algebraic independence of values of exponential and elliptic functions

http://dx.doi.org/10.1090/surv/019/01

INTRODUCTION

ALGEBRAIC INDEPENDENCE OF VALUES OF EXPONENTIAL AND ELLIPTIC FUNCTIONS

0. We present a review of the theory of transcendence and algebraic independence of numbers connected with the exponential, elliptic and Abelian functions. The present review was prepared in August, 1978 as an extended version of the report [Cll] for the International Congress of Mathematicians in Helsinki. It has been updated in 1981 to cover some recent progress. Some new developments (including e.g. irrationality of numbers and G-iunction method (cf. [C14]), zeroes of auxiliary functions [B17, M8], etc.) are not covered here as they deserve separate exposition. Let p ( z ) denote the Weierstrass elliptic function with algebraic invariants g 2 , g3 and f(z) the f-function, £'(z) — - jp(z). Let co, TJ denote any pair of periods and quasi-periods of p(z): f(z + dm + 1 and any integer N > 1 there exists Cx > 0 depending only on K, N,ul9...,um and A such that

(1)

| B0u0 + V , + • • • +Bmiim\>

ClH-H~C for Bt with algebraic entries of heights < H and C > 0 depending o n « l v . . , M m , A, N. The natural field for applications of bounds of linear forms in algebraic points on Abelian varieties, is, of course, diophantine equations. First of all, there is the Problem. To effectivize Siegel's theorem on the finiteness of the number of integer points on an algebraic curve & of genus g > 1. All the results obtained in this direction (archimedian as well as /?-adic) we call "conditional" because they are implicitly ineffective: they depend on knowledge of a basis of the Mordell-Weil group of the Jacobian J(Q) of 6. We call the curve 6 of genus g ^ l a CM-curve if its Jacobian J{&) is a CM-variety or weaker, if 6 admits a nonconstant rational map (defined over Q) into a CM-variety. As an application of the bounds (1), (3) to CM-curves, we have D. Masser's [M4] result. 2. Let Q be CM-curve defined over the number field K. If P is the point from 6(K) of height H(P) and denominator D(P\ then THEOREM

(4)

//(P) 0 depending only on K, 6, 6 > 0. In this direction the /?-adic results give stronger corollaries. Since 1976, D. Bertrand [B4] has developed methods to estimate linear forms in algebraic points of elliptic curves and Abelian varieties of CM-type in the ^-adic domain. He and Y. Flicker [B8, Fl] have proved precise results of the form (1), (3) in the /7-adic case. E.g.: THEOREM 3 [Bl]. Let Q be an elliptic CM-curve defined over K. / / P is a point from 6(K), H(P) is the height of P and pr D(P) is the greatest prime factor of the denominator D(P) of P, then

(5)

prZ)(P)>C4(log#(P))C5

for some C4 > 0, C5 > 0 depending on Q- and K {more precisely C5 depends only on the rank of/(6)(K)).

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4

ALGEBRAIC I N D E P E N D E N C E

Such results would be extremely useful if it were possible to make them completely effective in terms of 6 and K. There are two obstacles on the way: (a) the absence of an effective bound for generators of the Mordell-Weil group of J(Q)(K) in terms of 6, K; (b) the need for a lower bound for | A | in (2) in terms of the heights of ux,..., um. In the direction (b) first results were already obtained by Coates-Lang [C22] but the bound we really need has not been proved in general. We will formulate it as a CONJECTURE. In the previous notation, let ut be an algebraic point of A of degree < M and height < Ul'•:. i = 1,..., m. Then for any e > 0 (6)

| A | > e x p - Q l o g / / - fi l o g ^ 1 + e ) ,

where Q > 0 depends on M, N and e > 0 for matrices B0, Bu...,Bm, with algebraic entries of degrees ^ N and heights < H.

not all zero,

This conjecture is true for m — 2 for an arbitrary CM-variety A. Recently an answer to this conjecture was established by the author for an arbitrary m > 2 and an elliptic curve E with complex multiplication, with slightly different dependence on H in (6), cf. [C15b]. In direction (a), there are no unconditional positive results (and there is even some suspicion about the existence of effective bounds for generators of the Mordell-Weil group [M2]). The Birch-S winner ton-Dyer conjecture enables us to solve (a) [M2, 13]. For this one should estimate II ^ logUi9 where uv...Jum are generators of the Mordell-Weil group of A(K). According to the general form of the Birch - S winner ton-Dyer conjecture (see report of J. Coates [C23]), we need the bound for | Lxm\\) | , where LA(s) — 2 ann~s is 3. properly defined L-function of the Abelian variety A defined over K and m is the rank of Mordell-Weil group of A(K). For general A, it is unclear what are good upper bounds for | L^ w) (l) | . However it becomes possible to do this in case LA(s) satisfies a functional equation of predicted form (e.g. in the CM-case). Thus for an elliptic curve A = E defined over Q of conductor N we have, \L^(\)\«N1/4

(7)

m

assuming Weil conjecture for E, and

(8)

|£ ( E m) 0)l «NE

assuming also the Riemann hypothesis for LE(s). In the CM-case bounds (7)-(8) together with estimate (6) for m — 2 enable us to obtain bounds for integer points on some curves of positive genus. THEOREM

4. Let k be an integer such that \k\ is the power of one prime p and E

be the curve (9)

y2 = x3 + k.


ALGEBRAIC INDEPENDENCE

5

If LE(s) satisfies the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, then for integer solutions x, y of (9) we have (10)

max(| x | , | y |) ^ exp(C 7 /? 1 / 6+e )

for C7 > 0 depending only on e > 0. (This is a considerable improvement of H. Stark's general bound [S9].) Using the remark after the Conjecture, one can generalize the result (10) to an arbitrary elliptic curve E defined over Q and having complex multiplication. For example, if a CM elliptic curve E is defined over Q, having the conductor N, then the bound for the height H(P) of an integer point P of E(Q), is of the form H(P) < exp(c(e)7V l/12+e ) for arbitrary e > 0 and c(e) depending only on e and the rank of E(Q). Recently a paper appeared [G2], in which the author deduced bounds weaker than (10) based on the scheme above, but with bounds weaker than (6) or (8). Now, we shall deal with the "transcendental" part of linear independence results. The first such results were obtained by A. Baker [Bl, B2], and then by J. Coates [C21] and D. Masser [M3] for the product of two elliptic curves. Let A be an Abelian variety of dimension d defined over Q; with Abelian functions Ax(z),... ,Ad(z)9 B(z). Let H{(z),... ,Hd(z) be algebraically independent (over C(z)) quasi-periodic functions of A, i.e. such that for any period co of A, Hiix+a) = Ht(x) + ^ - ( 5 ) : / = 1,...,2.

The most interesting case is that of complex multiplication. COROLLARY

and

IT

2. If o) is a the period of ip(z) with complex multiplication, then & > * / « > * / « } > 2.

From (4) it can be deduced e.g. that for the modular invariant J(q), q — e2viz, that if J(q) is algebraic and 0 — q(d/dq\ then 0J(q) and 02J(q) are a.i. The last result as well as (3) was generalized to the/?-adic case by D. Bertrand [B7]. We can present an interesting generalization of (3) for the case of algebraic nonperiodic points of p(z) [Cll, C19]. THEOREM

Q. Then

4. Let u be an algebraic point of jp(z) linearly independent with co over

(5)

f(w) — rju/oo andri/o)

area.i.

E.g. from this follows COROLLARY 5. For u algebraic for #>(z), linearly independent with co over Q, f(w) — 7}u/co is transcendental.

This transcendence result was provable only by using the algebraic independence result. The result (5) also was proved in the /?-adic case by D. Bertrand [B7]. We have two more results of the author on algebraic independence for the case of complex multiplication. THEOREM

jp(z)m, then (6)

6. Let p(z) have complex multiplication and u be an algebraic point of u and$(u) are a.i.

This is the natural generalization of Corollary 2. THEOREM

(7)

7. Let jp(z) have complex multiplication by ft. Then for period co of TT/CO

andemi^ are a.i.

Also we have elliptic (and Abelian) analogues of the Lindemann-Weierstrass theorem (see below). Let us see how our results can be interpreted in terms of some problems of analysis and partial differential equations. We will present one application. Let u(x) be a periodic (with period T) potential that is n-band, or, in other words, a solution of a stationary nih order Korteweg-de Vries equation (S. Novikov [Nl], H. P. McKean [M9] and see [D3] for early results) 2ni=0 CjR^u] = 0.


8

ALGEBRAIC I N D E P E N D E N C E

Then the spectrum for the Schrodinger operator L — -d2/dx2 tt-band structure. (8)

+ u(x) has

X 0 < \ 1 < A 2 < A3 ^ A 4 < ••• < A 2 „ < +°°>

where A, are the first In + 1 points of T-periodic or 7"-antiperiodic spectrum L\p = At// and (A 2 / _ 1 ,A 2 / ): /*= 1,...,«, are « forbidden zonas. All Ay: y = 0,...,2w, are determined algebraically by the system of differential equations satisfied by u(x) and its derivatives. They are called "algebraic eigenvalues". However there are infinitely many (degenerate) points A,: / = In + 2 , . . . of the periodic and antiperiodic spectrum (i.e. A2*-i = X2i: i = n + 1, n + 2,...). They are sometimes called " transcendental", but it is possible to give precise sense to this word and to prove that such numbers are indeed transcendental. Let us take the best known example of an fl-band potential, the Lame potential, u(x) — n(n -f 1) p(x)9 where P(x) have algebraic invariants. COROLLARY 8 [CIO]. For the Lame potential u(x) = n(n + 1) p(x) the first 2n + 1 points of the periodic and antiperiodic spectrum are algebraic, while the others are transcendental. (E.g. all eigenvalues \it of L\p = [xt\p with periodic boundary conditions \p(0) — \p(T) — 0 for i = n + 1, n + 2 , . . . are transcendental.)

Another problem arising from Abelian varieties associated with Fermat curves is the problem of the transcendence of values of the T-function in rational, but not integer points. Of course, T(l/2) = JW is transcendental. From Corollary 2 we know also that

(9)

r(i/6), r(i/4), r(i/3), r(2/3), r(3/4), r(s/6)

are transcendental (and each of the numbers in (9) is a.i. with IT). These examples correspond to elliptic curves of CM-type, but in order to investigate general T(m/n), we must investigate the arithmetic nature of the periods and quasi-periods of Abelian varieties of CM-type in the Shimura sense. This problem turns out to be the most important and difficult in Transcendence Theory. It is very interesting that analytical difficulties in transcendence proof in this situation are tied up with algebraic difficulties. Let A be an Abelian variety defined over Q of dimension d. Then we have Abelian functions Ax(z\... 9Ad(z)9 B(z\ where Ax(z)9... 9Ad{z) are algebraically independent and B(z) is algebraic over Q[A]9. . . ,Ad]. The functions Ax(z),...9Ad(z)9 B(z) are assumed to be regular at z — 0; these functions are 2^/-periodic with common periods 5 „ . . . J 5 2 < / G C / . There are also d quasi-periodic functions H^(z)9... 9Hd(z)9 algebraically independent over C(z), with quasiperiods?]/7, Hjix+St)

=Hj(x)riij:

j= 1,..., o{x)o(u)

[R3]. Recently D. Bertrand-M. Laurent [BIO] obtained results on the transcendence of a Jacobi ^-function in the CM-case. 3. Algebraic independence in the exponential function case. In the exponential case the ultimate aim is to prove the SCHANUEL CONJECTURE.

over Q. Then among

Let a 1 ? ... ,an be complex numbers linearly independent «,,...,«„,

ea\...,ea"

there are at least n algebraically independent numbers.


12


It is now a common opinion (first formulated by S. Lang) that the Schanuel Conjecture implies all reasonable statements on the transcendence and algebraic independence of values of an exponential function. We have no idea how to prove this conjecture for general ax,..., an even if n = 2. For many years the most important case of the Conjecture has been the case of numbers of the form < / , . . . ,a^ , where a and /? are algebraic, a =£ 0,1 and /? is of degree d. Assuming the Schanuel Conjecture, these numbers are a.i. This set of numbers is a particular subclass of the following sets of numbers connected with the exponential function and has been the major target for investigation since 1949 [Gl]. Let a,,... ,aN and /?,,... ,/? M be two sequences of complex numbers linearly independent over Q. We define three sets of complex numbers: (1)

S,= {e^},

S2={pj,e"^}, (i= l,...,N;j=

Additional conditions imposed on au...,aN of linear independence, TV

N

(2)

2 "/«/

> exp

2

S3={ai,Pj,e'^} 1.....M). and /?,,...,fi M bound their measure M

2 VA

7=1

/ > exp ~rl

I

M

i=i

• andvu...9vMifl^\ul\>092f=l\vJ\>0. for integers ux In [Gl] Gelfond proposed a method that enabled researchers to examine the case #St > 1 or #St > 2 for / = 1,2,3. The investigation of this case (one or two algebraically independent numbers) was completed after Gelfond [Gl] in the papers of Brownawell, Tijdeman and Waldschmidt [B13, Tl, Wl]. The situation with three or more algebraically independent numbers is more difficult. The main obstacles, as usual, are algebraic-geometric. More precisely, in the course of the analytic part of the proof of the existence of n + 1 algebraically independent elements of S/5 one obtains a system of polynomials P / ( x , , . . . , x z ) G Z[xx,...,xn] of the height < Hi and degree < di9 i= 1,2,3,... that satisfy (3)

\P,{6„...,en)\ 4.

We see that our results give only some parts of the Schanuel Conjecture. However even these partial results give us hope of one day proving the whole conjecture (if nobody finds a counterexample first). In any case, analytical methods of Siegel-Gelfond-Schneider type should be considerably modified for this.


14


The number of problems connected with the simplest cases of the Schanuel Conjecture is innumerable. We cite the simplest example. PROBLEM 5. Let fi be a quadratic irrationality and log a l5 log a2 be logarithms of algebraic numbers, linearly independent over Q. One should try to prove that log ax and af are a.i. and that af, af are a.i. We do not know the solution of this problem, which is the simplest from the point of view of existing analytic methods (hint: in (3) n = 1, v — 1). Present techniques give us the possibility of only adding to the numbers #1? #2 a third number 63 (connected with an exponential or elliptic function) and then showing # { # ! , 02, 63} > 2. There are several results in this case for 0,, 02 from Problem 5 and we present some of them. THEOREM

(6)

(7)

6. In the notation of Problem 5 we have

#{loga1,af,af}>2, #{TTV',^}

>2,

PZQ(i).

Only the result (5) follows from Gelfond's [Gl] method. Results (6) and (7) are proved in this volume [C19], cf. [C4] using Baker's method. THEOREM 7. Let p(z) have complex multiplication by /?, and let co be a nonzero period of p(z) and u be an algebraic point of p(z). In the notation of Problem 5 we have

(8)

#{co,«f,af}^2,

(9)

#{^,af,«f}^2,

(I0 )

# (;.>sa^} >2 .

(11)

#{5,logai,af}>2,

(12)

#{w,loga,,af}

>2.

Results (8), (9) were obtained in 1975-1977 using a combination of exponential and elliptic functions; see [C19, Ch. 7]. There are similar results on the existence of two a.i. among three numbers connected with exponential and elliptic functions. For the proof of Theorem 7 cf. [C3] and [C19]. Multidimensional considerations are very important for further progress in Transcendence Theory. Lately they have become a subject of particular attention and have led to considerable progress in studies of the zeroes of auxiliary functions (Brownawell-Masser-Wustholz) [B17, M8] and transcendence proofs [B9, BIO, M8, W5].



15

In the multidimensional case one still needs the general form of the Schwarz lemma. Previous results [B12, C6, W2, W5] show that the Schwarz lemma depends on properties of singularities of hypersurfaces in Cn. We propose one algebraic-geometric problem. Let Q(S; K) = min{deg/>:/>(*) G C [ x ] , 9|>(vv) = 0 for a l U G N M k\< K9 w G s) for finite S C C " . QUESTION 8. For a given | S | < oo to describe all possible values of ti0(S) =

limK_„Q(S;

K)/K.

Using the properties of Q0(S) [C6] we find e.g. that for the meromorphic transcendental function f(z) in Cn of order of growth < p, the set S(f) of w G Q n with 3|/(vv) G Z for all k G N" is contained in a hypersurface of degree < np [C5, C6]. For further results see [C19].

4. Measures of transcendence and algebraic independence. Soon after the proof of transcendence of the first concrete numbers (Liouville numbers, e9ir) it became clear that the question of transcendence (or algebraic independence) is closely connected with the problem of diophantine approximation or, as people sometimes said, the "arithmetical nature" of transcendental numbers. The problem of the "arithmetic nature" of a given number (or sequence of numbers) is, roughly speaking, the problem of how this number is approximated by algebraic (rational, algebraic of bounded degree, etc.) numbers. All the well-known classifications of transcendental numbers are based on the property of approximations of numbers by algebraic ones. First of all, in order to distinguish which numbers are well approximated and which are badly approximated, we have the following corollary of Dirichlet's box principle. LEMMA 1. / / the numbers 0X9...90n are algebraically independent numbers, then for any N9 H there exists a polynomial P(x{9..., xn) G Z[xl9..., xn] of degree < N and height < H with

(1)

00. It is conjectured that for almost all (6V... ,0n) G R" (up to measure zero) the upper bound for \P(0l9...,On)\ (if P(xl9... 9xn) G Z[xl9... 9xn]) is not considerably different from the right side in (1). At least this conjecture is true for n — 1 (see [B2]). However to prove that bounds for measure of transcendence (n — 1) or algebraic independence (n > 2) of concrete numbers are of Dirichlet type is extremely difficult. Work in this directin started in 1899, when E. Borel considered the case n = 1, 6x=e.


16


During the last 80 years this direction has developed into a new branch of Transcendence Theory: Measures of transcendence. For numbers associated with functions of classical analysis we have only two general analytic methods to treat measures of transcendence. (A) Siegel's method [S7] for ^-functions, (B) The Gelfond-Schneider method and its variants: Baker's method, etc. Using (A) K. Mahler [Ml] obtained very good bounds for the measure of algebraic independence of ea\.. .,ea" for algebraic numbers ax,...,an linearly independent over Q. For ^ ( x , , . . . ,x w ) E Z[x l9 . . . , . x j , H(P) ^ H, d(P) < d, (2)

\P(ea\...,ea")\>H-c>dn

for c2 = c 2 (a 1 ,...,a w ). However this bound holds only for loglog H > c3d2n. Analogous bounds hold for other ^-functions [S7]/(z) instead of ez. See the general results of [C19, Ch. 5]. Method (B) was applied with great success to estimating the measure of transcendence of a single number connected with exponentials: ea, log a, a^, log a/log /?,... for algebraic a, /?. These measures of transcendence were obtained for independent degree and height (or even for the case when the degree is more essential than the height [C2b]). For the best possible results as of 1972 see the book of Cijsouw [C20]; for the situation as of 1978 see [W6], based on estimates of linear forms in logarithms of algebraic numbers, by Baker's method [B2]. For the case of a single number connected with the elliptic function p(z) (such as to, TJ, 77/co, TJ/CO, f(w), eu, w,... for an algebraic point of p(z)) we refer to papers of E. Reyssat [R2]. The work of [R2] contains all known bounds for the measure of transcendence of one number connected with an elliptic function. The main method of the proof is just the Schneider-Gelfond method together with knowledge of the number of zeroes of meromorphic functions of the form F(z) = P(/,(z), / 2 (z)), where /,(z), / 2 (z) have the form az + )8£(z) or p(yz). The bound for the number of zeroes of F(z) in a given disc | z | < R is a corollary of results of D. Masser [M3] and D. Brownawell-D. Masser [B17]. Thus, we can say that for now we have used all existing analytical methods for the investigation of the measure of transcendence of one number, connected with the exponential and elliptic functions. Some minor problems still remain, but the existing analytical methods give no hope of considerably improving the bounds for the measure of transcendence. For the case of two and more numbers the main difficulties are of an algebraic character. With the same sort of analytic arguments (construction of the auxiliary function F(z) — P(fx(z),... jn(z)) and investigation of its zeroes or small values) we now have very deep algebraic problems caused by the singularities of algebraic hypersurfaces. This explains why for a long time there were practically no known bounds for the measure of algebraic independence of two numbers (as well as no new examples of two algebraically independent numbers connected with exponential and elliptic functions).



17

In order to discuss measures of transcendence it is useful to introduce S. Lang's notion of type of transcendence [LI]. This notion corresponds to the situation when the degree and logarithmic height of the polynomial are of the same order. DEFINITION 2. Let 0 l 5 . . . ,0W be algebraically independent numbers. We say that ( 0 l 5 . . . ,0„) are of type of transcendence < T if there exists a constant C > 0 such that for every P(xl9.. .,*„) E Z [ x „ . . . , x j , H(P) ^ H, d(P)^d9 P ^ 0, we have (3)

| P ( 0 , , . . . , 0 j | > e x p ( - C ( l o g / / ( / > ) + )) T ).

It is clear from Lemma 1 that T > n + \ for any ( 0 l 9 . . . ,0W). We know that for n — 1 almost all numbers 0 have type of transcendence < 2 + e for any e > 0. It seems very natural that the following classical conjecture is true. CONJECTURE. For almost all ( 0 , , . . . ,0„) E C 1 , r/ze /y/?e of transcendence T < « + 1 + e /or a«y £ > 0. We know the proof of the part of the conjecture for n = 2 [VI]. From the results of [C20, R2] previously mentioned it follows that numbers 77, 77/co are of type < 2 + £ for any £ > 0, while ea,

TJ/CO are

of type < 3 + £ for any £ > 0, etc —

However up to now there are no examples of a number with type of transcendence exactly 2. Probably m and TT/W are such examples but we have no idea how to prove this. For the case of two or more numbers, up to 1975 we simply had no examples of several numbers and finite type of transcendence! Only in 1975 did the author prove that pairs of numbers (IT and co) and (TT/CO and e 77 ^), for co the period of p(z) with complex multiplications by /?, are two pairs of numbers with finite type of transcendence. Since 1975 we found a few more pairs of algebraically independent numbers with finite type of transcendence. However, as if mentioned before the main difficulties in the proof of measure of algebraic independence are algebraic geometrical. These difficulties occur as explained in §3, when, in the course of the proof of algebraic independence, one analyzes the system Pt(xu... ,xn) E Z[x l 5 ... ,xn]: i — 1,2, 3 , . . . , of polynomials (see (3) of §3), having small value's at the given point ( 0 , , . . . ,0W), for which we try to obtain the measure of algebraic independence. Both nonelementary and elementary methods give good bounds of the measure of algebraic independence of numbers, whose algebraic independence is established in §2. We present the list of results obtained by our methods. Also (for comparison) we present the previous estimates. Probably the most important result in this direction is the proof that the numbers (4)

TT/CO, TJ/CO

are of the type of transcendence < 3 + £ for any £ > 0.


18


The result (4) is "almost" nonimprovable according to Lemma 1. However it would be important to show that the type of transcendence is exactly 3. Here, as before p(z) is the Weierstrass elliptic function with algebraic invariants g 2 , g3. By co and TJ we denote a period and quasi-period associated with jp(z)andf(z): 2 f ( « / 2 ) = TJ. For simplicity, everywhere below we denote by P(JC, y) a nonzero polynomial from Z[JC, y] of height < H and of degree < d. We also put h = log H, t = h + d. THEOREM 3. The type of transcendence of 7r/co, e > 0. Moreover, if P(x, y) G Z[x, }>], //zew

(5)

| />(*/«>

T?/CO)

TJ/CO W

| > exp(-c 4 (log / / + dlogd)

at most 3 + e /or a«y

d2\og2(d

+ 1))

/or c4 > 0. Theorem 3 in its complete form [C14] is proved below [C19]. The history of Theorem 3 is as follows. After the proof in 1976 of the algebraic independence of 7r/co, TJ/CO, in 1977 the author found that 7r/co, TJ/CO have type of transcendence < 6 + c for any e > 0. Earlier, in 1975, the bound for type of transcendence < 6 -f e was obtained by the author in the case of complex multiplication (Corollary 2, §12). A little later the same result was obtained by D. Masser. In 1977 the author also proved the following bound for the measure of algebraic independence of ir/o,

(6)

|

P(TT/O), TI/O)

TJ/CO [C3]:

\> exp(-c 5 [log 7W 5 log 8 (log Hd) + (*/*>, i?/«) | > exp(-c 6 [log 7W3 5 log 8 (log Hd) + 0, and the same kind of technique gives us Theorem 3. Finally, in [Cll] we presented a bound (8)

| / > ( * / « . i?A>) l> exp(-c;(log H + d)d2\og3(d\og

//)).

This bound was subsequently improved in [C14] to its present " / / ~ c " form (5). The results that we obtained are almost the best possible from the point of view of the analytic part of the proof. As before in §2, we consider the following pairs of numbers that were previously proved by the author to be algebraically independent: 1) 77/(0, 7]/

axu + a2S(u),

£(u)--u,

for u, ut, Uj that are algebraic points of p(x), 2J8 | .II, + 2 Y | . ? ( K , )

in the c m . case;


o)


21

u/v for algebraic points u on ^P 1 (x)andt)on ^P2(x) with a.i. px(x), P2(x); p(ax),..., p(ocn) in the cm. case, where at, /?,, y. are algebraic numbers. We start the presentation of these results with a statement about the normality of periods of Abelian varieties. THEOREM 8. Periods (or quasi-periods) of Abelian varieties with real (complex) multiplications are normal transcendental numbers.

Transcendence results in Theorem 8 belong to [SI, B6]. 9. Periods of Abelian varieties of CM-type are normal transcendental numbers. In particular, for rational (noninteger) a and b the value of Euler's B-function is a normal transcendental number, COROLLARY

(19)

\P(B(a,b))\>H(Pyc(d^b\

Similarly, lower bounds for linear forms fixux + • • • +fimum + /?0 in algebraic points u!,..., um of an elhptic curve E have normal or "H~c" form \plul + --.+(imum + (i0\>H-c for H(/30) < H, as noted in §1. A large class of normal measures of algebraic independence is provided by a generalization of the Lindemann-Weierstrass theorem for elliptic and Abelian functions (§2, [C7, C19, Ch. 7]). Similarly, low bounds for linear forms jixux + • • • +$mum + /?0 in algebraic points ux,... ,um of an elliptic curve E have normal or " / / c " form as noted in §1. A large class of normal measures of algebraic independence is provided by a generalization of the Lindemann-Weierstrass theorem for elliptic and Abelian functions (§2, [C7, C19]). THEOREM 10. Let p(x) have complex multiplication and let a ^ 0 be an algebraic number. If P(x) G Z[x], P ^ 0, d(P) < d, H(P) < H, then

(20)

\P(p(a))\>H-c™d,

(/Tog log if ^ cX9d3 for c18, c19 > 0. 11. Let P(z) have complex multiplication by Q ( T ) and let a,ftbe algebraic numbers, linearly independent over Q ( T ) . If P(x, y) E Z[x, y], P ^ 0, d(P) c2Xd5 and c20, c2X > 0. For numbers p(a), in the case of P(z) without complex multiplication, we have THEOREM 12. Let a ^ 0 be an algebraic number and let p(z) have algebraic invariants. Then for P(X) G Z[x], P(x) £ 0, d(P) < d, H(P) < H we have

(22)

\P(p(a))\>H-c*d2

when log log H > c23d3 for c22, c23 > 0.


22


One can find the proof of Theorems 10-12 in [C7]. In general, we have the following elliptic generalization of Mahler's bound (2), see [C19, Ch. 5,7]. Let («,),...,

p(an))\>H(Pr^)d(Pr

where c 24 (a) > 0 depends only on [K(g 2 , g 3 , a , , . . . ,a„):Q] provided that loglogi/(P)>c 2 5 H~c*d2

for log log H > c21d5. Theorem 6 can be supplied with an additional normality statement (cf. [C14]). THEOREM

p(z)9 then

14. If p(z) has complex multiplication and u is an algebraic point of \P{u9${u))\>H-c^

with c2S(d) > 0 depending on d and u9 p(z) only. Theorem 14 for example implies that any number P(u9 f(w)) is normal. As a corollary (or, equally, as a corollary of Theorem 3), values of the T-function T(l/3), T(l/4), etc. listed in §2 are normal transcendental numbers. E.g. for IX1/3),

\T(\/3)-p/q\>\qr for some y > 0. The best value of y is unknown although a natural conjecture would be y < 2 + e, if \q\> g 0 ( £ )- For a simpler number T(l/2) — y/W9 the corresponding measure of irrationality is that of IT studied in [C17], The bounds from [C17] give for this y the value y < 39.7799 New methods based on Pade approximations [C16] give for T(l/2) the new value of the exponent y only as y 3, then the numbers 77, r ( l / / ) , . . . , r ( / i / / )

forn =

(l-l)/2

have type of transcendence exactly (I + l ) / 2 (or at least < (/ + l ) / 2 + e for any fi>0).



23

CONJECTURE 16. For p(z) of simple nature e.g. g2 = 0 or g3 = 0,

Mi)-/>/