =fP(f1(x1 .., Xg), .., fn(x1, ..., Xg)) hasa zero of order at least U(n-E)Ig at x = 0.Proof of Lemma 1.3: Let p(x, . x,) = Y-k,20 ...

Ik,20,kl+... +k,,D Pk, ...k"@XlXn , where Pkl... ,kn (k1 + .... +kn : D) are undetermined integers from K. Then, in thenotation of Theorem 1.1 we have the following expansionof R(x) at x = O:R(x) = Xmx {fk,20 ...* -kn 0,k+...+k -D

Pkl,.k.,k tam;kj,....kn} Hence, the system of linear equationson Pk,....k, equivalent to ordx=: R(x) 2 S has the formkO>1...I D Sm;kj,...,knPkj,...,kn = 0 [1.8]k.-O k,,.Ok,k+..+k-,,-D

for all m = (m1, ...,mg) with Iml < S, when S 2D. In Eq. 1.8all coefficients at Pk,.k are algebraic integers from K ofsizes bounded by S[l-g/(n-8)S/P.C5, according to the as-

sumptions of Theorem 1.1. The system of Eqs. 1.8 hasCS-i+g equations on (D + 1)' unknowns Pk,,...,k, (k1 + ... +k, - D). We can put S = [D(ne-)/g]. Then for sufficientlylarge D, we apply Lemma 1.2 and deduce the existence of asystem of algebraic integers Pkh.. k from K, not all zero, sat-isfying all Eqs. 1.8 and such that

max{[pk,,...,kn1:kl + ... + kn < D} s DC3DW"

Let us assume that R(X) # 0, where R(x) is defined as inLemma 1.3. Let M = ordy=:R(X) < a. If we put R(X) =Xmcmxm, then, by the definition of M, cm = 0, whenever Iml< M. Also there exists wno such that Imol = M and c Le C#o70. Following the formula given in the proof of Lemma 1.3,we have the following representation of the number c interms of coefficients of the expansions of fl(X)k1 ... fn(x)knand the coefficients of the polynomial p(xl, ..., xn:

C = Kk..D Pkl,.k, amo;kl.., [1.9]k1.O k,.O,kj+w...+k,-_D

Obviously, the denominator of c E K divides DM becauseM 2 D(n-E)Ig Also, c C'+, -H- cm for C4 = C4(a, g, n, E)> 0, where H bounds the sizes of the coefficients of thepolynomials P(x1, ..., Xn):

H -< Dc'D'n-E)1(8- f)

Consequently, we have

d(c) '-Dm,|c |< 'C4.D c5D(R- F)1(9-f .

To bound Icl from above, we use the estimates of Lemma1.1 and the representation of c:c = a' R(x)/mo!Io=;. Weremark that M 2 D(n-E)Ir, according to Lemma 1.3, and that(n - E)/g > 1, by the choice of E. Thus, for a sufficientlylarge D, D - D1(K, 8, g, n, E, a, y, co), the estimate 1.4implies

[1.10]

[1.111

ICI <- Cm6 (M/D)-M/P-DciD'n19-)

[1.12]

where we use the bound 1.10 on H.In case 2, together with the function R(X), we consider

functions R(')(X) ef p() (f) .. f (n)(X)), where P(v) (xl,Xn) is obtained from P(x1, .., xn) by application of the

isomorphism a- to all of its coefficients, C- = 1, ..., d. Thenfrom the definitions of functions f fWx), .., f)(x), it fol-lows that {d8R(3F)6=:}(0f) - d In particular, c(= R°R(Of)(jxl=:/mO!. Thus, under the assumptions of case2 on the uniformization of f (d°)(o, ..., f(0), we deducefrom Lemma 1.1 an inequality similar to 1.12:

[1.13]

for a- = 1, ..., d (true only under the assumptions of case 2).We use product formula d(c)d. IId=c(= |C ito c E K, c 7 0.In case 1, combining inequalities 1.11 and 1.12 together withthe bound on DM, we obtain M[1 -g(n- 4lM/p c 8 *(MID) -mP.DC9D 2 1. Since M 2D.

M[1-g1(n-5)]/p.C8 MCIoM 2: 01gl (n-E)]lp. [1.14]

For E < 8 and sufficiently largeM (or D), inequality 1.14 isimpossible. This shows that R(T) 0 and that the functionsf0AU f, fn(Y) are related over K by an algebraic relation oftotal degree at most D < M. Let us now consider case 2.We apply product formula using inequality 1.13 and thebound DM < M[l-g/(n-8)]MIP on d(c) C DM. We obtainMdrl-g/(n-6)]M/p c 7 (MID) -Mdp.D cdD( "( 1. This im-

Mathematics: Chudnovsky and Chudnovsky

Ic'o"I -< cm-(MID)-MIP-D ciD(n- ")1(9-

7 1)

Dow

nloa

ded

by g

uest

on

June

10,

202

0

2214 Mathematics: Chudnovsky and Chudnovsky

plies the bound 1.14, as above, with perhaps, different con-stants c8 and c10. Again for E < 8 and M sufficiently largewith respect to g, n, 8, E, p, C8, and c10, this implies R(T) 0.Thus Theorem 1.1 is established in cases 1 and 2. We ob-serve that, under the assumptions of case 3, any function ofthe form R(G) = P(f1(X), ..., f!(x)) with deg(P) c D, which isnot identically zero can have a zero at x = 0 of order at mostDC11. Here c1l is a certain effective constant depending onlyon the system of g algebraic differential equations satisfiedby fi(x), ..., fjl) (and on the maximal order of zeroes offMlx) at x = 0); see ref. 9. This result bounds M in terms ofDand c1l. Consequently, in both cases 1 and 2, under the as-sumptions of case 3, we need bounds on DM only for MO < Mc M1 for effective constants MO and M1.

Section 2. Formal Groups of Elliptic Curves Over Q

In this section, we give a simple solution to Tate's problemon the isogeny of two elliptic curves defined over Q thathave isomorphic Tate modules (see chapter 4 of ref. 3). Wewill use mainly Theorem 1.1 for g = 1 and Honda's result onformal group laws of elliptic curves defined over Q.We start with an arbitrary plane cubic model of an elliotic

curve E defined by the equation: y2 + alxy + a3y = x +a2x2 + a4x + a6. The Weierstrass elliptic function parametri-zation of this model ofE is given by the formulas 0P(u) = x +(a4 + 4a2)/12, 9P'(u) = 2y + aix + a3; ='(u)2- 49P(u)3 -g2'P(U) - g3, see ref. 10; so we have two meromorphic func-tions x = XE(U), y = YE(U) of u. An invariant differential w onE has the form co = dx/(2y + aix + a3) = d@P(u)/@P'(u) = du.Tate (10) has chosen the local parameter z = -x/y. Then theexpansion of the invariant differential c at z = 0 has the form(10) co- dzvf(z), f(z) = Ym=, bmzm-l, b1 = 1, where bmbelong to Z[a1, a2, a3, a4, a61:m = 1, 2, .... Thus du/dz = f(z)and we have the following expansion of the "elliptic loga-rithm" u:

def Iu(= L(z)) = j f(z)dz = > bmzm/m. [2.1]

m=l

Let E be defined over Q-i.e., ai E Z for i = 1, 2, 3, 4, 6.We assume, without loss of generality, that the E is in itsNeron's minimal model. We denote by cp for every p thetrace of the Frobenious isomorphism of E(p) = E(mod p).The L-function of E/Q has the form (10) L(E; s) =cmm-S, c1 = 1. To the L-function L(E; s) corresponds a for-mal logarithm 1(z) = 1m=1 cmzm/m. Two power series L(z)and P(z) define formal group laws G(x, y) = -1(l(x) + 1(y)),F(x, y) = L-'(L(x) + L(y)) in the neighborhood ofx = y = 0.Our main auxiliary result is Honda's (6).THEOREM 2.1. The twoformal group laws G(x, y) and F(x,

y) are strictly isomorphic over Z. This means that there ex-ists a power series hE(z) E Z[[z]], hE(z) = z + O(Z2) suchthat L-1(l(z)) hE(z).COROLLARY 2.2. If two curves E1/Q and E2/Q have the

same L-functions (or, equivalently, if they have the samenumber of solutions mod p for all p), then E1/Q and E2/Qare isogenous over Q.

Proof: Consider the two elliptic logarithms LE (z) andLE2(z), corresponding to E1 and E2 according to tq. 2.1.From Honda's theorem 2.1 and the assumptions of Corollary2.2, there exists a convergent power series f(z) E Z[[z]], f(z)= z + O(Z2) such that f(z) LIE(LE2(Z)). We can nowapply Theorem 1.1 with g = 1, n = 2, f1(x) = x, and f2(x) =f(x). We define

Zi(U) = -XE;(U)/YEi(U) [2.21

for the elliptic functions xE,(u), yE,(u) associated with Ej: i =1, 2. Then by Eq. 2.1, LE; (zi(u)) =U: i = 1, 2. Thus we obtain

the following uniformization of (x, f(x)): f(x) = zl(u), x =Z2(u), where z1(u), z2(u) are meromorphic functions of orderof growth two-i.e., p = 2 in Theorem 1.1. We also have or<oo and f = 0, since f(x) is an integral power series. ThusTheorem 1.1 implies that f(x) is an algebraic function. Inview of the parametrization and the representation 2.2, theWeierstrass elliptic functions 9P1(u) and 9P2(u) are algebra-ically dependent over Q. The lattices Yj = 2wiZ E 2 wiZ ofE,: i = 1, 2 are related by a rational transformation. Thistransformation induces the isogeny between E1 and E2.Theorem 1.1 (case 3) gives us bounds on the degree of the

isogeny between E1 and E2 and an effective bound on P,such that E1 and E2 are isogenous when they have the samenumber of solutions (mod p) for all p c P. The bound on Pwas established in ref. 11 assuming the generalized Riemannhypothesis. For an elliptic curve E/Q with the Weierstrassequation y2 = 4X3 - g2X - g3 (g2, 93 E Z) and the periodlattice £T we denote by A(E) the area of the fundamental par-allelogram of £ and H(E) = max(1g21"4, 1g31116), so that H(E)

yiA(E)-2 max(1, log H(E))"2.PROPOSITION 2.3. Let E1/Q and E2/Q be elliptic curves

defined over Q. Then for every E > 0, there exist an effectiveconstant ce > 0 depending only on E such that the followingconditions are satisfied. If E1 and E2 have the same numberof rational points (mod p) for any p ' c~max{l, A(El)-lA(E2)f1}l+, then E1 and E2 are isogenous over Q and,moreover, the degree of isogeny between E1 and E2 isbounded by cemax{1, A(Ej)-l1A(E2)'}1l+_.

Section 3. The Uniformization of Formal Groups

Let A be a (finitely generated over Z) subring of an algebraicnumber field K. A g-dimensional commutative formal grouplaw (12) over ring A is a g-tuple of power series F(x, y) =(F(1)(x, 7), ..., F(g)(x, y)) in 2g variables x = (xj, ..., xg); y =(y1, ..., yg) such that F(i)(x, y) xi + yi mod(deg 2); F(i)(F(x,37), ) F(i)(x, F( y, z)) and F(i)(x, y) F(i)(3y, x), i = 1, ..., g. One defines the homomorphisms [nIF(Q):F(X, Y) -* F(X, Y) as follows: [MAF(X) = X, [n]F(X)- F(X, [n - 1]F(X))g if n - 2; [OIF(X) = 0. [nIF(X) =i([-nIFGx)), if n < 0; and i(x) is the "minus" operators: FQx,i(x)) 0. One defines endomorphisms [m/n]F(X) over Z[1/n]09 A, looking at the inverse of [n]F(X).

In cases arising from abelian varieties, the formal grouplaw FGx, y) is an algebraic function in x, y. This allows us toevaluate the sizes and the denominators of expressions suchas amJ R(z (i1), ..., zg(u), z1(a77), ..., zg(ala)) for polynomialsRGx-, y). The corresponding device was presented in ref. 7.The assumption that the group law is defined algebraically isas follows.AssuMPTION 3.1. Assume that the formal group law F(X,

3Y) is an algebraic function over K(X, Y). This means thatthere exists an algebraic equation defining x0 as a functionof X:q(xo, X) = 0 and similarly defining yo as a function ofy:q(y0, y) = 0, such that (near x = 0, y = 0)

F(X, y) = P(k, y)/Q(x, y) [3.1]

forx = (xo, X) (=(xo, x1, ..., xg)) and y = (Yo, Y) (=(Yo, yi, ** *y)). Here P(R, y) = (P(1)(x, y), ..., P(g)(RI y)) and P(i)(x, y) (i- 1, ..., g), Q(x, y) are polynomials in 2(g + 1) variables x, yfrom KLx, y] with integral coefficients. For normalization,assume that the Taylor expansion ofthe branch xO = xO(X) ofthe root xO ofq(xo, X) = 0 with the initial condition xO(U) = 0in powers ofX (near X = 0) has integral coefficients from Kand that Q(O, 0) = 1.ASSUMPTION 3.2. Assume that there is a meromorphic uni-

formization of the group law 3.1 by functions of the variableU = (u1, ..., ug) in C9. This means that there are g + 1 func-

Proc. NatL Acad Sci. USA 82 (1985)

Dow

nloa

ded

by g

uest

on

June

10,

202

0

Proc. NatL Acad. Sci USA 82 (1985) 2215

tions zo(U) and z(U) = (z1(u), ..., zg(U)) that are analytic at U= 0 and meromorphic in C9 such that Z(U + V) F(7(u), Z(V))for 2(U) (zo(U), Z(U)). In the notation of Eq. 3.1,

Z(u + V) = P(m(1), 2(V))/Q(2(u), 2(V)) [3.2]

and xo(Z(U)) zo(U). The normalization ofthe uniformization3.2 is the following: zi(u) = uimod(deg 2) (i = 1, ..., g) andzo(U) = 0 mod(deg 2).Under these assumptions, the formal group law 3.1 is de-

fined over the ring A of integers of K.We can now modify the arguments of lemma 1.5 of chap-

ter 7, §11 of ref. 7. We start with a polynomial R(z, y) in 2gvariables 7 = (zl, ..., zg), Y = (yl, ..., yg) of total degree Do inz and of total degree D1 in y. We also consider a function

r(u) - R(z(u9/N), z(u1)) [3.3]

for an integer N - 1. Let us assume that a point uto is a zeroof order M of the function r(u). This means that for all mwith ImI < M, a " r(R)I=u = 0. We are interested in thenumber c = an r(7)1-,=-,0/n!, where Ini = M. We have first ofall, an r(-a)Il=, = N-M*adr(un + NW)IW=O. Applying thelaw of addition 3.2 to r(u + Nw) in 3.3, we get

r(- + Nw) = Q(z(u/N) Z(W))-Do.Q(t(ig), t(NW))-D-H(z-(uN), z(w), z(u), z-(N-w)),

def QZ ,Z )HQ9.. 4) = Q(U19)DO QQ3, z4) [3.4]

*R(P(Z1, tDI)Q(U, Z-2), P(Z3, t)Q(Z3, t4*

According to the assumption on M we get

an = NMQ(z(uo/N) O)'Q(t(io), O)-D1a n H(Z-(Uo/N)' t(W), t(,o), Z-(N-W)lW=;5'

provided that neither ThO nor i90/N is a singularity (pole) ofZi(a) (i = 1, f..., g). We put xi = zi(W) (i = 1,., g) and wedenote XN = (XO([N]F(X)), [N]F(X)). According to the unifor-mization, Z(NW) = XN whenever z(W) = x. Thus we obtain

an r(9)ju=aO Nm Q((9/iN), 0)-Do. Q(z(Ro), 0)Di*dXHQ(7(O1N), xc, A190), fCN)I-X=;U

From Eisenstein's theorem and Assumption 3.1 it followsthat all numbers a'yxO(X)/m!, a',mxO([N]F(V))/m!, a' [N]F(x)/m! at x = 0 are algebraic integers from K. ThusTHEOREM 3.1. Suppose that Assumptions 3.1 and 3.2 are

valid. Let R(Z, y) be a polynomial in 2g variables 7, y oftotaldegree DO in 7 and of total degree D1 in y. Define r(u) 'ejR(Z(U/N), 7(1)) for N 2 1 and assume that for 11O (which isnot a pole of zi(U1), zi(NU); i = 0, ..., g) we have a r(u)ju=uO= 0 whenever Iml < M. Then, for any n, Ini = M, we have therepresentation an r(11)|=uO/n! = N-M*Q(W(UO/N), O6)-DQ(2(1O), O)-D-Hn(i(U1o/N), (O)). Here Hn(21, 22) is a poly-nomial ofdegree at most Dod in 21 and ofdegree at most Djdin 22. Moreover, ifR(z, y) has integral coefficientsfrom K ofsizes bounded by H, then all coefficients of the polynomialHn(21, 22) are also integral numbers from K. The sizes ofthecoefficients of Hn(21, 22) are bounded from above byH(c12)Do DI'(C12 + C13N)M. If zi(Uo/N), zj(UO) (i = 0, ..., g)are algebraic numbers from K and if A1 is the denominatorof {zi(uo/N):i = 0 .... g} and A2 is the denominator of{z(): i = O0 ..., g}, then the number NM.A1,Dod A2 Did.Q(U(1O/N ), O)D'° Q(2(uo), 0)Dl.a* r(u)l = is an algebraic in-teger from K.

Section 4. The Effective Bounds on Isogenies

We now present a self-contained proof of an effective boundon the degree of K-isogeny that an elliptic curve defined overK can have, along the lines of Theorem 1.1.THEOREM 4.1. Let an elliptic curve E be defined over a

real algebraic field K and let there exist a cyclic subgroup Ain E of order n defined over K. Then n < C(E, K), whereC(E, K) is an effective constant depending only on the dis-criminant ofK and on the algebraic equation defining E.

Proof: Let E be given in its Weierstrass form by the equa-tion y2 = 4x3 - g2X - g3 for algebraic integers a4 = -g2/4,a6 = -g3/4 in K. We consider the uniformization of E byelliptic functions, presented in Section 2, and the uniformiza-tion of the formal group law F(x, y) (for g = 1), associatedwith E in Section 2, satisfying the assumptions of Section 3.Let 9P(u) be a Weierstrass elliptic function with parametersg2, g3 and let z(u) = -29P(u)/IP'(u). As in Section 3, z(u) is auniformization of F(x, y) and zo(u) is defined by the algebraicequation zO = z3 + a4zoz + a6z4 with an expansion zo = z3 +.... For the series [l/n]F(x) = x/n + ..., describing the divi-sion by n near the origin, we have [l/nIF = Y.'=, amxm,n2m- am E Z[a4, a6] (m = 1, ...). According to Eq. 2.2,

z(u) = g(u)/h(u), [4.1]

where g(u) and h(u) are entire functions of u of order ofgrowth 2, and

maxlUI=R max{lg(u)l, lh(u)l} ' exp{f3R2}, [4.2]

where P is an effective constant.Let 0 < 8 < 1/2. We assume that n is a sufficiently large

(with respect to sizes of g2, g3, A(K), [K:Q], and 8) positiveinteger such that there exists a cyclic subgroup A of order nin E, defined over K. Then A is generated by a torsion point(z(w/n), zo(w/n)) of order n exactly for a period w. Since K isreal and g2, g3 E K, the lattice £ = 2w1Z + 2aW2Z of periods-is invariant under the complex conjugation £* = TP. Thus,either Im(w1) = 0, Re(W2) = 0, or Ct2 = Cor. Since A is invari-ant under complex conjugation, w* = aw mod(n.T) for a, (a,n) = 1, and w can be taken as 2w1, 2o2, 2(wi + D2), or 2(i, -(02). In either case w is independent of n.LEMMA 4.2. Let D be a sufficiently large integer, D 2

DO(n, K, E, 8). Then there exists a nonzero polynomial P(XO,x1) E K[xo, x1] with integral coefficients ofdegree at most Din x1 and ofdegree less than n2 in x0 with thefollowing prop-erties. The sizes of coefficients of P(xO, x1) are bounded by4Dn226dfn , The function R(x) def P([l/n]F(x), x) has a zero of

order at least Dn2-8 at x = 0.Proof of Lemma 4.2: Let P(xo, x1) = Y.>n2- o Pm0 m,

xo0 xi1 , where Pmo~ml are undetermined integers from K. Inthe notation above: {[l/n]F(x)}i = -= 1 am) xm (i = 1, 2, ...).Then we have R(x) = =xm{0I{jO kcm ksD Pm0,kam-k}. Here, by the properties of the expansion of [l/n]F(x),n2ma(s) are integers from K (m, s = 1, ...). Hence, the systemof linear equations on Pmo.ml (mO = 0, ..., n2 _ 1; m1 = 0.D), equivalent to the condition ord,=o R(x) 2 5, is

n2- 1

1 > n2. amm~ *Pm.,k = 0mo=O O~kcmin(m.D)

[4.3]

m = 0 ..., S - 1. The coefficients at Pmo,k are algebraic inte-gers from K. To estimate their sizes we note that, in the ex-pansion [l/N]F(x) = .X'=l amxm, we have laml s cm for cl .1 (effectively depending on E and K only). Thus the sizes ofthe coefficients at Pm0.k in Eq. 4.3 are bounded from aboveby c n2n for C2 . 1 (effectively depending on E and K only)and D . n2. We set S = [D n2-S]. Then the system 4.3 has

Mathematics: Chudnovsky and Chudnovksy

Dow

nloa

ded

by g

uest

on

June

10,

202

0

2216 Mathematics: Chudnovsky and Chudnovksy

at most Dn 2-8 equations on n2(D + 1) unknown pmO,,l, (mi =0, ..., n2 - 1; ml = 0, ...,D). Thus for sufficiently large D wecan apply Lemma 1.2 and obtain a system of algebraic inte-gerspOml (mi = 0, ..., n - 1, ml = 0, ..,D) from K, not allzero, such that all Eqs. 4.3 are satisfied and such that

max{lmo,Mj:mo = 0, ..., n2 - 1; ml = 0, ..., D}S (CnD2Dn2 6)Dnf2 8/(Dn2-Dn2-) c 4Dn2-26

for a sufficiently large D, D . DO(n, K, E, 8) and n ' no(K,E, 8). Hence, all conditions of Lemma 4.2 are satisfied.We define r(u)Lef P(z(u/n), z(u)), where P(xo, xl) is a poly-

nomial satisfying all conditions of Lemma 4.2 and F(u) =h(u)Dh(u/n)h -1 r(u). In the notation of the proof of Lemma4.2, F(u) = Y. ' m=o Pmmm g(u/n)m0 h(u/n)fl-mog(u)mlh(u)Dml. Consequently, IF(u)l can be estimated viaEq. 4.2 in the following way:

IF(u)l c n2(D + 1)'H x exp{f3(D + 1)R2} [4.4]

for Jul = R, where H bounds the height (maximum of theabsolute values of the coefficients) of P(xo, xl and P3 de-pends only on 2wc, 2w2 (see Eq. 4.2). Let G(u) = F(u)/uS, 5- [Dn2-8]. According to Lemma 4.2, the function G(u) is anentire function (no poles at u = 0). Thus the maximum modu-lus principle lGlr c IGIR for r c R implies

lFtm) (u)/M!l C IFIr C IFIR.(r/R)S, S = [Dn2-8], [4.5]

where Jul < r - 1, r s R. Let us choose R = n'- 2 and asmall E > 0. Then for a sufficiently large integer n, n 2 nj(p,8, E) and for D 2 n2, it follows from Eqs. 4.4 and 4.5 andLemma 4.2 that, for Jul c n1-8/2-

Proc. Natl. Acad. Sci. USA 82 (1985)

zo(aw/n))g- (z(ag aw/n), zo(ag aw/n)), where ag E {1, ....n - 1} is defined mod(n). Let us denote the number in 4.8 byca, so that Ca is an algebraic integer from L. All numbersconjugate to Ca over K have the form caag:g E G,LEMMA 4.3. The total number of zeroes (counted with

their multiplicities) of r(u) inside the fundamental parallelo-gram generated by 2nw1 and 2nw2 is at most 3(n2D + n221).

Proof: The function z(u/n) does not satisfy any algebraicequation of degree less than n2 over C(z(u)). Consequently,r(u) 0 0. The number of poles and zeroes of r(u) incongruentmod(2w1Z + 2W2Z) is bounded by 3(n2D + n2 1).To prove that n is bounded, we introduce the following

nonzero integral number from K: c lef fa:<n Ca, where *means that the product is taken over all a such that awin isnot a half-period of 9P(u). Because A is defined over K, thenumber c belongs to K and is integral. We estimate the sizeof c by estimating the sizes of numbers Ca from 4.8 accordingto the description above jThus the size of c is bounded by1t1a<n {nMa. CM,+D . n4DnWe now represent c as the product c = HtIa<n1l-8/2-el)-1

CaHfIn/-2-f1w 1-a<n Ca. The first factor is bounded, accordingto the representation 4.8 by 4.7, and the second factor isbounded by the product of the sizes of Ca. This gives-EDn2- 8}1nlI-l8t2

Xli(a<n{1a.MalM+D . p4Dn22}Icd. Since c + 0 we have lCl[K:Q]Jl1Ci 2 1. Using the bound onthe size of c and on Icl we obtain

11 MSa<n{nMd Mca+D. n 4Dn2-26 }[K:Q]n- EDn2- 8/8J[n - 8/2- ell1ol-1]_1 1. [4.9]

Now let E be sufficiently small, E < 8/4. Then for a large nand D sufficiently large with respect to n, 4.9 implies

IF(M) (u)/M!l C , -eDn- 2 [4.6]

For u =aw and an integer a, 1 s a s n, let Mat denote thelargest integer m such that r(m )l____ = 0 for all m' < m.From this definition ofMa it follows that F(M-) (u) = h(u)D xh(u/n)n -lr(M )(u) for u = aw. Thus if u = aw for an integera, 1 s a ' n, awin is not a half-period of 9P(u) and if jawln 8/2-, then Eq. 4.6 implies that, for n . n2(pI3, Iwl, 8, e) andD - Dj(n),

lr(M~) (aw))/Ma!l c ,n-eDn (2- M) [4.7]We can apply now Theorem 3.1 with g = 1, R(z; y) e P(z,

y), UO = aw, M = Ma, and N = n. Let us denote by 26n a(rational) integer such that 26nz(aw/n) and £&,zo(aw/n) are al-gebraic integers: 1 c a ' n. Consequently,

rMa!w aw f/ ~ n2d2Ma! *n a I(O) Z( ); °, °) *n [4.8]

is an algebraic integer from L of the size bounded by nmC 1a+D n4Dn for an effective constant c14 and a sufficientlylarge D, D > D2(n).We now need some trivial information on the Galois group

Gn of the extension L = K(E(A)) = K({z(aw/n), zo(aw/n): a= 1, ..., n - 1}) over K. Since A is defined over K, the actionof g E Gn on (z(aw/n), zo(aw/n)) is the following: (z(aw/n),

I Moa> ci6Dn3-38/2-E,lra<n

[4.10]

for cl depending only on c14, I-1, [K:Q], 8, and e. Howev-er a<n Maf is a lower bound for the number of zeroes ofr(u) incongruent tnod(n-T). Comparing 4.10 with the bound ofLemma 4.3 we conclude that n is bounded whenever 38/2 +E < 1. Theorem 4.1 is proved.We dedicate this paper to our father.

1. Tate, J. (1965) Proceedings of the Purdue University Confer-ence 1963 (Springer, New York), pp. 93-195.

2. Tate, J: (1966) Invent. Math. 2, 134-144.3. Serre, J.-P. (1968) Abelian 1-adic Representations and Elliptic

Curves (Benjamin, New York).4. Baker, A. (1979) Transcendental Number Theory (Cambridge

University Press, Cambridge, England).5. Faltings, G. (1983) Invent. Math. 73, 349-366.6. Honda, T. (1970) J. Math. Soc. Jpn. 22, 213-246.7. Chudnovsky, G. V. (1984) Contributions To the Theory of

Transcendental Numbers (Am. Math. Soc., Rhode Island).8. Chudnovsky, D. V. & Chudnovsky, G. V. (1984) New York

Number Theory Seminar (Springer, New York).9. Schmidt, W. M. (1977) Acta Arith. 32, 275-296.

10. Tate, J. (1974) Invent. Math. 23, 179-206.11. Serre, J.-P. (1981) Publ. Math. IHES 54, 323-401.12. Hazewinkel, U. (1978) Formal Groups and Applications (Aca-

demic, New York).

Dow

nloa

ded

by g

uest

on

June

10,

202

0