DETERMINATION OF ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES IN POSITIVE CHARACTERISTIC

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1 DETERMINATION OF ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES IN POSITIVE CHARACTERISTIC CHIEH-YU CHANG AND JING YU Abstract As 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 p-th power relations 1 Introduction Let θ be a variable and A + be the set of all monic polynomials in F q [θ] Here F 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: (11) ζ C (n) := 1 a F q(( 1 )), n =1, 2, n θ a A + These zeta values were introduced in 1935 by L Carlitz [3], where he discovered the fact that there is a constant π algebraic over F q (( 1 θ )) such that ζ C(n)/ 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 [7], is a fundamental period of the socalled Carlitz module Call the positive integer n even provided it is a multiple of The ratios ζ C (n)/ π n for even n involve what are now called Bernoulli-Carlitz numbers, just as the case of Riemann zeta function the ratios ζ(n)/π n for even positive integer n can be expressed in terms of the classical Bernoulli numbers In late 1980 s, Anderson-Thakur [1] is able to relate the zeta values ζ C (n) to the n-th tensor powers of the Carlitz module for all positive integers n As a result of this big step forward, the second author is able to prove the transcendence of the zeta values ζ C (n) for all positive integers n [9], in particular for odd n (ie, n is not divisible by ) Later in [10], the second author is 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, p-th power (Frobenius) relations certainly are also there for all positive integers n and m: (12) ζ C (p m n)=ζ C (n) pm Date: February 10, Mathematics Subject Classification Primary 11J93; Secondary 11M38, 11G09 Key words and phrases Algebraic independence, Zeta values, t-motives The second author was supported by NSC Grant No M

2 2 CHIEH-YU CHANG AND JING YU Our purpose is to prove that these, ie, Euler-Carlitz relations and p-th 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/p(q 1) +1 Our tool for proving algebraic independence is a fundamental theorem of Papanikolas [5] which is 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 positive characteristic transcendence theory The t-module structure is the key for proving almost all the interesting linear independence results The break through in passing from 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 [2] For convenience, we simply call t-motives what have been called rigid analytically trivial dual t-motives in terminology of [2] and [5] In [5] Papanikolas succeeds to prove algebraic independence of Carlitz logarithms of algebraic functions that are linear independent over the rational function field Following his method closely we shall prove here, for any positive integer n, the algebraic independence of n-th Carlitz polylogarithms of algebraic functions that are linearly independent over the rational function field (Corollary 36) In the 1990 paper of Anderson-Thakur ([1], Theorem 383), there is yet another consequence of their interpretation of the value ζ C (n) in terms of the n-th tensor power of the Carlitz module This is a beautiful formula expressing ζ C (n) explicitly as linear combination of simple n-th Carlitz polylogarithms with rational coefficients This formula is not used in the previous works dealing with linear independence, but turns out to be the crucial in final determination of all algebraic relations among these zeta values Acknowledgements 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 21 Notations 2 Notations and Preliminaries 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 (( 1 θ )), the completion of k with respect to the place at infinity

3 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 3 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 formal Laurent series f = i a it i C ((t)) we define the σ-twist by the rule σ(f) :=f ( 1) = i a i t i The 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 ( 1) by the rule B ( 1) ij = B ( 1) ij In particular, for any matrix B with coefficients in C we have B ( 1) q ij = B 1 ij 213 Entire power series A power series f = i=0 a it i C [[t]] satisfies i lim i ai =0 and [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 214 Carlitz s theory: the logarithms and exponential Let L 0 := 1, L i := i j=1 (θ )(i =1, 2, ) θqj The Carlitz logarithms is the series z qi log C (z) := L i which converges -adically for all z C with z < θ q It satisfies the functional equation θlog 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 z qi exp C (z) := D i Here we set i=0 i=0 D 0 := 1 D i := i 1 j=0 (θqi θ qj )

4 4 CHIEH-YU CHANG AND JING YU The Carlitz exponential is an entire power series in z satisfying the functional equation exp C (θz) =θexp C (z) + exp C (z) q Moreover one has the product expansion (21) exp C (z) =z (1 z a π ), a 0 A where (22) π = θ( θ) 1 (1 θ 1 qi ) 1 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 n-th Carlitz polylogarithm The n-th 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 n-th polylogarithm of α It can be easily checked that these polylogarithms are always nonzero In transcendence theory we are interested in those polylogarithms of α k, as analogues of classical logarithms of algebraic numbers 22 Review of Papanikolas theory We follow Papanikolas [5] (cf also [2]) 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 left k[t]-module and left k[σ]-module, and satisfying for integer N sufficiently large (t θ) N M σm Let m Mat r 1 (M) be a k[t]-basis of M Multiplication by σ on M is given as σ(m) =Φm, for some matrix Φ Mat r (k[t]) It is also convenient to bring in left k(t)[σ, σ 1 ]-modules which are finite-dimensional over k(t) These are called pre-t-motives They form an abelian category Each Anderson t-motive M corresponds to a pre-t-motive k(t) k[t] M, with σ(f m) :=f ( 1) σm, f k(t),m M We are interested in rigid analytically trivial Anderson pre-t-motives For such a pre-t-motive there exists Ψ GL r (L) such that (23) σ(ψ) = Ψ ( 1) =ΦΨ, where Φ represents multiplication by σ with respect to a basis of this pre-t-motive, and L is the fraction filed of the Tate algebra T in the variable t This Ψ is said to be nq

5 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 5 a rigid analytic trivialization of the matrix Φ Rigid analytic 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 By Tannakian duality, for each t-motive M, 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 (23) is viewed as analogue of system 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 lie in finite extensions of k, one can substitute θ for the variable t We then call Ψ(θ) GL r (k ) a period matrix of the motive M The entries of Ψ(θ) 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 ([5], Theorem 117) to compute the transcendence degrees which are of interests to us Theorem 21 Let M be a t-motive with Galois group Γ M Suppose that Φ GL r ( k(t)) Mat r ( k[t]) represents multiplication by σ on M and that detφ = c(t θ) s,c k LetΨbe a rigid analytic trivialization of Φ in GL r (T) Mat r (E) and let L be the field generated by the entries of Ψ(θ) over k Then dim Γ M =trdeg k L Papanikolas [5] has furthermore developed a Galois theory for the σ-semilinear equations, in analogy with the classical differential Galois theory This provides vital description of the algebraic group Γ M in terms of a fundamental solution Ψ of (23) 222 Review of Galois theory of systems of σ-semilinear equations A σ-algebra is a commutative k(t)-algebra Σ that also has a compatible structure as a left k(t)[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 k(t)[σ, σ 1 ]-submodule Suppose Φ GL r ( k(t)) gives multiplication of σ on t-motive M with rigid analytic trivialization Ψ GL r (L) satisfying σ(ψ) = ΦΨ Let X := (X ij )beanr r matrix whose entries are independent variables X ij, and set := det(x) We let Σ 0 := k(t)[x, 1 ]=k(t)[x ij, 1 ] Elements h Σ 0 will be denoted h(x) :=h(x ij ) Furthermore, one makes Σ 0 into a σ-algebra in the unique way so that k(t) acts on Σ 0 by usual left multiplication; σ acts on Σ 0 by setting σx := ΦX Since σ(ψ) = ΦΨ, the following k(t)-algebra homomorphism ν Ψ is a σ-algebra homomorphism Let ν Ψ : Σ 0 L X ij Ψ ij p Ψ := ker(ν Ψ )

6 6 CHIEH-YU CHANG AND JING YU Note that p Ψ is a σ-ideal in Σ 0 Thus we have an isomorphism of σ-algebras, Σ 0 /p Ψ k(t)[ψ, (Ψ) 1 ]=:Σ Ψ The ring Σ Ψ is called a Picard-Vessiot extension of k(t) for Φ By [5] 522, its base extension Σ Ψ k(t) k(t) is always an integral domain Given a σ-algebra Σ, fix any F q (t)-algebra R, weletσ (R) := R Fq(t) Σ with σ-action given by σ(c h) =c σ(h), c R, h Σ Note that Σ (R) Ψ is a σ-algebra and one always regards R as a σ-subalgebra of Σ(R) 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 P (R) Ψ = R F q(t) p Ψ and Σ (R) 0 /P (R) Ψ Σ(R) Ψ as σ-algebra isomorphism The group GL r (R) acts on Σ (R) 0 by (24) γ h := h(xγ), for γ GL r (R),h Σ (R) 0 This induces σ-automorphism of Σ (R) Ψ leaving k(t) (R) fixed: κ R (γ) : Σ (R) Ψ Σ (R) Ψ h γ h Finally set (25) Γ Ψ (R) :={γ GL r (R); γ h P (R) Ψ for all h P(R) Ψ } We collect the results proved by Papanikolas ([5], Theorem 442, Theorem 446, Theorem 5212, Theorem 5214 and Theorem 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 κ R : Γ Ψ (R) Aut σ (Σ (R) Ψ /k(t)(r) ) γ κ R (γ) 3 Polylogarithms and t-motives 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 n-th Carlitz polylogarithms of α 1,,α m Following Papanikolas [5] one proceeds to construct a t-motive M = M(α 1,,α m ) with the polylogarithms in question as periods The

7 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 7 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 Ω(t) := (t θ qi )=( θ) q (1 t ) k [[t]] C [[t]] θ qi One checks that Ω(t) E Furthermore, Ω(t) satisfies the functional equation (31) Ω ( 1) (t) =(t θ)ω(t) and Ω(θ) = 1 π, where π is Carlitz period The function Ω gives rigid analytic trivialization of the Carlitz motive C This is the t-motive with underlying k(t)-vector space k(t) itself and σ acts by σf := (t θ)f ( 1) for f C Forn 1, the n-th tensor power C n := C k(t) k(t) C of the Carlitz motive also has k(t) 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 From Tannakian theory, it is easy to check that the Galois group of the n-th tensor power of the Carlitz motive C n is isomorphic to G m over F q (t) This is equivalent to the well-known fact that π n is transcendental over k by Theorem 21 Finally we introduce the direct sum of t-motives s n=1c n The defining matrix for this t-motive is the diagonal (t θ) (t θ) 2 0 Φ:= 0 0 (t θ) s It has rigid analytic trivialization Ω Ω Ω s Its Galois group still has dimension one by Theorem 21 By (25), it is not difficult to show this algebraic Galois group is G m embedded into the diagonal torus of GL s via x x 2 0 (32) x 0 0 x s

8 8 CHIEH-YU CHANG AND JING YU 312 The series L α,n and its functional equation Given n N and any α k nq with α < θ, we consider the following power series (33) L α,n (t) :=α + α qi (t θ q ) n (t θ q2 ) n (t θ qi ) n Substitute θ for t, one sees that L α,n (θ) is exactly the n-th Carlitz polylogarithm of α From the defining series one has that L α,n ( 1) (t) =α ( 1) + L α,n(t) (t θ) n By (31), we obtain (34) (Ω n L α,n ) ( 1) = α ( 1) (t θ) n Ω n +Ω n L α,n In general, given m nonzero algebraic numbers α 1,,α m k with α i < nq θ, we let L αi,n be the series as in (33), i =1,,m We define Φ=Φ(α 1,,α m ):= (t θ) n 0 0 α ( 1) 1 (t θ) n 1 0 GL m+1(k(t)) Mat m+1 (k[t]), α ( 1) m (t θ) n 0 1 Ω n 0 0 Ω n L α1,n 1 0 Ψ=Ψ(α 1,,α m ):= GL m+1(t), Ω n L αm,n 0 1 by (34), then we have (35) Ψ ( 1) =ΦΨ Note that all the entries of Ψ are inside E 313 The t-motive M Φ From now on in this Section, we fix α 1,,α m k nq with α i < θ and define M = M Φ to be the pre-t-motive whose underlying k(t)-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 =Φ(α x 1 1,,α m ) x m x m Lemma 31 Fix any α 1,,α m k with α i < θ M is a t-motive nq, then the pre-t-motive

9 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 9 Proof (cf Proposition 713 of [5]) 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 α ( 1) 1 (t θ) n+1 (t θ) 0 (t θ)φ =, α ( 1) m (t θ) n+1 0 (t θ) 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 σm = (t θ) n+1 y 0, (t θ)y 1,,(t θ)y m k[t] and hence (t θ) N M σm for N n + 1 Thus M is an Anderson t-motive By using the functional equation of (35), we have (ΩΨ) ( 1) =(t θ)φ(ωψ) Therefore, M is a rigid analytically trivial Anderson t-motive Next, we observe that the following map k(t) k[t] M C k(t) M 1 y 0 1 x 0 1 y 1 1 x 1 1 y m 1 x m is an isomorphism of left k(t)[σ, σ 1 ]-modules and hence C k(t) M is a t-motive Tensoring with the dual of the Carlitz motive, we conclude that M is a t-motive 314 Determining the Galois group Γ MΦ We proceed to determining 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 G(R) = 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 y m

10 10 CHIEH-YU CHANG AND JING YU We label the nontrivial coordinates of G as X 0,X 1,,X m Since the n-th tensor power of the Carlitz motive C n is contained in M, by Tannakian category argument, we have a surjection of algebraic groups over F q (t) π :Γ Ψ G m More precisely, the surjection π coincides with the natural projection on the X 0 - coordinate of G, since by Theorem 22 we see that the restriction of the action of any element γ Γ Ψ (F q (t)) to Σ (Fq(t)) Ω := F q(t) k(t)[ω n n Fq(t), Ω n ] is equal to the action of the X 0 -coordinate of γ 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 G m a with coordinates X 1,,X m For any a F q (t), we can choose u Γ Ψ (F q (t)) such that π(u) =a We note that given any [ ] 1 0 γ = V (F v I q (t)), m we have [ ] u γu = V (F av I q (t)) m Hence V (F q (t)) is a vector space over F q (t) Moreover, we have the following proposition (cf [5], Proposition 723) Proposition 32 With notations as above, the group V is a linear subspace of G m a over F q (t) Proof We claim that the induced tangent map dπ : Lie Γ Ψ Lie G m is nontrivial To prove this, we take a maximal torus T of Γ Ψ Fq(t) F q (t) which is of dimension 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), dim Fq(t)Lie Γ Ψ = dim Γ Ψ, dim Fq(t)Lie G m =1 We note that Ker dπ = Lie V Dimension argument implies that 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)

11 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 11 Since V is smooth over F q (t), Hilbert s Theorem 90 provides an exact sequence 1 V (F q (t)) Γ Ψ (F q (t)) G m (F q (t)) 1 by ([8], section 185) Let b 0 F q (t) \ F q, and fix a matrix b b (36) γ = Γ Ψ(F q (t)) 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 [5], Proposition 725) Proposition 33 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 (36) 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, are defining polynomials for Γ Ψ 32 Algebraic relations among polylogarithms We consider ν Ψ : k(t)[x 0,X0 1,X 1,,X m ] L X 0 Ω n X i Ω n L αi,n,,,m, p Ψ := Ker ν Ψ, Σ Ψ := Im ν Ψ Let Z Ψ := SpecΣ Ψ By [5] Theorem 5214, Z Ψ k(t) k(t) is a principal homogeneous space for Γ Ψ Fq(t) k(t) overk(t) It follows in particular that Z Ψ and Γ Ψ are isomorphic over k(t) Another key consequence (cf [5] 731) is that one can also get explicit defining equations for the variety Z Ψ over k(t): Proposition 34 Suppose G i, i = 1,,s are defining linear polynomials for Γ Ψ over F q (t) 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 k(t) Proof Since Γ Ψ is an affine linear space, Z Ψ is also an affine linear space Moreover, as Z Ψ is defined over k(t), Z Ψ ( k(t)) Thus we can choose a a ξ = Z Ψ(k(t)) a m 0 1 such that Z Ψ (k(t)) = ξ Γ Ψ (k(t)) Therefore, one has that the linear polynomials in k(t)[x 0,X0 1,X 1,,X m ], H i := G i X 0 G i (a 0,a 1,,a m )/a 0, i =1,,s,

12 12 CHIEH-YU CHANG AND JING YU are defining polynomials for Z Ψ, ie, p Ψ =(H 1,,H s ) The following theorem extends Papanikolas theorem on k-linear relations among Carlitz logarithms Theorem 35 Let α 1,,α m k satisfy α i < θ for i = 1,,m Let M = M Φ be the t-motive constructed from Φ(α 1,,α m ), with rigid analytic trivialization Ψ for Φ We have (1) Let F = c 1 X c m X m, c 1,,c m F q (t), be a defining linear form for V so that G =(b 0 1)F F (b 1,,b m )(X 0 1), b 0,b 1,,b m F q (t),b 0 / F q, is a defining polynomial for Γ M Then the following relations holds: m m (b 0 (θ) 1) c i (θ)l αi,n(θ) c i (θ)b i (θ) π n =0 (2) Every k-linear relation among π n,l α1,n(θ),,l αm,n(θ) is a k-linear combination of the relations from part (a) (3) Let N be the k-linear span of π n, L α1,n(θ),,l αm,n(θ) Then dim Γ M = dim k N Proof (cf [5], Theorem 732) By Proposition 34, we can choose f k(t) so that H := G fx 0 p Ψ We have 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 F (b 1,,b m )Ω n = σ(fω n )=f ( 1) (t θ) n Ω n Thus, (t θ) n f ( 1) f = G((t θ) n 1,α ( 1) 1 (t θ) n,,α ( 1) m (t θ)n ) F (b 1,,b m ) k[t] We note that f and f ( 1) are regular at t = θ Hence m m f(θ) = G( 1, 0,, 0) t=θ + c i (θ)b i (θ) = c i (θ)b i (θ) We therefore obtain 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 Dividing through by Ω n and evaluating at t = θ, we arrive at part (1) Part (2) is deduced from part (1) and (3) 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 {L α1,n(θ),,l αm,n(θ), π n } nq

13 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 13 whose coefficients give us a matrix P Mat s (m+1) (k) The s 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 Any k-linear relation not from (a) would imply dim k N<m+1 s = dim Γ Ψ which contradicts part (3) For part (3), part (1) implies that dim k N m +1 s = dim Γ Ψ On the other hand, dim k N trdeg k k( π n,l α1,n(θ),,l αm,n(θ)) = dim Γ Ψ = dim Γ M by Theorem 21 and Theorem 22 One can now extend algebraic independence of Carlitz logarithms (cf [5], Theorem 742) to algebraic independence to polylogarithms: Corollary 36 Let α 1,,α m k satisfy α i < θ nq for i =1,,m If L α1,n(θ),,l αm,n(θ) are linear independent over k, then they are algebraic independent over k In particular for any α k nq with α < θ, the polylogarithm L α,n (θ) is transcendental Proof Let Φ = Φ(α 1,,α m ) and M = M Φ be the t-motive associated to these polylogarithms L ai,n(θ), i =1,,m, as in 313, with Galois group Γ M Let L = k( π n,l a1,n(θ),,l am,n(θ)), and N = k-linear span of { π n,l a1,n(θ),,l am,n(θ)} 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 a1,n(θ),,l am,n(θ) are linearly independent over k, then they are algebraically independent over k Otherwise by hypothesis we have L = k(l α1,n(θ),,l αm,n(θ)) which has transcendence degree m over k Thus in either case L α1,n(θ),,l αm,n(θ) are algebraically independent over k Another consequence of Theorem 35 is the following Corollary: Corollary 37 Let α 1,,α m k satisfy α i < θ for i =1,,mLet f k[x 1,,X m+1 ] be a nonconstant polynomial If the value f(l α1,n(θ),,l αm,n(θ), π n ) is nonzero then it must be transcendental over k Proof Take a k-linearly independent subset {L αj1,n(θ),,l αjl,n(θ), π n } {L α1,n(θ),,l αm,n(θ), π n } Let L = k( π n,l α1,n(θ),,l αm,n(θ)) which contains β = f(l α1,n(θ),,l αm,n(θ), π n ) nq

14 14 CHIEH-YU CHANG AND JING YU We have trdeg k L = trdeg k k(lαj1,n(θ),,l αjl,n(θ), π n )=l +1 by Theorem 21 and the above Theorem On the other hand one can also write β = g(l αj1,n(θ),,l αjl,n(θ), π n ) for some non-constant polynomial g k[x 1,,X l+1 ] Since {L αj1,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 [4] and [6]) ζ C (n) k k 41 Special zeta values and π We recall first the factorials of Carlitz Writing down the q-adic expansion i=0 n iq i of n, and let Γ n+1 = D ni 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) = z n B n Γ n=0 n+1 By taking logarithmic derivative of (21), Carlitz ([3]) 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 (41) ζ C (n) = B n π n Γ n+1 For those values ζ C (n) with n not divisible by, 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 [1], the proof of Theorem 383) relates the values ζ C (n) to the n-th tensor power of the Carlitz nodule, thereby obtain the following formula connecting ζ C (n) with the n-th polylogarithms of 1,θ,,θ ln with l n < nq 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 (42) Γ n ζ C (n) = l n i=0 i=0 h n,i L θi,n(θ) These polynomials h n,i come from a generating function identity with variables x, y: (1 j (θqj y qi ) j=0 D j x qj ) 1 := H n(y) n=0 Γ n+1 x n H n 1 (y) := i h n,iy i It is very important that for all n, the coefficients h n,i vanish for i nq so that the powers of θ needed for the polylogarithms in question fall in the right range Having

15 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 15 expressed ζ C (n) as linear polynomial of these n-th polylogarithms with coefficients from k by (42), one then derives from Corollary 37 the following main Theorem of [9] 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 integers 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 [9] Given 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 dim k N n 2 since ζ C (n) and π n are linear independent over k For each such n we fix once for all a finite set of distinct exponents 0 i 0 (n),,i mn (n) l n such that both { π n,l θ i 0 (n),n (θ),,l θ imn (n),n (θ)} and { π n,ζ C (n),l θ i 1 (n),n (θ),,l θ imn (n),n (θ)} are bases of N n over k This can be done because of Theorem 42 To each such odd integer n, take M Φn to be the t-motive defined by the matrix Φ n =Φ(θ i0(n),,θ imn (n) ) The Galois group of this motive has dimension m n + 2 by Theorme 35 which by Theorem 21 also equals to the transcendence degree over k of k( π n,l θ i 0 (n),n (θ),,l θ imn (n),n (θ)) = k( π n,ζ C (n),l θ i 1 (n),n (θ),,l θ imn (n),n (θ)) In particular, the elements π n,ζ C (n),l θ i 1 (n),n (θ),,l θ imn (n),n (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 face that ζ C (n) equals to the n-th 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 basing 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 θ i 1,n(θ),,L (n) θ imn (n),n (θ)), where 0 i 1 (n),,i mn (n) l n is a chosen finite set of exponents with bound l n coming from (42)

16 16 CHIEH-YU CHANG AND JING YU From Section 312, rigid analytic trivialization of M Φn is given by the matrix Ω n 0 0 Ω n L θ i 0 Ψ n := (n),n 1 0 Ω n L θ imn (n),n 0 1 Thus the following functional equation holds Ψ ( 1) n =Φ n Ψ n Given positive integer s, we set U(s) :={1 n s p n, q 1 n} Define diagonal block matrices Φ(s) := n U(s) Φ n, Ψ(s) := n U(s) Ψ n The matrix Φ(s) defines a t-motive M(s) :=M Φ(s) which is the direct sum of the t-motives M Φn with n U(s) Clearly Ψ(s) gives 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 Ψ(s) by Theorem 22 In particular, we contend that Theorem 45 Fix any s N, we have dim Γ(s) =1+ n U(s) (m n +1) The proof of this Theorem occupies the next Section Apply Theorem 21 to Γ(s), we find that 1 + n U(s) (m n + 1) is exactly the transcendence degree over k of the following field: k( π, n U(s) {L θ i 0 (n),n (θ),,l θ imn (n),n (θ)}) = k( π, n U(s) {ζ C(n),L θ i 1 (n),n(θ),,l θ imn (n),n (θ)}) It follows that the set { π} n U(s) {ζ C (n),l θ i 1 (n),n (θ),,l θ imn (n),n (θ)} is algebraically independent over k, hence also {ζ C (n) n U(s)} is algebraically independent over k Counting cardinality of U(s) 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/p(q 1) The Galois group Γ Ψ(s) Let l(s) := n U(s) (m n + 2) Define G(s) tobe the algebraic subgroup of GL l(s) over F q (t) which consists of all diagonal block matrices of the form [ ]

17 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 17 where the block matrix at the position corresponding to n U(s) has the size m n + 2 Thus G(s) is isomorphic to the direct product of n U(s) (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 As in Section 314 we have a surjective map of algebraic groups over F q (t) π :Γ Ψ(s) G m which coincides with the natural projection on the upper left corner position of the first 2 2 matrix Let V (s) be the kernel of π so that one has an exact sequence of algebraic groups over F q (t) (43) 1 V (s) Γ Ψ(s) G m 1 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 U(s) it is defined by (44) a 1 0 a n 1 0, for a G m(f q (t)) 0 1 a n 0 1 From the fact that n U(s) C n is contained is M(s), writing down the coordinates for the Galois group as in (32) implies 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 U(s), let Γ n be the Galois group of M Φn,we also have an exact sequence of algebraic groups over F q (t) 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 mn+1 a In view of the fact that M Φn is contained in M(s) too, this leads to the commutative diagram: (45) 1 V (s) Γ Ψ(s) 1 ϕ n V (s) 1 Vn Γd Gm 1, where ϕ n is the natural projection, ϕ n V (s) is the restriction of ϕ n to V (s) and χ d : 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 U(s) To complete the proof of the equality V (s) =G(s), suppose that V (s) is of codimension r>0ing(s) We identify G(s) with the product space G n := G mn+1 a, n U(s) ϕ n n U(s) with block matrices at the position corresponding to n U(s) identified with points in G mn+1 a having coordinates (x n0,,x nmn ) For each selection J of r double indices ij with i U(s) and1 j m n + 1, we define W (J) to be the linear subspace of G(s) of codimension r consisting of points whose coordinates x ij G m χ n

18 18 CHIEH-YU CHANG AND JING YU vanishes for all indices ij selected to J Let the J-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 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) tow (J) induces on V (s) an isomorphism of vector groups For each n U(s), 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 U(s) Then at the position corresponding to n the coordinate space G n W (J) G n The following Lemma implies that f n maps G n W (J) to zero for all n n in U(s) 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 Given distinct n 1,n 2 N with p n 2 We let H i be the vector group over F q (t) with G m -action of weight n i, ie, a x := a ni 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 f(a v) =a f(v), for a G m,v H 1, then f 0 Proof It suffices to consider the case where both H 1 and H 2 are one-dimensional Suppose that f is nontrivial Sine f is a morphism of additive groups, we may write d f(x) = c i X pi, c i F q (t) 1 i d, c d 0 i=0 Fix v F q (t), we have that d c i (a n1 v) pi = We define i=0 F (Z) := d a n2 c i v pi for all a F q (t) i=0 d (c i (Z n1 v) pi Z n2 c i v pi ) F q (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 F (a) = 0 for all a F q (t), we obtain a contradiction Therefore, f 0 Corollary 48 Fix any s N Then we have exact sequence of algebraic groups over F q (t): 1 V (s) Γ(s) G m 1 where V (s) is isomorphic to the vector group G mn+1 a n U(s) Furthermore, the conjugation action by G m on V (s) has multi-weight (n) n U(s)

19 ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 19 Finally we note that one can also use the Galois theory for systems of σ-semilinear equation to write another proof of Theorem 45 We do not discuss the details here References 1 Greg W Anderson and Dinesh SThakur, Tensor powers of the Carlitz module and zeta values Ann of Math 132 (1990), Greg W Anderson, W Dale Brownnawell and Matthew A Papanikolas, Determination of the algebraic relations among special Gamma-values in positive characteristic, Ann of Math 160 (2004), L Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math J 1 (1935), David Goss, Basic structures of function field arithmetic, Springer-Verlag, Berlin, Matthew A Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Dinesh S Thakur, Function field arithmetic, World Scientific Publishing, River Edge NJ, L I Wade, Certain quantities transcendental over GF(pn,x), Duke Math J 8 (1941), William C Waterhouse, Introduction to affine group schemes, Springer-Verlag, New York, Jing Yu, Transcendence and special zeta values in characteristic p, Ann of Math (2) 134 (1991), Jing Yu, Analytic homomorphisms into Drinfeld modules, Ann of Math (2) 145 (1997), Department of Mathematics, National Tsing Hua University, Hsinchu City 300, Taiwan ROC address: d917202@oznthuedutw Department of Mathematics, National Tsing Hua University and National Center for Theoretical Sciences, Hsinchu City 300, Taiwan ROC address: yu@mathctsnthuedutw