Determination of algebraic relations among special zeta values in positive characteristic

Advances in Mathematics 216 2007) 321 345 wwwelseviercom/locate/aim Determination of algebraic relations among special zeta values in positive characteristic Chieh-Yu Chang, Jing Yu 1 Department of Mathematics, National Tsing Hua University and National Center for Theoretical Sciences, Hsinchu City 300, Taiwan, ROC Received 7 July 2006; accepted 21 May 2007 Available online 2 June 2007 Communicated by Michael J Hopkins Abstract As an analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζ C n) at positive integers n By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler Carlitz relations and the Frobenius pth power relations 2007 Elsevier Inc All rights reserved MSC: primary 11J93; secondary 11M38, 11G09 Keywords: Algebraic independence; Zeta values; t-motives 1 Introduction Let θ be a variable and A + be the set of all monic polynomials in F q [θ]heref q is the finite field of q elements with characteristic p The object of this paper is to explain all the algebraic relations among the following zeta values: ζ C n) := )) 1 1 a n F q, n= 1, 2, 11) θ a A + * Corresponding author E-mail addresses: cychang@mathctsnthuedutw C-Y Chang), yu@mathctsnthuedutw J Yu) 1 The author was supported by the NSC Grant No 94-2115-M-007-004 0001-8708/$ see front matter 2007 Elsevier Inc All rights reserved doi:101016/jaim200705012

322 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 These zeta values were introduced in 1935 by L Carlitz [4], where he discovered the fact that there is a constant π algebraic over F q 1 θ )) such that ζ Cn)/ π n falls in F q θ) if n is divisible by q 1 The constant π, which later on showed to be transcendental over F q θ) by Wade [9], is a fundamental period of the so-called Carlitz module Call the positive integer n even provided it is a multiple of q 1 The ratios ζ C n)/ π n for even n involve what are now called Bernoulli Carlitz numbers, just as the case of the Riemann zeta function the ratios ζn)/π n for even positive integer n can be expressed in terms of the classical Bernoulli numbers In the late 1980s, Anderson Thakur [2] were able to relate the zeta values ζ C n) to the nth tensor power of the Carlitz module for all positive integers n As a result of this big step forward, the second author of the present paper was able to prove the transcendence of the zeta values ζ C n) for all positive integers n [11], in particular for odd n ie, n is not divisible by q 1) Later in [12], the second author was also able to determine all linear relations among these Carlitz zeta values and powers of the fundamental period π As expected the Euler Carlitz relations for n divisible by q 1 are the only linear relations with algebraic coefficients) among these transcendental values Because it is in the characteristic p world, pth power Frobenius) relations certainly are also there for all positive integers n and m: ζ C p m n ) = ζ C n) pm 12) Our purpose is to prove that these, ie, Euler Carlitz relations and pth power relations, account for all the algebraic relations among the zeta values ζ C n), n = 1, 2, 3, The main result in this article can be stated as: Main Theorem For any positive integer n, the transcendence degree of the field F q θ) π,ζ C 1),, ζ C n) ) over F q θ) is: n n/p n/q 1) + n/pq 1) + 1 In the classical case, among the Riemann zeta values at positive integers, conjecturally one expects that the Euler Bernoulli relations, ie, ζn)/π n Q for even positive integer n, are the only algebraic relations That is, given integer n>2, the transcendence degree of the field Q ζ2), ζ3),,ζn) ) over Q should be n n/2 On the other hand, for a fixed monic prime v in F q [θ], letting k v be the v-adic completion of k := F q θ), we are also interested in the v-adic zeta values ζ C,v n) k v,n N, introduced by Goss [5] Following Goss, first, through analytic continuation for integer m 0 one defines ζ C m) = { i=0 a A + deg a=i } 1 a m

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 323 These ζ C m) are actually in k One then puts ζ C,v m) := 1 v m) ζ C m), m 0, and ζ C,v n) := lim ζ C,v n q deg v 1 ) p i), n N i Goss [6] shows that ζ C,v n) = 0foreven integer n Those even integers are so-called trivial zeros of ζ C,v The transcendence of ζ C,v n) for odd positive integer n is proved by the second author of the present paper see [11]) Comparing with our main theorem, conjecturally one expects that among the v-adic zeta values at positive integers, there are no non-trivial algebraic relations: Conjecture For any positive integer n, the transcendence degree of the field F q θ) ζ C,v 1),,ζ C,v n) ) over F q θ) is: n n/p n/q 1) + n/pq 1) Now we turn to the Carlitz zeta values; our tool for proving algebraic independence is a fundamental theorem of Papanikolas [7] which is a function field version of Grothendieck s conjecture on periods of abelian varieties The basic structure which comes up in this theory is the t-motive introduced by Anderson and their motivic algebraic) Galois groups The t-motive is a notion which is dual to the notion of t-module which has been treated extensively by Yu in previous works of the positive characteristic transcendence theory The t-module structure is the key for proving almost all the interesting linear independence results The breakthrough in passing from the linear independence to algebraic independence by way of t-motives, is achieved by the efforts of Anderson, Brownawell and Papanikolas, in particular, the ABP criterion of [3] For convenience, we simply call t-motives what have been called rigid analytically trivial dual t-motives in terminology of [3] In [7] Papanikolas succeeds to prove the algebraic independence of Carlitz logarithms of algebraic functions that are linear independent over the rational function field Following his method of constructing t-motives, for any positive integer n, it is not difficult to prove the algebraic independence of nth Carlitz polylogarithms of algebraic functions which are linearly independent over the rational function field Corollary 32) Having such algebraic independence at hands one goes back to the 1990 paper of Anderson Thakur [2, Theorem 383], where they have a beautiful formula expressing ζ C n) explicitly as a linear combination of simple nth Carlitz polylogarithms with rational coefficients This formula originated from interpreting the value ζ C n) in terms of the nth tensor power of the Carlitz module, is not used in the previous works dealing with the linear independence, but turns out to be crucial in the final determination of all algebraic relations among these zeta values

324 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 2 Notations and preliminaries 21 Notations 211 Table of symbols F q := the finite field of q elements, q is a power of a prime number p A := F q [θ], the polynomial ring in the variable θ over F q A + := the set of monic elements of A k := F q θ), the fraction field of A k := F q θ 1 )), the completion of k with respect to the place at infinity k := a fixed algebraic closure of k k := the algebraic closure of k in k C := the completion of k with respect to canonical extension of the place at infinity := a fixed absolute value for the completed field C T := {f C t ; f converges on t 1} This is known as the Tate algebra L := the fraction field of T G a := the additive group GL r := the group of invertible r r square matrices G m := GL 1, the multiplicative group 212 Twisting Given n Z, for any formal Laurent series f = i a it i C t )) we define the n-fold twist of f by σ n : σ n f ) := f n) = i a q n i t i The n-fold twisting operation is an automorphism of the Laurent series field C t )) stabilizing several subrings, eg, k t, k[t], and T More generally, for any matrix B with entries in C t )) we define twisting B n) by the rule B n) ij = B ij n) In particular, for any matrix B with coefficients in C we have B n) ij = B ij q n 213 Entire power series A power series f = i=0 a i t i C t that satisfies and i lim ai = 0 i [ k a 0,a 1,a 2,): k ] < is called an entire power series As a function of t, such a power series f converges on all C and, when restricted to k, f takes values in k The ring of the entire power series is denoted by E

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 325 214 Carlitz s theory: the logarithm and the exponential Let L 0 := 1, i L i := θ θ q j ) j=1 i = 1, 2,) The Carlitz logarithm is the series log C z) := which converges -adically for all z C with z < θ It satisfies the functional equation θ log C z) = log C θz) + log C z q ) whenever these values in question are defined The formal inverse to the Carlitz logarithm is the Carlitz exponential It is the series Here we set exp C z) := i=0 i=0 z qi L i z qi D i D 0 := 1, i 1 D i := θ q i ) θ qj j=0 The Carlitz exponential is an entire power series in z satisfying the functional equation Moreover one has the product expansion where q exp C θz) = θ exp C z) + exp C z) q exp C z) = z π = θ θ) 1 a A\{0} 1 z a π ), 21) 1 θ 1 q i ) 1 22) i=1

326 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 is a fundamental period of Carlitz Throughout this paper we will fix a choice of θ) 1 so that π is a well-defined element in k 215 The nth Carlitz polylogarithm The nth Carlitz polylogarithm is the series z qi L n i=0 i which converges -adically for all z C with z < θ Its value at particular z = α 0 is called the nth polylogarithm of α It can be easily checked that these polylogarithms are always nonzero In the transcendence theory we are interested in those polylogarithms of α k, as analogues of classical logarithms of algebraic numbers 22 Review of Papanikolas theory nq We follow Papanikolas [7] cf also [1,3]) in working with t-motives 221 Review of t-motives Let k[t,σ] be the polynomial ring in variables t and σ such that for all c k: ct = tc, σt = tσ, and σc= c 1/q σ An Anderson t-motive is a left k[t,σ]-module M which is free and finitely generated both as a left k[t]-module and left k[σ ]-module, and satisfying for integer N sufficiently large t θ) N M σ M 23) Let m Mat r 1 M) be a k[t]-basis of M Multiplication by σ on M is given as σm) = Φm, for some matrix Φ Mat r k[t]) Note that 23) implies det Φ = ct θ) N for some c k and some N N It is also convenient to bring in left kt)[σ,σ 1 ]-modules which are finite-dimensional over kt) These are called pre-t-motives They form an abelian category Each Anderson t-motive M, m,φ)as above corresponds to a pre-t-motive M := kt) k[t] M with the canonical kt)-basis m = 1 m Mat r 1 M): forf Mat 1 r kt)), and σf m) = f 1) Φ m, σ 1 f m) = f 1) Φ 11) m

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 327 We are interested in rigid analytically trivial pre-t-motives For such a pre-t-motive M of dimension r over kt) there exists Ψ GL r L) such that σψ)= Ψ 1) = ΦΨ, 24) where Φ GL r kt)) represents multiplication by σ on M with respect to a basis of M over kt), and L is the fraction field of the Tate algebra T in the variable tthisψ is said to be a rigid analytic trivialization of the matrix Φ Note that if Ψ is also a rigid analytic trivialization for Φ, then Ψ 1 Ψ GL r F q t)) cf [7, Lemma 426b)]) Rigid analytically trivial pre-t-motives that can be constructed from Anderson t-motives using direct sums, subquotients, tensor products, duals and internal Hom s, are called t-motives These t-motives form a neutral Tannakian category T over F q t) By the Tannakian duality, for each t-motive M of dimension r over kt), the Tannakian subcategory generated by M is equivalent to the category of finite-dimensional representations over F q t) of some algebraic subgroup Γ M GL r defined over F q t) This algebraic group Γ M is called the Galois group of the t-motive M The σ -semilinear equation 24) is viewed as an analogue of systems of linear differential equations with column vectors of Ψ as solutions Suppose that the solutions are in fact everywhere convergent power series in t with coefficients lying in finite extensions of k, one can substitute θ for the variable t We then call Ψθ) 1 GL r k ) a period matrix of the motive M, where r is the dimension of M over kt) The entries of Ψθ) generate a field over k whose transcendence degree over k is an analytic invariant of this motive M We make use of the fundamental theorem of Papanikolas [7, Theorem 117] to compute the transcendence degrees which are of interests to us Theorem 21 Let M be a t-motive and let Γ M be its Galois group Suppose that Φ GL r kt)) Mat r k[t]) represents multiplication by σ on M and that det Φ = ct θ) s,c k Let Ψ be a rigid analytic trivialization of Φ in GL r T) Mat r E) and let L be the field generated by the entries of Ψθ)over k Then dim Γ M = trdeg k L Papanikolas [7] has furthermore developed a Galois theory for the σ -semilinear equations, in analogy with the classical differential Galois theory This provides a vital description of the algebraic group Γ M in terms of a fundamental solution Ψ of 24) 222 Review of the Galois theory of systems of σ -semilinear equations A σ -algebra is a commutative kt)-algebra Σ that also has a compatible structure as a left kt)[σ,σ 1 ]-module Moreover, we require that the σ -action is a ring homomorphism A σ -ideal of Σ is an ideal that is also a kt)[σ,σ 1 ]-submodule Suppose that Φ GL r kt)) gives multiplication of σ on t-motive M with rigid analytic trivialization Ψ GL r L) satisfying σψ)= ΦΨ Let X := X ij ) be an r r matrix whose entries are independent variables X ij, and set Δ := detx) Welet Σ 0 := kt) [ X, Δ 1] = kt) [ X ij,δ 1]

328 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 Elements h Σ 0 will be denoted hx) := hx ij ) Furthermore, one makes Σ 0 into a σ -algebra in the unique way so that kt) acts on Σ 0 by usual left multiplication; σ acts on Σ 0 by setting σx:= ΦX Since σψ)= ΦΨ, the following kt)-algebra homomorphism ν Ψ is a σ -algebra homomorphism Let ν Ψ : Σ 0 L X ij Ψ ij p Ψ := kerν Ψ ) Note that p Ψ is a σ -ideal in Σ 0 Thus we have an isomorphism of σ -algebras, Σ 0 /p Ψ = kt) [ Ψ, ΔΨ ) 1 ] =: Σ Ψ The ring Σ Ψ is called a Picard Vessiot extension of kt) for Φ By [7, 522], its base extension Σ Ψ kt) kt) is always an integral domain Given a σ -algebra Σ, fixing any F q t)-algebra R, weletσ R) := R Fq t) Σ with σ -action given by σc h) = c σh), c R, h Σ Note that Σ R) Ψ is a σ -algebra and one always regards R as a σ -subalgebra of ΣR) Ψ with trivial σ -action We let P R) Ψ be the kernel of the following σ -algebra homomorphism Σ R) 0 L R) X ij Ψ ij Since R is a vector space over F q t), one obtains and P R) Ψ = R F q t) p Ψ Σ R) 0 /P R) Ψ = Σ R) Ψ as σ -algebra isomorphism The group GL r R) acts on Σ R) 0 by γ h := hxγ ), for γ GL r R), h Σ R) 0 25) This induces σ -automorphism of Σ R) Ψ leaving kt) R) fixed:

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 329 κ R γ ) : Σ R) Ψ ΣR) Ψ h γ h Finally set Γ Ψ R) := { γ GL r R); γ h P R) Ψ } for all h PR) 26) We collect the results proved by Papanikolas [7, Theorems 442, 446, 5212, 5214 and 5410] as the following theorem Theorem 22 The functor Γ Ψ is representable by an affine algebraic group scheme which is isomorphic to the motivic Galois group Γ M over F q t) Moreover this group scheme is geometrically connected and smooth over F q t) Also for any F q t)-algebra R, the following is an isomorphism functorial in R 3 Polylogarithms and t-motives κ R : Γ Ψ R) Aut σ Σ R) Ψ /kt)r)) γ κ R γ ) Let n be a fixed positive integer Given α 1,,α m k such that nq α i < θ, for 1 i m, our aim is to determine all the algebraic relations among the nth Carlitz polylogarithms of α 1,,α m Following Papanikolas [7] one proceeds to construct a t-motive M = Mα 1,,α m ) with the polylogarithms in question as periods The motivic Galois group of this t-motive M is the key and the miracle is that this algebraic Galois group can be explicitly computed 31 Constructing t-motives from polylogarithms 311 The entire function Ω and its functional equation We define the power series Ωt) := θ) q i=1 1 t θ qi Ψ ) k t C t One checks that Ωt) E Furthermore, Ωt) satisfies the functional equation and where π is Carlitz period Ω 1) t) = t θ)ωt) 31) Ωθ) = 1 π,

330 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 The function Ω gives a rigid analytic trivialization of the Carlitz motive C Thisisthetmotive with underlying kt)-vector space kt) itself and σ acts by σf := t θ)f 1) for f C For n 1, the nth tensor power C n := C kt) kt) C of the Carlitz motive also has kt) as underlying space, but with σ -action σf := t θ) n f 1) for f C n Thus its rigid analytic trivialization is provided by the function Ω n Since π is transcendental over k, by Theorem 21 the Galois group Γ C n of the nth tensor power of the Carlitz motive C n is of dimension one inside GL 1 and hence Γ C n is isomorphic to G m over F q t) Finally we introduce the direct sum of t-motives s n=1 C n The defining matrix for this t-motive is the diagonal t θ) 0 0 0 t θ) 2 0 Φ := 0 0 t θ) s It has rigid analytic trivialization Ω 0 0 0 Ω 2 0 0 0 Ω s Its Galois group still has dimension one by Theorem 21 By 26), it is not difficult to show that this algebraic Galois group is G m embedded into the diagonal torus of GL s via 312 The series L α,n and its functional equation Given n N and any α k with α < θ L α,n t) := α + x 0 0 0 x 2 0 x 32) 0 0 x s i=1 nq, we consider the following power series α qi t θ q ) n t θ q2 ) n t θ qi ) n 33) Substituting θ for t, one sees that L α,n θ) is exactly the nth Carlitz polylogarithm of α From the defining series one has that L 1) α,n t) = α 1) + L α,nt) t θ) n

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 331 By 31), we obtain Ω n L α,n ) 1) = α 1) t θ) n Ω n + Ω n L α,n 34) More generally, given m nonzero algebraic numbers α 1,,α m k with α i < θ we let L αi,n be the series as in 33), i = 1,,m We define t θ) n 0 0 α 1) 1 t θ) n 1 0 Φ = Φα 1,,α m ) := GL ) ) m+1 kt) Matm+1 k[t], α m 1) t θ) n 0 1 Ω n 0 0 Ω n L α1,n 1 0 Ψ = Ψα 1,,α m ) := GL m+1t), Ω n L αm,n 0 1 by 34), then we have Note that all the entries of Ψ are inside E nq, Ψ 1) = ΦΨ 35) 32 The t-motive M Φ nq Given α 1,,α m k with α i < θ,letφ = Φα 1,,α m ) We define M := M Φ to be the pre-t-motive whose underlying kt)-vector space is of dimension m + 1 with basis x 0,x 1,,x m on which σ acts by the following rule x 0 x 0 x 1 σ = Φα x 1 1,,α m ) x m x m By 35), the matrix Ψα 1,,α m ) is a rigid analytic trivialization of M Φ ThisM Φ is in fact a t-motive Moreover, following Papanikolas [7], the algebraic dimension of its Galois group can be easily computed from the polylogarithms L α1,nθ),,l αm,nθ): Theorem 31 Let α 1,,α m k satisfy α i < θ for i = 1,,m Let M = M Φ be the t-motive constructed from the matrix Φα 1,,α m ), with Galois group Γ M Then nq dim Γ M = dim k N, where N is the k-linear span of π n,l α1,nθ),, L αm,nθ)

332 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 We leave the detailed proof to the appendix As a consequence one can extend the algebraic independence of Carlitz logarithms cf [7, Theorem 742]) to algebraic independence of polylogarithms: Corollary 32 Let α 1,,α m k satisfy α i < θ for i = 1,,mIfL α1,nθ),, L αm,nθ) are linearly independent over k, then they are algebraically independent over k In particular for any α k nq with α < θ, the polylogarithm L α,n θ) is transcendental Proof Let Φ = Φα 1,,α m ) and M = M Φ be the t-motive associated to these polylogarithms L αi,nθ), i = 1,,m, as in 32, with Galois group Γ M Let and nq L = k π n,l α1,nθ),, L αm,nθ) ), N = k-linear span of { π n,l α1,nθ),, L αm,nθ) } By assumption, we see that m dim k N m + 1 By Theorem 21 and the above theorem, trdeg k L = dim Γ M = dim k N If π n,l α1,nθ),, L αm,nθ) are linearly independent over k, then they are algebraically independent over k Otherwise by hypothesis we have L = k L α1,nθ),, L αm,nθ) ) which has transcendence degree m over k Thus in either case L α1,nθ),, L αm,nθ) are algebraically independent over k Another consequence of Theorem 31 is the following corollary: Corollary 33 Let α 1,,α m k satisfy α i < θ k[x 1,,X m+1 ] be a nonconstant polynomial If the value nq for i = 1,,m Let f f L α1,nθ),,l αm,nθ), π n) is nonzero then it must be transcendental over k Proof Take a k-linearly independent subset { Lαj1,n θ),, L αjl,n θ), π n} { L α1,nθ),,l αm,nθ), π n} Let L = k π n,l α1,nθ),, L αm,nθ)) which contains β = f L α1,nθ),,l αm,nθ), π n)

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 333 We have trdeg k L = trdeg k k L αj1,nθ),,l αjl,nθ), π n) = l + 1 by Theorem 21 and the above theorem On the other hand one can also write β = g L αj1,nθ),,l αjl,nθ), π n) for some nonconstant polynomial g k[x 1,,X l+1 ] Since { Lαj1,n θ),, L αjl,n θ), π n} is a transcendental basis for L over k, β has to be transcendental over k 4 Algebraic independence of the special zeta values Let n range through positive integers We are interested in the zeta values cf [6,8]) 41 Special zeta values and π ζ C n) k k We recall first the factorials of Carlitz Write down the q-adic expansion i=0 n i q i of n, and let Γ n+1 = i=0 D n i i The Bernoulli Carlitz numbers B n in A are then given by the following expansion from the Carlitz exponential series z exp C z) = n=0 B n z n Γ n+1 By taking logarithmic derivative of 21), Carlitz [4] arrives at the Euler Carlitz relations among the zeta values and powers of π: Theorem 41 For all positive integer n divisible by q 1, one has ζ C n) = B n Γ n+1 π n 41) For those values ζ C n) with n not divisible by q 1, ζ C n)/ π n are obviously not in k so not in k, as can be seen from the infinite product expansion 22) To study these mysterious values, Anderson Thakur cf [2, the proof of Theorem 383]) relate the values ζ C n) to the nth tensor power of the Carlitz module, thereby obtain the following formula connecting ζ C n) with the nth polylogarithms of 1,θ,,θ l n with l n < nq

334 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 Theorem 42 Anderson Thakur) Given any positive integer n, one can find a finite sequence h n,0,,h n,ln A, l n < nq, such that the following identity holds Γ n ζ C n) = l n i=0 h n,i L θ i,nθ) 42) These polynomials h n,i come from a generating function identity with variables x,y: j i=1 1 θ ) 1 qj y qi ) H x qj n y) := x n, h n,i y i Γ n+1 j=0 D j n=0 H n 1 y) := i Having expressed ζ C n) as a linear polynomial of these nth polylogarithms with coefficients from k by 42), one then derives from Corollary 33 the following main theorem of [11] Corollary 43 For any positive integer n, ζ C n) is transcendental over k We can now prove the algebraic independence of π with the value ζ C n), for positive integer n not divisible by q 1 This is far more stronger than the transcendence of the ratio ζ C n)/ π n proved also in [11] Given positive integer n not divisible by q 1, set N n := k- span { π n,l 1,n θ), L θ,n θ),, L θ ln,n θ)} By 42) we have ζ C n) N n and m n + 2 := dim k N n 2 since ζ C n) and π n are linearly independent over k For each such n we fix once for all a finite subset {α n0,,α nmn } { 1,θ,,θ l } n such that both { π n, L n0 θ),,l nmn θ) } and { π n,ζ C n), L n1 θ),,l nmn θ) } are bases of N n over k, where L nj t) := L αnj,nt) for j = 0,,m n This can be done because of Theorem 42 To each such odd integer n,takem Φn to be the t-motive defined by the matrix Φ n = Φα n0,,α nmn ) The Galois group of this motive has dimension m n + 2 by Theorem 31 which by Theorem 21 also equals to the transcendence degree over k of k π n, L n0 θ),, L nmn θ) ) = k π n,ζ C n), L n1 θ),,l nmn θ) )

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 335 In particular, the elements π n,ζ C n), L n1 θ),, L nmn θ) are algebraically independent over k Therefore, π and ζ C n) are algebraically independent over k This completes the proof of the following Theorem 44 For any positive integer n, q 1 n, π and ζ C n) are algebraically independent over k Finally we note that if n q 1, then formula 42) reduces simply to the fact that ζ C n) equals to the nth Carlitz polylogarithm of 1 in this very special case One has l n = 0 and dim k N n = 2 for all n<q 1 42 Direct sums and the main theorem For each n not divisible by q 1, we have defined a t-motive M Φn in the previous section based on Theorem 42 The Galois group of this t-motive for the positive integer n has dimension m n + 2 which is the same as the transcendence degree over k of the field k π n,ζ C n), L n1 θ),,l nmn θ) ) From Section 312, a rigid analytic trivialization of M Φn is given by the matrix Thus the following functional equation holds Ω n 0 0 Ω n L n0 1 0 Ψ n := Ω n L nmn 0 1 Ψ 1) n = Φ n Ψ n Given positive integer s, wesetus):= {1 n s p n, q 1 n} Define diagonal block matrices Φ s) := Φ n, n Us) Ψ s) := n Us) Ψ n The matrix Φ s) defines a t-motive M s) := M Φs) which is the direct sum of the t-motives M Φn with n Us) Clearly Ψ s) gives a rigid analytic trivialization for M s) We are interested in the Galois group Γ s) of this t-motive which is identified with the algebraic group Γ Ψs) by Theorem 22 In particular, we contend that

336 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 Theorem 45 Fix any s N, we have dim Γ s) = 1 + n Us) m n + 1) The proof of this theorem occupies the next section Applying Theorem 21 to Γ s), we find that 1 + n Us) m n + 1) is exactly the transcendence degree over k of the following field: k π, n Us) { Ln0 θ),, L nmn θ) }) = k π, n Us) { ζc n), L n1 θ),, L nmn θ) }) It follows that the set { π} n Us) { ζc n), L n1 θ),, L nmn θ) } is algebraically independent over k, hence also {ζ C n) n Us)} is algebraically independent over k Counting cardinality of Us) we obtain Corollary 46 For any positive integer n, the transcendence degree of the field k π,ζ C 1),,ζ C n) ) over k is n n/p n/q 1) + n/pq 1) + 1 43 The Galois group Γ Ψs) Let ls) := n Us) m n + 2) Define G s) to be the algebraic subgroup of GL ls) over F q t) which consists of all diagonal block matrices of the form 1 0 0 [ ] 1 0 1 0 1 0 1 where the block matrix at the position corresponding to n Us) has the size m n + 2 Thus G s) is isomorphic to the direct product of n Us) m n + 1) copies of G a canonically Since the Carlitz motive C is contained in M Φ1, it is also contained in M s) for any positive integer s By the Tannakian category argument, we have a surjective map of algebraic groups over F q t) π : Γ Ψs) G m which coincides with the natural projection on the upper left corner position of the first 2 2matrix, since by Theorem 22 we see that the restriction of the action of any element γ Γ Ψs) F q t))

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 337 to Fq t) ) ΣΩ := F q t) Fq t) kt) [ Ω,Ω 1] is equal to the action of the upper left corner of γ LetV s) be the kernel of π so that one has an exact sequence of algebraic groups 1 V s) Γ Ψs) G m 1 43) From the σ -semilinear equation satisfied by Ψ s) we find that V s) G s) We claim that in fact V s) = G s) To prove this claim we introduce a G m -action on G s) On the block matrix at the position corresponding to n Us) it is defined by 1 0 0 1 0 0 1 0 a a n 1 0, 0 1 a n 0 1 for a G m Fq t) ) 44) From the fact that n Us) C n is contained in M s), writing down the coordinates for the Galois group shows that the restriction of the above G m -action to V s) agrees with the conjugation of G m on V s) coming from 43) On the other hand for each n Us), letting Γ [n] be the Galois group of M Φn,wealsohave an exact sequence of algebraic groups coming from the fact that C n is contained in M Φn : 1 V [n] Γ [n] G m 1 By our construction each V [n] is isomorphic to G m n+1 a In view of the fact that M Φn is contained in M s) too, this leads to the commutative diagram: 1 V s) ϕ n Vs) Γ Ψs) ϕ n G m χ n 1 45) 1 V [n] Γ [n] G m 1, where ϕ n is the natural projection, ϕ n Vs) is the restriction of ϕ n to V s) and χ n : G m G m is the character defined by a a n Because here n is always prime to the characteristic, we deduce from 45) that ϕ n V s) ) = V [n] for all n Us) To complete the proof of the equality V s) = G s), suppose that V s) is of codimension r>0 in G s) We identify G s) with the product space n Us) G [n] := n Us) G m n+1 a, with block matrix at the position corresponding to n Us) identified with points in G m n+1 a having coordinates x n0,,x nmn ) For each selection J of r double indices ij with i Us)

338 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 and 0 j m i, we define W J ) to be the linear subspace of G s) of codimension r consisting of points whose coordinates x ij vanish for all indices ij selected to J LettheJ -coordinate space in G s) be the J -dimensional vector subgroup consisting of points whose coordinates all vanish except those x ij with ij selected to J, where J is the cardinality of J Since V s) is a closed subgroup having codimension r, we can select J so that V s) has a zero-dimensional intersection with the J -coordinate space Since this intersection is closed under the G m -action 44), it must be zero It follows that with such a selection of J, the natural projection from G s) to W J ) induces on V s) an isomorphism of vector groups For each n Us), composing the inverse of this isomorphism with the morphism ϕ n in 45) gives surjective morphism f n from W J ) onto V [n] which is furthermore a surjective G m -morphism of vector groups The selected set of indices J must contain some ij,sayi = n Us) Then at the position corresponding to n the coordinate space G [n] W J ) G [n] The following lemma implies that f n map G [n ] W J ) to zero for all n n in Us) Therefore it must map G [n] W J ) onto V [n] which has dimension greater than G [n] W J ) This is impossible Hence we conclude that V s) = G s) holds Lemma 47 Take distinct n 1,n 2 N with p n 2 WeletH i be the vector group over F q t) with G m -action of weight n i, ie, a x := a n i x for a G m,x H i,i= 1, 2 If f : H 1 H 2 is a G m -morphism of vector groups, ie, f satisfies then f 0 fa v) = a fv), for a G m,v H 1, Proof It suffices to consider the case where both H 1 and H 2 are one-dimensional Suppose that f is nontrivial Since f is a morphism of additive groups, we may write fx)= Fix v F q t),wehavethat We define d c i X pi, c i F q t), 0 i d, c d 0 i=0 d c i a n 1 v ) p i = i=0 FZ):= d a n 2 c i v pi for all a F q t) i=0 d ci Z n 1 v ) p i Z n ) 2 c i v pi Fq t)[z] i=0 We note that F is a polynomial of positive degree since c d 0, v 0, n 1 n 2 and p n 2 From Fa)= 0 for all a F q t), we obtain a contradiction Therefore, f 0

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 339 Corollary 48 Fix any s N Then we have the exact sequence of algebraic groups: where V s) is isomorphic to the vector group 1 V s) Γ s) G m 1 n Us) G m n+1 a Furthermore, the conjugation action by G m on V s) has multi-weight n) n Us) Acknowledgments We wish to thank Matthew A Papanikolas and Dinesh S Thakur for many helpful discussions and comments concerning the contents of this paper The first author also thanks Ming-Lun Hsieh for helpful discussions Appendix A In this appendix, we would give the detailed proof of Theorem 31 by following closely Papanikolas ideas A1 The t-motive M Φ From now on in this appendix, we fix α 1,,α m k with α i < θ and let Φ := Φα 1,,α m ), Ψ := Ψα 1,,α m ) and M = M Φ be the pre-t-motive defined as in 32 Lemma A1 Fix any α 1,,α m k with α i < θ t-motive nq nq, then the pre-t-motive M is a Proof Cf Proposition 713 of [7]) We let M be a free k[t]-module of rank m + 1 with basis y 0,y 1,,y m We define the k[t,σ]-module structure of M by setting y 0 y 0 y 1 y 1 σ = t θ)φ y m Let C be the Anderson t-motive corresponding to the Carlitz motive C, ie, C is k[t] with σ -action given by σf := t θ)f 1) for f C Since t θ) n+1 0 0 α 1) 1 t θ) n+1 t θ) 0 t θ)φ =, α m 1) t θ) n+1 0 t θ) y m

340 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 we have the short exact sequence as k[t,σ]-modules 0 C n+1) M C m 0, where C m is the direct sum of m copies of C Since C n+1) and C are finitely generated over k[σ ], we have that M is finitely generated over k[σ ] Moreover, we observe that σ M = t θ) n+1 y 0,t θ)y 1,,t θ)y m k[t] and hence t θ) N M σ M for N n + 1 Thus M is an Anderson t-motive by Proposition 432 of [3] By using the functional equation of 35), we have ΩΨ ) 1) = t θ)φωψ) Therefore, M is a rigid analytically trivial Anderson t-motive Next, we observe that the following map kt) k[t] M C kt) M 1 y 0 1 x 0 1 y 1 1 x 1 1 y m 1 x m is an isomorphism of left kt)[σ,σ 1 ]-modules and hence C kt) M is a t-motive Tensoring with the dual of the Carlitz motive, we conclude that M is a t-motive A2 Determining the Galois group Γ MΦ We proceed to determine the algebraic Galois group of the motive M = M Φ constructed as above from the given α 1,,α m k Let G be the algebraic subgroup of GL m+1 over F q t) such that for any F q t)-algebra R, {[ ] } 0 GR) = GL I m+1 R) m By Theorem 22, this Galois group Γ M GL m+1 can be identified as Γ Ψ for given rigid analytic trivialization Ψ of Φ From the definition of Γ Ψ, one deduces immediately that Γ Ψ G We label the nontrivial coordinates of G as X 0,X 1,,X m Since the nth tensor power of the Carlitz motive C n is contained in M, we have a surjection of algebraic groups over F q t) which will be referred as the natural projection π : Γ Ψ G m

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 341 This map coincides with the natural projection on the X 0 -coordinate of G as in Section 43 We let V be the kernel of π so that we have an exact sequence of algebraic groups over F q t), 1 V Γ Ψ G m 1 The group V is a subgroup of the group of unipotent matrices of G, which itself is naturally isomorphic to G m a Thus we can think of V Gm a with coordinates X 1,,X m For any a F q t), we can choose u Γ Ψ F q t)) such that πu) = a We note that given any [ ] 1 0 γ = V F v I q t) ), m we have [ ] u 1 1 0 γu= V F av I q t) ) m Hence VF q t)) is a vector space over F q t) Moreover, we have the following cf [7, Proposition 723]) Lemma A2 With notations as above, the group V is a linear subspace of G m a over F qt) Proof We claim that the induced tangent map dπ :LieΓ Ψ Lie G m is nontrivial To prove this, we take a maximal torus T of Γ Ψ Fq t) F q t) whichisofdimension 1 Let π be the base extension of π to F q t) The restriction of π to T is an isomorphism as π is the projection on the position of upper left corner This implies that d π is nontrivial, hence dπ is nontrivial Since Γ Ψ and G m are smooth over F q t) Theorem 22), We note that The dimension argument implies that dim Fq t) Lie Γ Ψ = dim Γ Ψ, dim Fq t) Lie G m = 1 Ker dπ = Lie V dim Fq t) Lie V = dim V and hence V is smooth over F q t) Since V is defined over F q t), V is a linear subspace of G m a over F q t) Since V is smooth over F q t), Hilbert s Theorem 90 provides an exact sequence 1 V F q t) ) Γ Ψ Fq t) ) G m Fq t) ) 1

342 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 by [10, Section 185] Let b 0 F q t) \ F q, and fix a matrix b 0 0 0 b 1 1 0 γ = Γ Ψ Fq t) ) A1) b m 0 1 Now as Papanikolas observes that the Zariski closure in Γ Ψ of the cyclic group generated by γ is the line in G connecting γ to the identity matrix Since Γ Ψ is connected and of dimension 1 greater than the dimension of V, it follows that Γ Ψ equals to the affine linear space spanned by V and the line in G connecting γ to the identity matrix Moreover, it yields the following explicit description of Γ Ψ cf [7, Proposition 725]) Theorem A3 Suppose F 1,,F s F q t)[x 1,,X m ] are linear forms defining V, and suppose that γ Γ Ψ F q t)) is given as in A1) Then the following linear polynomials in F q t)[x 0,X 1,,X m ], G i := b 0 1)F i F i b 1,,b m )X 0 1), i = 1,,s, define the algebraic group Γ Ψ A3 Algebraic relations among polylogarithms We can now collect all the algebraic relations of given nth Carlitz polylogarithms of algebraic elements in k We consider ν Ψ : kt) [ X 0,X 1 0,X ] 1,,X m L X 0 Ω n X i Ω n L αi,n, i = 1,,m, p Ψ := Ker ν Ψ, Σ Ψ := Im ν Ψ Let Z Ψ := Spec Σ Ψ By [7, Theorem 5214], Z Ψ kt) kt) is a principal homogeneous space for Γ Ψ Fq t) kt) over kt) It follows in particular that Z Ψ and Γ Ψ are isomorphic over kt) One can also get explicit defining equations for the variety Z Ψ over kt) cf [7, 731]): Lemma A4 Suppose G i, i = 1,,s are defining linear polynomials for Γ Ψ over F q t) as in Theorem A3 Then the ideal p Ψ is generated by polynomials of the form H i := G i f i X 0, i = 1,,s, with f i kt) Proof Since Γ Ψ is an affine linear space, Z Ψ is also an affine linear space Moreover, as Z Ψ is defined over kt), Z Ψ kt)) Thus we can choose

C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 343 a 0 0 0 a 1 1 0 ξ = Z ) Ψ kt) a m 0 1 such that Z Ψ kt) ) = ξ ΓΨ kt) ) Therefore, one has that the linear polynomials in kt)[x 0,X 1 0,X 1,,X m ], H i := G i X 0 G i a 0,a 1,,a m )/a 0, are defining polynomials for Z Ψ, ie, p Ψ = H 1,,H s ) i = 1,,s, Finally we are ready for the algebraic relations among Carlitz polylogarithms: Theorem A5 Let α 1,,α m k satisfy α i < θ for i = 1,,m Let M = M Φ be the t-motive constructed from Φα 1,,α m ), with rigid analytic trivialization Ψ for Φ, and natural projection π : Γ Ψ G m Let V be the kernel of π Choose b 0 F q t) \ F q, and fix We have 1) Let nq b 0 0 0 b 1 1 0 γ = Γ Ψ Fq t) ) b m 0 1 F = c 1 X 1 + +c m X m, c 1,,c m F q t), be any linear form vanishing on V, then the following relation holds: b0 θ) 1 ) m m c i θ)l αi,nθ) c i θ)b i θ) π n = 0 i=1 2) Every k-linear relation among π n,l α1,nθ),, L αm,nθ) is a k-linear combination of the relations from part 1) Note that Theorem 31 follows immediately from part 1) of this theorem Indeed 1) implies that dim k N dim Γ Ψ On the other hand, i=1 by Theorems 21 and 22 dim k N trdeg k k π n,l α1,nθ),, L αm,nθ) ) = dim Γ Ψ = dim Γ M

344 C-Y Chang, J Yu / Advances in Mathematics 216 2007) 321 345 Proof of Theorem A5 Cf [7, Theorem 732]) By Lemma A4, we can choose f kt) so that H := G fx 0 p Ψ Wehave H Ω n,ω n L α1,n,,ω n L αm,n) = G Ω n,ω n L α1,n,,ω n L αm,n) fω n = 0 Then σg Ω n,ω n L α1,n,,ω n ) L αm,n = Ω n G t θ) n 1,α 1) 1 t θ) n,,α m 1) t θ)n) + fω n Fb 1,,b m )Ω n Thus, = σ fω n) = f 1) t θ) n Ω n t θ) n f 1) f = G t θ) n 1,α 1) 1 t θ) n,,α m 1) t θ)n) Fb 1,,b m ) k[t] We note that f and f 1) are regular at t = θ Hence We therefore obtain i=1 m m fθ)= G 1, 0,,0) t=θ + c i θ)b i θ) = c i θ)b i θ) i=1 m m b 0 1) c i Ω n L αi,n c i b i Ω n 1 ) fω n = H Ω n,ω n L α1,n,,ω n L αm,n) = 0 i=1 Dividing through by Ω n and evaluating at t = θ, we arrive at part 1) Part 2) is deduced from part 1) and Theorem 31 Let s = m dim V Pick s defining linear forms for V such that the matrix B V Mat s m F q t)) which is formed with the coefficients of these defining linear forms of V has rank s From 1), we obtain sk-linear relations among { Lα1,nθ),, L αm,nθ), π n} i=1 whose coefficients give us a matrix P Mat s m+1) k)thes m matrix which is formed with the first m columns of P is a k-multiple of B V θ) Since t and θ are independent variables, it follows that rank P = s and hence dim k N m + 1 s Anyk-linear relation not from part 1) would imply dim k N<m+ 1 s = dim Γ Ψ which contradicts Theorem 31

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