Algebraic independence of the values of elliptic function at algebraic ...

Inventionesmath. 61, 267-290 (1980)

Iylverltiorle$ mathematicae 9 by Springer-Vcrlag 1980

Algebraic Independence of the Values of Elliptic Function at Algebraic Points Elliptic Analogue of the Lindemann-Weierstrass Theorem G. Chudnovsky Department of Mathematics,ColumbiaUniversity,New York, NY 10027, USA

Introduction As it is well known, the only general statement about algebraic independence of n values of the exponential function is the Lindemann-Weierstrass theorem ILl], I-L2] (1882): if algebraic numbers ~ , . . . , ~ , are linearly independent over ~, then e'l,...,e ~" are algebraically independent over Q. Since then such general results were obtained only for E-functions, Siegel [-Si], like Jo(Z). However the methods of [Si] use the structure of E-functions as entire functions, satisfying a linear differential equation with rational coefficients and having certain special arithmetical properties of the coefficients of their Taylor expansion. On the other hand, the general Gelfond-Schneider method [G], [Sc], [L] can be applied to a wide class of functions satisfying algebraic differential equations and/or a law of addition. Nevertheless there are no general results on the algebraic independence of two or more numbers (like the LindemannWeierstrass theorem) which are obtained by Gelfond-Schneider method. Of course, in the exponential and elliptic case we have some results on the algebraic independence of numbers of the form ~J, ~3(cofl), or of some constants connected with the periods [Chl], [Ch2], [Ch3], [MW2]. This is quite natural because in the Gelfond-Schneider method the problem of algebraic independence is often reduced to an algebro-geometrical problem about the structure of singular points of intersections of hypersurfaces near fixed transcendental points. Unlike Siegel's method, it is impossible to construct independent systems of linear forms and complete the proof by showing that some determinant is not zero. Fortunately we now have a comparatively elementary method to treat the problem of algebraic independence of several numbers via the GelfondSchneider method [Ch3]. In general to treat an arbitrary number of algebraically independent elements, we must inevitably use the ideas underlying Hironaka's resolution of singularities. Dedicated to 50-th anniversaryof A.I. Vinogradov.

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In principle, these algebraic methods, combined with some technical improvements in the Gelfond-Schneider method give us the possibility to generalize the Lindemann-Weierstrass result for the elliptic functions. At present, we have not succeeded in overcoming all the technical difficulties required to do this in general. We are able to give here only the first results in this direction.

I. Formulation of the Problem We use standard notations of transcendental number theory and elliptic functions. For P(x 1 ..... xn)E~E[x 1.... ,xn] we denote by H(P), the height of P(x I ..... Xn), i.e. the maximum of the absolute value of the coefficients of P(x 1 .... ,xn), and by d(P) the total degree of P ( x l , ...,xn) in xl, ...,x n. By dx,(P ) we denote the degree of P(x I .... ,xn) in x i. The type t(P) of the polynomial P(x 1 ..... xn) is defined as log H(P) + d(P). We consider elliptic curves over an algebraic number field. We take a uniformization of an elliptic curve by the Weierstrass elliptic function ~3(z). Thus we are considering a Weierstrass elliptic function ~3(z), satisfying ~3/z =4~3 3 --g2 ~ - - g 3 , Where the invariants g2, g3 are algebraic numbers. Let co~, (02 be a pair of fundamental periods of ~(z) and L = Z c o 1 @7l co2 be the corresponding lattice of ~(z). We write throughout the paper IF __~((D1/fD2) , if 091/09 2 is imaginary quadratic: -[Q, otherwise.

(1)

As usual, if the first case in (1) is satisfied, we say that ~(z) has complex multiplication. Now we consider algebraic numbers 0~1~"", ~n

which are linearly independent over IF. The problem is the following: are the numbers ~(~1) ..... ~(~.) algebraically independent over ~ ? Of course, the assumption that al, ..., a n are linearly independent over IF cannot be in general reduced to the assumption of linear independence over Q. We shall prove below some positive results in this direction. For n < 4 we give complete results in the case of complex multiplication. Another problem arising here is to estimate the measure of algebraic independence of the numbers ~(~l),...,~(~n) (whenever they are algebraically independent). In this connection, we recall that for the exponential function e ~, the Lindemann-Weierstrass theorem also has a quantitative version (due to K. Mahler [ K M ] ) : if e l , . . . , 0~, are linearly independent over Q and P ( X l , . . . , xn) e Z [Xl .... ,xn], P ~ 0 , then IP(e ~',..., e'")[ > H(P)-c, a(l,)(2)

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for some constant cl depending only on ~ .... ,e, with H(P) sufficiently large with respect to d(P). Up to the constant c~ the estimate (2) is the best possible and close to the upper bound given by Dirichlet's box principle. We may ask the same question for ~(e~) .... ,~(a,). For n = 1 E. Reyssat [ER] obtained the bound IP(~3(~))[ > H(P)-c2a(v)3 for any algebraic a 4:0 and P(x) e iE [xJ, P(x) ~ 0 (his results also cover the case of independent H(P) and d(P)). We will prove here the best possible results for the case of complex multiplication and n = l, 2 I/]1(~3(~))1 > H(P1)-c3a(e,); IP2(~]3(cQ,~(fl))l > H(P2)-~aWz)~,

(3) (4)

where ~3(z) has complex multiplication, ct and fl are algebraic numbers linearly independent over IF, P1(x)e Z [x l, P2(x, y)~ Z Ix, y];/]1 ~ 0, P2 ~ 0, c 3 > 0, c 4 > 0 and H(P~) is sufficiently large with respect to d(P~). The estimate (3) is particularly interesting, as it provides (in the case of complex multiplication) the first example, after e", of a value of a classical function at an algebraic point having very bad approximation by algebraic numbers of bounded degree. Perhaps this indicates that for ~3(~) (as for e") there are some special continued fraction expansions, etc. At the end of this introduction we must say that our results can be generalized in two directions. Firstly, the number n of algebraically independent numbers can be increased without big difficulties (up to n < 6). Second of all, these results can be generalized to Abelian functions and their values at algebraic points.

2. Main Results

Let as before ~3(z) have algebraic invariants and field IF of endomorphisms. Theorem2.1. Let ct1,~2,~ 3 be three algebraic numbers, linearly independent over if). Then among

~3(~,), ~(~2), ~(~3) there are two algebraically independent numbers. Theorem 2.2. Let ~1 .... , as be five algebraic numbers, linearly independent over ~. Then among

there are three algebraically independent numbers.

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Corollary. Assume that ~3(z) admits complex multiplication. I f ~1, ~2 are algebraic numbers which are linearly independent over IF, then ~3(~1), ~3(~2) are algebraically independent over ff~. To deduce the Corollary, we simply apply Theorem 1 to the three numbers ~1,~tE, Z~l, where z is an element of IF which is not in ~. Remark. It is clear that Theorems 2.1-2.2 cannot be improved in general, e.g. in the case of complex multiplication of ~3(z) by z, for the system of numbers ~1 = 1, ct2=z, ~3 = ] / 2 , ct4=z]/~, cts = ] / 3 , we have

:~ {~(0~1)' ~(0C2)' ~(~3)} = 2 ; =~={~[~(0~1)..... ~[~((X5)} = 3. Here ~ S for a finite S ~ denotes the degree of the transcendence of II~(S). In fact, our results are proved parallely to strong transcendence measures for , ~(e) and algebraic non-zero ~. Theorem 2.3. Assume that ~3(z) admits complex multiplication, and let ~ be a nonzero algebraic number. Then there exist positive constants C1, C 2, depending only on ~(z) and ~; as follows: for all non-zero polynomials R ( x ) e Z [x] with degree at most d and height at most H such that loglogH=> C 1 d 3, we have IR(~3(~))I > H -c2n. Finally, our method is capable of considerable generalization. We present, without detailed proofs, a number of more general results we have obtained in w

3. The Method of Proof This was first announced by the author in March 1977 at Oberwolfach, as a more complicated means of establishing the Lindemann-Weierstrass theorem using now the Gelfond-Schneider method. Our motivation for giving such a proof was that we knew it .would extend to the elliptic case. This is exactly what we show in the present paper. It seems worthwhile, however, to begin by giving the proof in the exponential case, as the ideas underlying the proof are less buried in technical details. Let el .... , e, be n > 1 algebraic numbers which are linearly independent over II~. Write lI for the ring generated over Z by the 2 n numbers ej, e "j (1 < j 1, we have ~(m z)=

A,,(~(z))

Bm(~(z))

(4.3)

where Am(X), BIn(X) are polynomials of respective degrees m 2 and m 2 - 1 , whose m2 coefficients are algebraic integers in ~(gE,ga) with sizes at most c 1 . Lemma4.2. Let rh--(ml, . . . , m k ) 6 T z k \ { o } and Zo=m 1 ct1 + ... +mktX k. Then there exist polynomials T,~(6), ~), T~(O, ~), U~(O, ~) from (9 [0, O] of total degree in 6) al most k

c4 Em , i=l

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274

and degree in 0 at most d - 1, and of sizes at most

exp(c,Zm t k

i=1

/

such that the following conditions are satisfied: Tr

L~(o,0)

O)

~(Zo)= u~(o, o)'

(4.4)

~'(Zo) = u~(o, - - 0)

This lemma follows immediately from the law of addition for ~3(z), multiplication formula (4.3) and the definition of O. Remark 4.3. We have simple lower and upper bounds for quantities ]T,~(O,O)[, IT'(O,,9)[, [U,I(O,O)[ from Lemma4.2. First of all by majoration we have:

Zm:) k

max l

o,O l,

exp

i=l

For the bound below we use formula (4.4). It follows from the result of [M] that min Ic~- 121> exp( - c 7 log 2 H(c0)

(4.6)

~L

for cv=cv(g2,g3,d(e))>O and algebraic e ~ 0 of degree _ 1 and the constant M sufficiently large with respect to N and L 1 (e.g. M > N k+l) the function F(z) satisfies the following conditions IF~

% + . . . + nk ak)l < exp(-- M S X k log Lo)

(4.37)

for all rational integers ni, 0 _-e x p ( - Yo" log H(ct) [log log H(ct)] 8) where ~o>0 depends only on d(~). This bound in combination with (4.36) establishes (4.33) under conditions (4.18)-(4.19) on L o and X. Suppose now that N is sufficiently large in (4.37). This corresponds to the assumption (4.34) with %8 = M and %9 = N. Now we apply the bound (4.38) for s = 0 and the following system of numbers z: 1 z(ff, r ) = n l ~ +n2 022 + - -

(4.39)

r

where nl, n2, r are rational integers and O__0. For any r > 1, let P,(w) be the polynomial

(

1

P,(w)=P w , ~ 3 ( - I t = \rll

Lo

LI

~to=O

;q=o

~ w a~ ~, Paoxl~ '

(4.43)

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Then we apply Lemma 1.5 [M] to the polynomials P~(w) with w=z(~,r) for fixed r; where the number of different z (a, r) is > 4 L o > 4 dw(P,(w)). We obtain H(P~(w))=max 2o

~ P~o.2~ 4t = 0

-c42 for distinct ffl=(n[,n~) and ~2=(n2,n 2) with constant c42 = min {if21: f2eL}. Now for every ) . o = 0 , . . . , L o we apply L e m m a l . 5 [M] to each of the polynomials LI P2o ( z ) =

Z P2o, 2, z)'' ,~=0

a t 2 L I points of the form z = ~ ( ! ) . W e obtain immediately H(P) < e x p ( - c42 N k+ 1 S X k log Lo) and Lemma 4.9 is proved. The following lemma is the outcome of our analyti6 arguments: Corollary 4.10. For sufficiently large Lo, Li, S, X satisfying (4.18)-(4.19) we have either (i) there exists a polynomial P ( x l , . . . , xm)~Z [x 1.... ,xm] such that - c44 S X k log L o < log iP(01,..., 0m)l< - c45 S X k log Lo; t(P)cr d 5 for constants c44>0 , c4s >0 depending only on ~(z) and ~, The elementary method immediately gives us Theorem 5.2 along the same

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lines as in the proof that type of transcendence of --,r~ _t/ is 3 + e for any e > 0 o)

6o

[Ch3], [Ch6]. Moreover, the proof is even more simple as the degree is always bounded (i.e. types of polynomials involved are sufficiently large with respect to their degrees).

6. Concluding Remarks To conclude this paper we state results that can be obtained by refinment of our methods and formulate some conjectures. First of all our methods (we mean the algebraic part, because the analytical part of w remains almost without changes) can be considerably generalized. We give only three examples: Theorem A1. Let ~3(z) have algebraic invariants and complex multiplication by Q(z). I f ct1..... ct, are linearly independent over Q(z) and n < 6 then ~(~,) ..... ~(~.)

are algebraically independent. We can naturally generalize the results of w for Abelian functions (see [Ch5]): Theorem A2. Let A be a simple Abelian variety defined over ~ and ~ l (Z--)..... rid(z-),N( z-) be corresponding Abelian functions, regular at ~=6, where ~1(~) .... ,~d(z-) are algebraically independent and dim A = d . Let ~1, ...,a, be vectors from I~a (with algebraic coordinates) linearly independent over I~. Let D be a number such that n

-+I>D 2 and D