Math. Ann. 272, 91-98 (1985) tmam �9 1985

A Six Exponentials Theorem in Finite Characteristic

Jing Yu 1

Institute of Mathematics, Academia Sinica, Taipei, Taiwan, R.O.C.

1. Introduction

The purpose of this article is to further illustrate the analogy between algebraic number fields and algebraic function fields. Starting from any algebraic curve over a finite field, we contend that there is a transcendence theory. We have initiated in [7] an approach to the problems of transcendence in finite characteristic via Schneider's method. Here, we shall carry through that method obtaining theorems analogous to the well-known six exponentials theorem in classical transcendental number theory [4, Chap. II].

Let ~r be an arbitrary smooth projective, geometrically irreducible curve over F~, q = p". We fix a rational point c~ on ~, and consider the ring A of functions on regular away from oo. We set k to be the function field of ~r and koo its completion at 00. Elements in the algebraic closure ~?oo which are transcendental over k will be called transcendental numbers.

The analogue of the exponential function in our context are the functions introduced by Drinfeld in [1]. He begins with a lattice M, i.e. a finitely generated discrete A-module contained in/?~o, and he considers the following "exponential function"

to*O

There are lots of arithmetics connected with these functions which illustrate the analogy between number fields and function fields, [1, 2].

We shall fix an embedding of the algebraic closure k-c k-| We are interested only in those functions eu(z) developed on the completion ofk-~ which have all its Taylor coefficients in E. A special case of the main result is the following

Theorem. Let M be a lattice of rank 1 in ~ such that the function eu(z) has all its Taylor coefficients in ~. Let fit, t2 E l~ be linearly independent over k and let or, ~ l~

I This work was partially supported by N.S.C.R.O.C.

9 2 J. Yu

( v = 1, 2, 3) be also linearly independent over k. Then at least one of the following six exponentials is transcendental

eu(fll~v), eu(~2aO, v = 1,2, 3.

Our method works for lattices of arbitrary rank. Thus what we obtain is in fact an 4d + 2 exponentials theorem. We will formulate and prove a general theorem concerning the algebraic values of entire functions. Both the above result and the main theorem in [7] are corollaries of this general theorem. The classical analogue of this general theorem is due to Schneider [5]. The theorem on the six exponentials in the classical case is due to Lang [4].

2. Background

We let ~(a) to be the additive valuation of a e k which measures the order of pole of a at o9. We extend this valuation to the algebraic closure k-| By the size of~ e k-we mean the maximum of the valuations of all the conjugates of a. We let den ~, the denominator of a, to denote any non-zero element a in A which is of minimal valuation among those having the property that aa is integral over A. We call elements in ~- algebraic integers if they are integral over A.

We will use the following version of Siegel's lemma [6, 7].

Lemma 2.1. Let K / k be a finite extension. Let m, n be integers with 0 < m < n and let

o~]i6K , l < j < m , l<_i<n,

be algebraic integers in K which size at most B. Then there exist algebraic integers x l , ..., Xr,, not all O, satisfying

~,x , = O, j = 1 . . . . . m i = 1

and size of xi, i= 1 . . . . ,n, is at most

cn +Bm

n - - m

Here c denotes a constant depending only on the field K. Let f ( z ) be an entire function on the completion of k-o~, say,

f(z)= b,z h, h = O

To study their growth, we consider for rational r the following

Mr(f ) = Max (~(bh) + hr)

= Sup t~(f(z)) *(z)_~r

= Sup t~(f(z)). , (z ) = r

The non-archimedean function theory gives, [,']:

Six Exponentials Theorem in Finite Characteristic 93

Lemma 2.2. Let R > r be rational numbers and f an entire non-zero function. Then M,(f) < MR(f ) - -v , (R- r) where v~ is the number of zeros (counting multiplicities) of f inside the disc r~(z) < r.

In characteristic p, we define the order of an entire function f to be the smallest real number Q such that for given e > 0

M , ( f ) = p tQ+')" for r sufficiently large.

In [7], we have proved the following:

Lemma 2.3. Let M be a lattice of rank d in ~ . Then the function eM(z) has order equals to d logq/logp.

Finally, we summarize the essentials of Drinfeld's theory. Let F = F 1 be the function z ~ z q. We let k-~{F} be the non-commutative

polynomial ring generated by F over ~-~ under composition. By a Drinfeld A-module of rank d we mean an F~-linear ring homomorphism ~b: A ~ E ~ { F } such that for each a e A , there exists al .. . . . a , in/?~, m=da(a), a~,4:0 and such that

(b(a) = aF ~ + a lF j + ... + am Fm.

If Kck-o~ is a subring and the image of ~b is contained in K{F}, we "say that the Drinfeld A-module of ~b is defined over K.

For a given lattice M in /?~ (which is a torsion-free A-module, hence is projective and has a fixed rank), the function e~(z) always induces an Fq-linear isomorphism:

One finds

eM(az) = (~M(a) (eu(z)) , a ~ A ,

= aeu(z) + aleM(z) ~ + ... + a,,eM(z) ~m, m=da(a) .

That is to say, the above isomorphism transforms the obvious action of A on l~oo/M into a more interesting polynomial action on k-~. The polynomials ~bM(a) in ~?oo {F} then give us a Drinfeld A-module of rank d. If ~u is defined over ~? one finds that all the Taylor coefficients of eM(z) lie in k-.

Drinfeld proved more in [1], he showed that all Drinfeld A-modules arise from lattices in the above manner.

We remark also that in proving our theorems about Drinfeld's functions we may assume that the Drinfeld A-modules in questions are defined over the ring of algebraic integers in ~. This is because our Dedekind domain A is always a finitely generated F~-algebra and there is the following relation between modules corresponding to homothetic lattices:

~ 6 ,u(a) = aF ~ + a l ~ - 1F1 + ... + amct~ m- 1Fro,

e M(z) = o~oe,~ ,M(o~o l z) .

94 J. Yu

3. Algebraic Independence of Drinfeid's Functions

Let M be a lattice of rank d in/?~, with corresponding entire function eM(z). We consider the set of all elements 2 in k-~ such that eM(z) and eM(2Z) are algebraically dependent functions. This set forms a subfield KM ofE~. We shall calt it the field of multiplications of M.

We let R~ C k-~ be the ring consisting of all elements ~ in J~ such that aM C M. By an order in a finite extension field K/k we mean a subring of K which contains A and is a finitely generated A-module of rank [K : k]. We have the following.

Theorem 3.1. (i) The field of multiplications KM is an "imaginary" algebraic extension of k of degree < d. Here imaginary means K~nkoo = L (ii) The ring R~ is an order in Kst.

Proof. Suppose e~(z) and eM(2Z) are algebraically dependent over ~ . As they both give additive homomorphisms into k-oo, we may apply a well-known theorem of Artin [3, Chap. VIII, Sect. 12], which says that there are ~,/~j in k-~ such that the following holds

1 ~, ~eu(Z) p' + ~ #jeu(2Z) p~ = O.

i=0 j = o

It follows that if co e M, 09 :~ 0 and a e A then eu(2aeJ) must be one of the finitely many roots of the additive polynomial

m

~o #jXpj = 0. j =

Since eu is additive, there are a 4= 0 in A such that a2~ e M. Consequently, we can find a0 ~ A such that ao2M C M. This implies that 2 must be algebraic over k of degree < d. If the whole field Ku is of degree larger than d, then we have ~-1 ..... 2d § in K~ linearly independent over k. We may choose a~eA such that a~21o9 ..... at2~+ ~co all lie in M. This contradicts to the fact that M generates a vector space of dimension d over k.

Now, as M is discrete, M also generate a vector space of dimension d over k| This implies that if 2ek~oc~KM and a 2 ~ M for some aeA, o~4:0 in M, then ,~ must be in k. This completes the proof of (i).

To prove (ii) it remains only to show that Ru_-_. Ku. This follows immediately from the following identity for a ~ RM:

eM(~z)=~eM(z) o~,~M/M (1 eM(O~) j ~ q.e.d. ~ 0

It follows from our definition of KM that if/~,/72 e k-~ then ~,/~2 are linearly independent over K~ if and only if the functions e~([31z) and eM(~z) are algebraically independent over k-~.

Corollary 3.2. Let M be a rank 1 lattice and ~ , ~ ~ E~. Then ~ and ~ are linearly independent over k if and only if e~r(~z) and eM([3~z) are algebraically independent functions.

Proof. For rank 1 lattice M, we have K u = k. q.e.d.

Six Exponentials Theorem in Firtite Characteristic 95

It is reasonable to conjecture that more is true, namely, fl~, ...,fit ~ l~ are linearly independent over K ~ if and only if eM(fl~z) .. . . , eu(fl~z) are functions algebraically independent.

Finally, we note that the proof of Theorem 3.1 also works in the following situation.

Theorem 3.3. Let M~, M~ be two lattices in lco~ which do not generate the same k-vector space inside Icon. Then the functions eut(z ) and eu2(z ) are algebraically independent over ~ .

In particular, if M~, Mz are of different rank, then eu,(z) and e~:(z) are algebraically independent over k-~.

4. Theorems

Let BN and B~ be two real-valued sequences. If there is a constant c > 0 such that B~ <= cB~ for all N we will then write Bs <~ B~.

Theorem 4.1. Let K be a fixed finite extension of k and let f l . . . . . f~ ( s > 2), be entire functions algebraically independent over 1~, of orders 01 logq/logp . . . . Q~ logq/logp, respectively. Suppose there is a positive real number t and a sequence S~ of finite subsets of k-~ such that the following conditions are satisfied

(i) Max Size (fi(~o)) <~ q ~ , j = 1 . . . . . s. r ~Sl~r

(ii) Max t)(denf~(oJ)) ~ qOS, j = 1 . . . . . s. r

(iii) qtN ~ Card SN, and Max r~(oJ) < N, for N large. o~S~r

(iv) f~(SN)C K for all j and all N. Then

0 1 + ... + 0 s t < s--1

A corollary of this theorem is the following

Theorem 4.2. Let M be a lattice of rank d in [coo such that the corresponding Drinfetd A-module is defined over ~. Let ill, g2 ~ [coo be linearly independent over KM, the field of multiplications of M. Let also a, ~ l~o, v = 1,..., 2d + 1 be linearly independent over k. Then at least one of the following 4d + 2 numbers is transcendental

eu(fll0tv), e~t(fl2~v), v = 1, 2, ..., 2d + 1.

Proof. Let 0 ~ = 02 = d, s = 2, t = 2d + 1. Let SN consists of all elements of the form b1~1+ ... +b2d+10qd+l with u(b~)<N for v= 1 .. . . , 2 d + 1. The Riemann-Roch theorem for the curve r~ implies that the condition (iii) in Theorem 4.1 is satisfied. SUppose that the values eMfflff~) are all algebraic, say contained in field extension K. We consider the functions f l(z)=eM(fl lz ) and f2(z)= eM(fl2z) which certainly satisfy condition (iv). Iftr is any embedding of K into k-~o, we conjugate the Drinfeld A-raodule e u (i.e. applying the automorphism or to the coefficients of the

96 J. Yu

polynomials in K{F}). Let e~M(Z) be the entire function associated to the conjugated Drinfeld A-module. Then all functions e~(fl~z) have order equals to d logq/logp. This leads to an estimate of size which gives condition (i). If we can also bound the denominators as in condition (ii) we will have a contradiction to Theorem 4.1. Thus eM(flff~) cannot all be algebraic over k.

To complete the proof we recall that for a e A

eM(az) = aeu(z) + aleu(z) ~ + ... + ameM(z) e", m = d~(a).

We remarked in the end of Sect. 3 that we may assume all the a~ are algebraic integers in E. Hence the valuation of the denominators of fj(co), e~ e SN is bounded by

2 d + 1

E qa~o(deneM(flff~)), q.e.d. v = l

Another corollary is the main theorem in [-7].

Theorem 4.3. Let (~M be a Drinfeld module o f rank d defined over k, with corresponding lattice M. Let f ie ~oo and let col, ..., co z be l k-linearly independent elements in [~ at which the function eM(flZ ) assumes values also in l~. Then

l<d.

Proof. Let QI=O, 02=d, f l ( z )=z , f2(z)=eu(flz), q.e.d.

5. Proof of Theorem 4.1

Suppose the theorem is not true. We may assume that

t 01+ ..- +Q~ Max Q~<Q+- < t , where ~ - l ~ j ~ s S S

Because if Qs >= g + t_, then we have the inequalities $

0~+ ... +Q~_~+t 0 ~ s - 1

t > 01+ ... + 0 s - t + QI+ .-. + 0 s - l + t s-- 1 (s-- 1) 2 '

( ( s - 1) 2 - 1 ) >s(Q1 + ... +Q~- 1),

~1 jr "'" "~-Qs-I s > 2 , and t >

s - 2

By induction on s we find that this is impossible. We may therefore fix positive constants 61, 62 such that 62 e Q and

t + 3 ~ t t > Q + - - =

s 1 + 262

Six Exponentials Theorem in Finite Characteristic 97

We then replace SN, if necessary, by a subset, so that the sequence S N becomes an increasing sequence of finite sets with the estimation:

qtN ,~ Card SN ~ q(t + ~l)N .

We shall denote by ct, c2 . . . . , constants which do not depend on N. We are going to construct an auxiliary function using Lemma 2.1.

Lt(N) Ls(N) FN(Z)---- Z "'" Z T()~I . . . . . 2 s ) f i ( z ) ; q ' " f s ( g ) ;~s,

t i = 0 Zs=O

[ (q Q/+t+2'I~N-] where Lj(N) = Lq~ - --7--] j and Ix] denotes the greatest integer not exceeding real number x. The integer N is chosen so large that the following holds

M. ( f j ) < q(q'+ ~ ) " , l<=j<s, R a N .

We require that FN(co) = 0 for all co e Ss. This gives a system of homogeneous linear equations in y(21 . . . . . 2~). We multiply each equation FN(co)=0 by the element

f i (denZ(co)) LJ(N) �9 j = l

The resulting system has integral coefficients, with size at most

c l q \ s / .

The number of equations is at most c2q (t+~')N. The number of unknowns is c3q (~+2~')N. It follows that y(21, ..., 2~) can be chosen from integral elements of k-

with size at most c4q~ ~/ , as soon as N is sufficiently large. This leads to the following estimate of the growth of F:

(q +t § 3,;~ R MR(FN)<csq~ s / , R > N .

For each integer Q >__ N we are going to show that

(1)Q FN(CO) = 0 for all co e S o.

(II)q M , Q ( F N ) < - q re, where ro=(1+Sz)Q.

We first check that (I)e =:-(II)q. Applying Lemma 2.2 with R=RQ ---(1+2(52)Q, we obtain:

M,Q(FN) < csq tQ- c652Qq tQ �9

The part (II)Q => (I)Q+ 1 follows from the following

D(x)__> - ( l - 1) size x - / n ( d e n x ) ,

Where i is the separable degree over k of x 4= 0 in ~. We let x = FN(co), with o2 ~ S o + r Then by (II)Q

~(x)< -q'~.

98 J. Yu

On the other hand, from our hypothesis on SQ + 1 we have

(o+'+2~Q D(denx),r q~ --q--)

and

size x <~ q~Q + - - ~ - ) ~ .

t+361 We have chosen 61 so that t>O+ - - , hence FN(co) =0.

s Since (II)o holds for all Q large, Fs must be 0. This contradicts to the

algebraic independence of f~(z), q.e.d.

References

1. Drinfeld, V.G.: Elliptic modules (in Russian). Math. Sb. 94, 594-627 (1974) [-Engl. transl., Math. USSR Sb. 23, No. 4 (1974)]

2. Hayes, D.: Explicit class field theory in global function fields. In: Studies in algebra and number theory. Rota, G.-C. (ed.). New York: Academic Press 1979

3. Lang, S.: Algebra. New York: Addison Wesley 1965 4. Lang, S.: Introduction to transcendental numbers. New York: Addison Wesley 1966 5. Schneider, T.: Ein Satz fiber ganzwertige Funktionen als Prinzip ffir Transzendenzbeweise.

Math. Ann. 121, 131-140 (t949-1950) 6. Wade, L.I.: Transcendence properties of the Carlitz ~-furtctions. Duke Math. J. 13, 79-85

(1946) 7. Yu, J.: Transcendence theory over function fields. To appear in Duke Math. J. (1985)

Received November 19, 1984