Algebra & Number Theory

Transcription

1 Algebra & Number Theory Volume No 1 Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu mathematical sciences publishers

2 msp ALGEBRA AND NUMBER THEORY 5:1(2011) Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu By analogy with the Riemann zeta function at positive integers, for each finite field p r with fixed characteristic p, we consider Carlitz zeta values ζ r (n) at positive integers n Our theorem asserts that among the zeta values in the set r=1 {ζ r(1), ζ r (2), ζ r (3), }, all the algebraic relations are those relations within each individual family {ζ r (1), ζ r (2), ζ r (3), } These are the algebraic relations coming from the Euler Carlitz and Frobenius relations To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed 1 Introduction 11 Motivic transcendence theory Classically, Grothendieck s period conjecture for abelian varieties predicts that the dimension of the Mumford Tate group of an abelian variety over should be equal to the transcendence degree of the field generated by its period matrix over Conjecturally, the Mumford Tate group is the motivic Galois group from Tannakian duality, and therefore Grothendieck s conjecture provides an interpretation of the algebraic relations among periods in question by way of motivic Galois groups We are concerned with the algebraic independence of special zeta values over function fields with varying finite constant fields in positive equal characteristic In the positive characteristic world, there is the concept of t-motives introduced by Anderson [1986], dual to the concept of t-modules The third author showed that the structure of t-modules is key for proving many interesting linear independence results about special values in this setting [Yu 1997] The breakthrough in passing from linear independence to algebraic independence by way of t-motives began The first author was supported by NSC and NCTS The second author was supported by NSF Grants DMS and DMS The third author was supported by NSC Grant No M MSC2000: primary 11J93; secondary 11M38, 11G09 Keywords: Algebraic independence, Frobenius difference equations, t-motives, zeta values 111

3 112 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu with Anderson, Brownawell and the second author, and in particular with the linear independence criterion of [Anderson et al 2004] (the ABP criterion) By introducing a Tannakian formalism for rigid analytically trivial pre-t-motives and relating it to the Galois theory of Frobenius difference equations, the second author [Papanikolas 2008] has shown that the Galois group of a rigid analytically trivial pre-t-motive is isomorphic to its difference Galois group Furthermore, the second author has successfully used the ABP criterion to show that the transcendence degree of the field generated by the period matrix of an ABP motive (that is, the pre-t-motive comes from a uniformizable abelian t-module) is equal to the dimension of its Galois group More generally, we say that a rigid analytically trivial pre-t-motive has the GP (Grothendieck period) property if the transcendence degree of the field generated by its period matrix is equal to the dimension of its Galois group (for more details of terminology, see Section 2) Using a refined version of the ABP criterion proved by the first author [Chang 2009], we observe that there are many pre-t-motives that are not ABP motives but that have the GP property This motivates us to introduce a method of uniformizing the Frobenius twisting operators with respect to different constant fields for those pre-t-motives that have the GP property The pre-t-motive we obtain in this way is defined over a larger constant field, but still has the GP property (see Corollary 224) This technique is very useful when dealing with the problem of determining all the algebraic relations among various special values of arithmetic interest in a fixed positive characteristic It is used in this paper to study special zeta values For another application to special arithmetic gamma values, see [Chang et al 2010] 12 Carlitz zeta values Let p be a prime, and let p r [θ] be the polynomial ring in θ over the finite field p r of p r elements Our aim is to determine all the algebraic relations among the zeta values ζ r (n) := a p r [θ] a monic 1 a n p r ((1/θ)) p((1/θ)), where r and n vary over all positive integers Each ζ r (n) lies in p ((1/θ)), since it is fixed by the automorphism ( a i (1/θ) i a p i (1/θ)i) : p ((1/θ)) p ((1/θ)) The study of these zeta values was begun by Carlitz [1935] For a fixed positive integer r, he discovered that there is a constant π r, algebraic over p r ((1/θ)), such that ζ r (n)/ π r n lies in p(θ) if n is divisible by p r 1 The quantity π r arises as a fundamental period of the Carlitz p r [t]-module C r, and Wade [1941] showed that π r is transcendental over p (θ) We say that a positive integer n is (p, r)-even if it is a multiple of p r 1 Thus the situation of Carlitz zeta values at ( p, r)-even positive integers is completely

4 Difference equations and algebraic independence of zeta values 113 analogous to that of the Riemann zeta function at even positive integers For these (p, r)-even n, we call the p (θ)-linear relations between ζ r (n) and π r n the Euler Carlitz relations Because the characteristic is positive, there are also Frobenius p-th power relations among these zeta values: for positive integers m, n, ζ r (p m n) = ζ r (n) pm Anderson and Thakur [1990] and Yu [1991; 1997] made several breakthroughs in understanding Carlitz zeta values Using the t-module method, the transcendence of ζ r (n) for all positive integers n, and in particular for odd n (that is, n not divisible by p r 1), was proved, and it was also proved that the Euler Carlitz relations are the only p (θ)-linear relations among {ζ r (n), π r m ; m, n } In [Chang and Yu 2007], the first and third authors used ABP motives instead of t-modules to show that for fixed r, the Euler Carlitz relations and the Frobenius p-th power relations account for all the algebraic relations over p (θ) among the Carlitz zeta values π r, ζ r (1), ζ r (2), ζ r (3), To complete the story of Carlitz zeta values, the next natural question is what happens if r varies Denis [1998] proved the algebraic independence of all fundamental periods { π 1, π 2, π 3, } as the constant field varies Thus, in view of [Chang and Yu 2007], one expects that for the bigger set of zeta values, {ζ r (1), ζ r (2), ζ r (3), }, r=1 the Euler Carlitz relations and the Frobenius p-th power relations still account for all the algebraic relations This is indeed the case, as we find from the following theorem (stated subsequently as Corollary 452) Theorem 121 Given any positive integers s and d, the transcendence degree of the field ( d ) p (θ) { π r, ζ r (1),, ζ r (s)} r=1 over p (θ) is d ( s r=1 s p s p r 1 + s p(p r 1) ) Outline Our strategy is to construct a pre-t-motive that has the GP property and whose period matrix accounts for the Carlitz zeta values in question In [Chang and Yu 2007], an ABP motive has already been constructed for Carlitz zeta values with respect to a fixed constant field The problem here is one concerning varying the constant fields in a fixed characteristic, and one has to uniformize Frobenius powers in order to apply the method developed in [Papanikolas 2008]

5 114 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu This paper is organized as follows In Section 2, we review Papanikolas theory and investigate the pre-t-motives that have the GP property Here we introduce the mechanism of uniformizing Frobenius twisting operators while taking direct sums Section 3 includes discussions about rigid analytically trivial pre-t-motives of type SV, that is, whose Galois groups are extensions of split tori by vector groups The heart of this section is Theorem 322, where we determine the dimensions of Galois groups of direct sums of pre-t-motives of type SV The pre-t-motive for Theorem 121 is constructed in Section 4, and we prove that it satisfies the conditions of Theorem 322 Finally, we calculate its dimension explicitly in Theorem 451, which then has Theorem 121 as direct consequence 21 Notation 2 t-motivic Galois groups 211 Table of symbols p := the finite field of p elements, p a prime number k := p (θ) := the rational function field in the variable θ over p k := p ((1/θ)), completion of k with respect to the infinite place k := a fixed algebraic closure of k k := the algebraic closure of k in k := completion of k with respect to the canonical extension of the infinite place := a fixed absolute value for the completed field with θ = p [[t]] := the ring of power series in the variable t over ((t)) := the field of Laurent series in the variable t over := { f [[t]] f converges on t 1} This is known as the Tate algebra := the fraction field of σ := σ p := ( i a it i i a1/p i t i) : ((t)) ((t)) a := the additive group GL r /F := for a field F, the F-group scheme of invertible r r square matrices m := GL 1, the multiplicative group 212 Block diagonal matrices Let A i Mat mi ( ) for i = 1,, n, and m := m m n We define n i=1 A i Mat m ( ) to be the canonical block diagonal matrix, that is, the matrix with A 1,, A n down the diagonal and zeros elsewhere 213 n-fold twisting For n and a formal Laurent series f = i a it i ((t)), we define the n-fold twisting f (n) := σ n ( f ) := i a pn i t i The n-fold twisting operation is an automorphism of the Laurent series field ((t)) that stabilizes several subrings, for example, k[[t]], k[t], and More generally, for any matrix B with entries in ((t)), we define B (n) by the rule B (n) i j = B (n) i j

6 Difference equations and algebraic independence of zeta values Entire power series A power series f = i=0 a it i [[t]] that satisfies i ai = 0 and lim 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 and, when restricted to k, f takes values in k The ring of the entire power series is denoted by 22 Pre-t-motives and the GP property For r a positive integer, let k(t)[σ r, σ r ] be the noncommutative ring of Laurent polynomials in σ r with coefficients in k(t), subject to the relation σ r f := f ( r) σ r for all f k(t) A pre-t-motive M of level r is a left k(t)[σ r, σ r ]-module that is finite-dimensional over k(t) Letting m Mat n 1 (M) comprise a k(t)-basis of M, multiplication by σ r on M is represented by σ r (m) = m for some matrix GL n ( k(t)) Furthermore, M is called rigid analytically trivial if there exists GL n ( ) such that σ r ( ) := ( r) = Such a matrix is called a rigid analytic trivialization of the matrix We also say that is a rigid analytic trivialization of M (with respect to m) Note that if GL n ( ) is also a rigid analytic trivialization of, then by [Papanikolas 2008, 416] we have 1 GLn ( p r (t)) (1) Moreover, if we put m := Bm for any fixed B GL n ( k(t)), then := B ( 1) B 1 represents multiplication by σ r on M with respect to the k(t)-basis m of M, and := B is a rigid analytic trivialization of Definition 221 Suppose we are given a rigid analytically trivial pre-t-motive M of level r that is of dimension n over k(t) If there exists a k(t)-basis m Mat n 1 (M) such that there exists GL n ( ) Mat n ( ) that is a rigid analytic trivialization of M with respect to m and satisfies tr deg k(t) k(t)( ) = tr deg k k( (θ)), then we say that M has the GP property, where k(t)( ) (resp k( (θ))) is the field generated by all entries of (resp (θ)) over k(t) (resp k) The GP property is independent of the choices of for a fixed m because of (1)

7 116 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu Given a rigid analytically trivial pre-t-motive M of level r with (m,, ) as above, for any s we define its s-th derived pre-t-motive M (s), which is a pret-motive of level rs: the underlying space of M (s) is the same as M, but it is now regarded as a left k(t)[σ rs, σ rs ]-module Letting := ( (s 1)r) ( r), we have σ rs m = m and σ rs := ( rs) =, and hence is also a rigid analytic trivialization of M (s) Proposition 222 Let M be a rigid analytically trivial pre-t-motive of level r that has the GP property For any positive integer s, the s-th derived pre-t-motive M (s) of M is also rigid analytically trivial and has the GP property Theorem 223 [Chang 2009, Theorem 12; Papanikolas 2008, Theorem 522] Suppose Mat n ( k[t]) defines a rigid analytically trivial pre-t-motive M of level r with a rigid analytic trivialization Mat n ( ) GL n ( ) If det (0) = 0 and det (θ 1/pri ) = 0 for all i = 1, 2, 3,, then M has the GP property By [Anderson et al 2004, Proposition 313], the condition det (0) = 0 implies Mat n ( ) Combining Theorem 223 and Proposition 222, we have: Corollary 224 Given an integer d 2, we let l := lcm(1,, d) For each 1 r d, let l r := l/r and let r Mat nr ( k[t]) GL nr ( k(t)) define a pre-tmotive M r of level r with a rigid analytic trivialization r Mat nr ( ) GL nr ( ) Suppose that each r satisfies the hypotheses of Theorem 223 for r = 1,, d Then the direct sum d (l M := M r ) r is a rigid analytically trivial pre-t-motive of level l that has the GP property Proof For each 1 r d, we define r=1 Moreover, if we define r := ( (l r 1)r) r r ( r) r then we have := d r=1 r, := d r=1 r, ( l) = Note that the matrix representing multiplication by σ l on M with respect to the evident k(t)-basis is given by Our task is to show that satisfies the hypotheses of Theorem 223 (with respect to the operator σ l ), whence the result It is obvious that det (0) = 0

8 Difference equations and algebraic independence of zeta values 117 since det r (0) = 0 for each 1 r d Suppose that det (θ 1/plj ) = 0 for some j This implies that there exists 1 r d and 0 m l r 1 such that However, this is equivalent to det ( rm) r (θ ( lj) ) = 0 det r (θ ( (lj rm)) ) = 0 (2) Since 0 m l r 1 and r l, we have that (lj rm) > 0 and r (lj rm) Thus, (2) contradicts the hypothesis that det r (θ ( rh) ) = 0 for all h = 1, 2, 3, 23 Difference Galois groups and transcendence In this section, we review the related theory developed in [Papanikolas 2008] Let r be a fixed positive integer The category of pre-t-motives of level r forms a rigid abelian p r (t)-linear tensor category Also, the category of rigid analytically trivial pre-t-motives of level r forms a neutral Tannakian category over p r (t) Given an object M in, we let M be the strictly full Tannakian subcategory of generated by M That is, M consists of all objects of isomorphic to subquotients of finite direct sums of M u (M ) v for various u, v, where M is the dual of M By Tannakian duality, M is representable by an affine algebraic group scheme Ɣ M over p r (t) The group Ɣ M is called the Galois group of M and it is described explicitly as follows Suppose that GL n ( k(t)) provides multiplication by σ r on M with respect to a fixed basis m Mat n 1 (M) over k(t) Let GL n ( ) be a rigid analytic trivialization for Let X :=(X i j ) be an n n matrix whose entries are independent variables X i j, and define a k(t)-algebra homomorphism ν : k(t)[x, 1/ det X] such that ν(x i j ) = i j for all 1 i, j n We let := im ν = k(t)[, 1/ det ], Z := Spec Then Z is a closed k(t)-subscheme of GL n /k(t) Let 1, 2 GL n ( k(t) ) be the matrices satisfying ( 1 ) i j = i j 1 and ( 2 ) i j = 1 i j for all 1 i, j n Let := We have an p r (t)-algebra homomorphism µ: p r (t)[x, 1/ det X] k(t) such that µ(x i j ) = i j for all 1 i, j n Furthermore, we define := im µ, Ɣ := Spec (3)

9 118 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu Theorem 231 [Papanikolas 2008, Theorems 4211, 431, 4510] The scheme Ɣ is a closed p r (t)-subgroup scheme of GL n / p r (t), which is isomorphic to the Galois group Ɣ M over p r (t) Moreover, Ɣ has the following properties: (a) Ɣ is smooth over p (t) and is geometrically connected (b) dim Ɣ = trdeg k(t) k(t)( ) (c) Z is a Ɣ -torsor over k(t) In particular, if M has the GP property, then (d) dim Ɣ = trdeg k k( (θ)) We call Ɣ the Galois group associated to the difference equation ( r) = This Ɣ is independent of the analytic trivialization, up to isomorphism over p r (t) Throughout this paper we always identify Ɣ M with Ɣ, and regard it as a linear algebraic group over p r (t) because of Theorem 231(a) Remark 232 Let n 1, n 2 be positive integers and 0 := 0 n1 n 2 be the zero matrix of size n 1 n 2 Suppose that the matrix [ ] 1 0 := GL 3 n1 +n 2 ( k(t)) 2 defines a rigid analytically trivial pre-t-motive M of level r Then one can always find its rigid analytic trivialization of the form [ ] 1 0 := GL 3 n1 +n 2 ( ) 2 By (3), we have that {[ ]} 0 Ɣ GL n1 +n 2 / p r (t) Let N be the sub-pre-t-motive (of level r) of M defined by 1 GL n1 ( k(t)) with rigid analytic trivialization 1 ; then by the Tannakian theory we have a natural surjective morphism π : Ɣ ( p (t)) Ɣ 1 ( p (t)), γ π(γ ) In fact, π(γ ) comes from the restriction of the action of γ to the fiber functor of N (which is a full subcategory of M ) Precisely, π(γ ) is the matrix cut out from the upper left square of γ with size n 1 (for detailed arguments, see [Papanikolas 2008, 622]) (4)

10 Difference equations and algebraic independence of zeta values A dimension criterion 31 Pre-t-motives of type SV Let r be a fixed positive integer and let {n 1,, n h } be h nonnegative integers We say that a pre-t-motive M of level r is of type SV (its Galois group being an extension of a split torus by a vector group) if the multiplication by σ r on M is represented by the matrix := a i 0 0 h a i1 1 0 A i, A i := GL 1+n i ( k(t)), a ini 0 1 i=1 which has rigid analytic trivialization h := F i, F i := i=1 f i 0 0 f i1 1 0 GL 1+n i ( ) f ini 0 1 Let T be the Galois group associated to the difference equation f f h ( r) a 1 0 = 0 a h f 1 0, 0 f h and note that by (3), T is a subtorus of the h-dimensional split torus in GL h / p r (t) By the same reason as (4), we have the natural projection of Galois groups given in terms of coordinates by i=1 Ɣ T, (5) x i 0 0 h x i1 1 0 [x 1] [x h ] x ini 0 1 One has also the exact sequence of linear algebraic groups 1 V Ɣ T 1, (6)

11 120 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu where V is a vector group contained in the ( h i=1 n i) -dimensional coordinate vector group G, which is defined as the set of matrices h x i1 1 0 i=1 x ini 0 1 with usual multiplication This subgroup V is the unipotent radical of the (solvable) Galois group Ɣ, and dim Ɣ = dim V + dim T Definition 311 Let M be a rigid analytically trivial pre-t-motive of level r that is of SV type given as above We say that its Galois group Ɣ is full if dim V = h i=1 n i, that is, V = G 32 Criterion for direct sum motives to have full Galois group We continue with the notation of Section 31 For each i, 1 i h, the 1 1 matrix [a i ] defines a sub-pre-t-motive (of level r) of M, which is one-dimensional over k(t) Its rigid analytic trivialization is given by f i satisfying [ f i ] ( r) = [a i ][ f i ] We call one such sub-pre-t-motive of level r a diagonal of the pre-t-motive M By Theorem 231, the Galois group of a diagonal of M is m if and only if the corresponding trivialization f i is transcendental over k(t) In this situation, the canonical projection T m on the i-th coordinate of T is surjective There is a canonical action of T on G given in terms of coordinates by x i1 1 0 h ([x 1 ] [x h ]) h i=1 := x i x i1 1 0 i=1 x ini 0 1 x i x ini 0 1 This action induces an action of T on V compatible with the one coming from (6) Hence, we have that given any element γ V whose x i j -coordinate is nonzero, the orbit inside V given by the action of T on γ must be infinite if f i is transcendental over k(t) Definition 321 Let M i be a rigid analytically trivial pre-t-motive of level r that is of type SV for i = 1,, d We say that this set of pre-t-motives {M i } d i=1 is diagonally independent if for any 1 i, j d, i = j, the Galois group of N i N j is a two-dimensional torus over p r (t), where N i is any diagonal of M i and N j is any diagonal of M j

12 Difference equations and algebraic independence of zeta values 121 Theorem 322 Given any positive integer r, let M 1,, M d be rigid analytically trivial pre-t-motives of level r that are of type SV and have full Galois groups Suppose that the set {M i } i=1 d is diagonally independent Then the Galois group of M := d i=1 M i is also full Proof We first explain that without loss of generality, we may assume d = 2 We prove the result by induction on d Consider M = ( d 1 i=1 M i) Md Then the induction hypothesis implies that the Galois group of ( d 1 i=1 M i) is full Since the following argument of the case d = 2 can be applied to the computation of the Galois group of the direct sum of ( d 1 i=1 M i) and Md, we may assume d = 2 Now we set d = 2 and let the pre-t-motive M i be defined by a matrix i, for i = 1, 2, with = 1 2 a rigid analytic trivialization of M, and let Ɣ, Ɣ 1, Ɣ 2 be the Galois groups The unipotent radicals of these groups are denoted by V, V 1, V 2 respectively Let G, G 1, G 2 be the coordinate vector groups containing V, V 1, V 2, respectively Suppose the matrix 1 has h diagonal blocks, and let the coordinates of G 1 be denoted by x i j, i = 1,, h, j = 1,, n i Similarly, let y i j, i = 1,, l, j = 1,, m i denote the coordinates of G 2 Any subspace W G obtained by setting some of these coordinates to 0 is called a linear coordinate subspace The hypothesis that the Galois group of M i is full means exactly that G i = V i, for i = 1, 2 We are going to prove that G = V Suppose V has codimension s in the coordinate vector group G We can find a linear coordinate subspace W G of dimension s such that W V is of dimension 0 Since W V is invariant under T, it must be equal to the neutral element of G, because the hypothesis that M 1 and M 2 are diagonally independent implies in particular that the Galois group of any diagonal of M is m Let W G be the linear coordinate space given by those coordinates disjoint from those of W Then the natural projection from G to W induces on V an isomorphism of vector groups Composing the inverse of this isomorphism with the surjective morphism π 1 in the diagram 1 V Ɣ T 1 1 G 1 π 1 Ɣ 1 π 1 T 1 1, we obtain a morphism π 1 from W onto G 1 that is furthermore a T -morphism We contend that under the hypothesis that M 1 and M 2 are diagonally independent, π 1 maps G 2 W to zero This contention results from the following basic lemma by taking any diagonal N 1 (resp N 2 ) of M 1 (resp M 2 ) and considering the restriction of the above morphism π 1 to a single block Now since G = G 1 G 2, it follows that π 1 (G 1 W ) = G 1 Thus G 1 W Similarly we also have G 2 W, and hence G = W and G = V

13 122 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu Lemma 323 Let G 1 (resp G 2 ) be the vector group with coordinates x 1 1 0, resp y x n 0 1 y m 0 1 Let 2 m act on G 1 (resp G 2 ) by 2 m (x, y) : x i xx i, 1 i n, (resp y j yy j, 1 j m) If π 1 : G 2 G 1 is a 2 m -morphism, then π Application to zeta values Throughout this section, we use ri for two subscripts (for example, D ri depends on the two parameters r and i), and use, rn for three subscripts (for example, L α,rn depends on the three parameters α, r and n) 41 Carlitz theory Throughout Section 4 we fix a positive integer r Recall the Carlitz p r [t]-module, denoted by C r, which is given by the p r -linear ring homomorphism C r = (t (x θ x + x pr )) : p r [t] End p r ( a ) Note that when we regard C r as a Drinfeld p [t]-module, it is of rank r [Goss 1996; Thakur 2004] One has the Carlitz exponential associated to C r : Here we set exp Cr (z) := i=0 z pri D ri i 1 D r0 := 1, D ri := (θ pri θ pr j ), i 1 Now exp Cr (z) is an entire power series in z satisfying the functional equation j=0 exp Cr (θz) = θ exp Cr (z) + exp Cr (z) pr Moreover one has the product expansion exp Cr (z) = z where 0 =a p r [θ] π r = θ( θ) 1/(pr 1) ( 1 z ), a π r ( ) 1 θ 1 p ri 1 i=1

14 Difference equations and algebraic independence of zeta values 123 is a fundamental period of C r We fix once and for all a choice of ( θ) 1/(pr 1) so that π r is a well-defined element in p ((1/θ)) We also choose these roots in a compatible way so that when r divides r, the number ( θ) 1/(pr 1) is a power of ( θ) 1/(pr 1) The formal inverse of Carlitz exponential is the Carlitz logarithm It is the power series z log Cr (z) = pri, L ri where i=0 L r0 := 1, L ri := i (θ θ pr j ) As a function in z, log Cr (z) converges for all z with z < θ pr /(p r 1) It satisfies the functional equation j=1 θ log Cr (z) = log Cr (θz) + log Cr (z pr ) whenever the values in question are defined For a positive integer n, the n-th Carlitz polylogarithm associated to C r is the series z Plog rn (z) := pri Lri n, (7) which converges -adically for all z with z < θ npr /(p r 1) Its value at a particular z = α = 0 is called the n-th polylogarithm of α associated to C r In transcendence theory we are interested in those polylogarithms of α k, as analogous to classical logarithms of algebraic numbers 42 Algebraic independence of special functions For any positive integer r, let ( r (t) := ( θ) pr /(p r 1) 1 t ) k [[t]] ((t)) θ pri i=1 r, since θ pri Furthermore, r satisfies the functional equation i=0 ( r) r (t) = (t θ) r (t), (8) and its specialization at t = θ gives r (θ) = 1/ π r By (8), the function r provides a rigid analytic trivialization of the Carlitz motive r of level r that has the GP property (see Theorem 223) This is the pre-t-motive with underlying space k(t) itself and σ r acts by σ r f = (t θ) f ( r) for f r For any d, we let l := lcm(1,, d) and l r := l/r for r = 1,, d We let (l r ) r be the l r -th derived pre-t-motive of r that is a rigid analytically

15 124 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu trivial pre-t-motive of level l (see Section 22) Then we define the direct sum M = M d := d r=1 (l r ) r By Corollary 224, M also has the GP property We note that the canonical rigid analytical trivialization of M is the diagonal matrix Mat d ( ) GL d ( ) with diagonal entries 1,, d Lemma 421 Given any positive integer d 2, we let l := lcm(1,, d) Let M = M d be the rigid analytically trivial pre-t-motive of level l with rigid analytic trivialization defined as above Then we have dim Ɣ = d In particular, the functions 1,, d are algebraically independent over k(t) and the values π 1,, π d are algebraically independent over k Proof Suppose dim Ɣ < d Since is a diagonal matrix with diagonal entries 1,, d, by (3) we have that Ɣ T, where T is the split torus of dimension d in GL d / p l(t) We let X 1,, X d be the coordinates of T and χ j the character of T that projects the j-th diagonal position to m Note that {χ j } d j=1 generates the character group of T Hence Ɣ is the kernel of some characters of T, that is, canonical generators of the defining ideal for Ɣ can be of the form X m 1 1 X m d d 1 for some integers m 1,, m d, not all zero By (3) we have that and hence ( m 1 1 m d d ) ( m 1 1 m d d ) = 1 k(t), β := m 1 1 m d d k(t) (9) We recall that r has zeros on {θ pr j } j=1 Since β k(t), it has only finitely many zeros and poles, and hence ord t=θ ph (β) = 0 for h 0 Choose a prime number p sufficiently large that p > d, and the order of vanishing of β at t = θ p p is zero These conditions imply that m 1 = 0 from (9) Iterating this argument, we conclude that m 1 = = m d = 0, a contradiction 43 Algebraic independence of polylogarithms Given n and α k with α < θ npr /(p r 1), we consider the power series L α,rn (t) := α + i=1 α pri (t θ pr ) n (t θ pri ) n, which as a function on converges on t < θ pr We note that L α,rn (θ) is exactly the n-th polylogarithm of α associated to C r, that is, L α,rn (θ) = Plog rn (α)

16 Difference equations and algebraic independence of zeta values 125 Given a collection of such numbers α, say α 1,, α m, we define (t θ) n 0 0 α ( r) 1 (t θ) n 1 0 rn (α 1,, α m ) := GL m+1( k(t)) Mat m+1 ( k[t]) α m ( r) (t θ) n 0 1 and n r 0 0 r n rn (α 1,, α m ) := L α 1,rn 1 0 GL m+1( ) Mat m+1 ( ) r n L α m,rn 0 1 Then one has [Chang and Yu 2007, 312] rn (α 1,, α m ) ( r) = rn (α 1,, α m ) rn (α 1,, α m ) (10) Hence, rn (α 1,, α m ) defines a rigid analytically trivial pre-t-motive of level r that has the GP property In [Chang and Yu 2007], we followed Papanikolas methods to generalize the algebraic independence of Carlitz logarithms to algebraic independence of polylogarithms Precisely, by [Chang and Yu 2007, Theorem 31] and Theorem 223 we have: Theorem 431 Given any positive integers r and n, let α 1,, α m k satisfy α i < θ npr /(p r 1) for i = 1,, m Then dim p r (θ) N rn = tr deg k k( π n r, L α 1,rn(θ),, L αm,rn(θ)) = tr deg k(t) k(t)( n r, L α 1,rn,, L αm,rn), where N rn := p r (θ)- Span{ π n r, L α 1,rn(θ),, L αm,rn(θ)} 44 Formulas for zeta values 441 Euler Carlitz relations Fix a positive integer r for this subsection Carlitz [1935] introduced the power sum ζ r (n) := a p r [θ] a monic 1 a n p((1/θ)) which are the Carlitz zeta values associated to p r [θ] (n a positive integer),

17 126 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu Writing down a p r -adic expansion i n ri p ri of n, we let Ɣ r(n+1) := i=0 D n ri ri We call Ɣ r(n+1) the Carlitz factorials associated to p r [θ] The Bernoulli Carlitz numbers B rn in p (θ) are given by the following expansions from the Carlitz exponential series z exp Cr (z) = We state the Euler Carlitz relations: n=0 B rn Ɣ r(n+1) z n Theorem 442 [Carlitz 1935] For all positive integers n divisible by p r 1, one has ζ r (n) = B rn π r n Ɣ (11) r(n+1) We call a positive integer n (p, r)-even if it is divisible by p r 1; otherwise we call it (p, r)-odd Thus, when p = 2 and r = 1, all positive integers are even 443 The Anderson Thakur formula Anderson and Thakur [1990] introduced the n-th tensor power of the Carlitz p r [t]-module C r, and they related ζ r (n) to the last coordinate of the logarithm associated to the n-th tensor power of C r for each positive integer n More precisely, they interpreted ζ r (n) as p r (θ)-linear combinations of n-th Carlitz polylogarithms of algebraic numbers: Theorem 444 [Anderson and Thakur 1990] Given any positive integers r and n, one can find a sequence h 0,rn,, h lrn,rn p r (θ), l rn < np r /(p r 1), such that l rn ζ r (n) = h i,rn Plog rn (θ i ), (12) where Plog rn (z) is defined as in (7) In the special case of n p r 1, i=0 ζ r (n) = Plog rn (1) Definition 445 Given any positive integer r, for each n, (p r 1) n, with l rn as given by (12), we fix a finite subset such that {α 0,rn,, α mrn,rn} {1, θ,, θ l rn } { π n r, 0,rn(θ),, mrn,rn(θ)} and { π n r, ζ r(n), 1,rn (θ),, mrn,rn(θ)} are p r (θ)-bases for N rn, where j,rn (t) := L α j,rn(t) for j = 0,, m rn This can be done because of (12) (see [Chang and Yu 2007, 41]), and note that m rn + 2

18 Difference equations and algebraic independence of zeta values 127 is the dimension of N rn over p r (θ) In the case of p = 2 and r = 1, the p (θ)- dimension of N 11 is 1 and we set m 11 := 1 Definition 446 Given any positive integers s and d with d 2, for each 1 r d, we define {Ur (s) = {1} if p = 2 and r = 1; U r (s) = {1 n s; p n, (p r 1) n} For each n U r (s), we define that if p = 2 and r = 1, otherwise rn := (t θ) GL 1 ( k(t)), rn := 1 GL 1 ( ), otherwise rn := rn (α 0,rn,, α mrn,rn) GL (mrn +2)( k(t)), rn := rn (α 0,rn,, α mrn,rn) GL (mrn +2)( ) By Theorem 431, we have that dim Ɣ rn = tr deg k(t) k(t)( rn ) = m rn + 2 Put r := n U r (s) rn; then r defines a rigid analytically trivial pre-t-motive M r of level r with rigid analytic trivialization r := n U r (s) rn (see (10)) Also, M r is of type SV, and by Theorem 223, M r has the GP property The main theorem of [Chang and Yu 2007] is the following: Theorem 447 [Chang and Yu 2007, Theorem 45] For any positive integers s and r, the Galois group Ɣ Mr over p r (t) is full, that is, dim Ɣ Mr = 1 + (m rn + 1) n U r (s) 45 Proof of Theorem 121 Given any integer d 2, we put l := lcm(1,, d) and l r := l/r for r = 1,, d For each 1 r d, let M r := M (l r ) r be the l r -th derived pre-t-motive of M r defined as above Note that M r is a rigid analytically trivial pre-t-motive of level l that is still of type SV By Proposition 222, each M r has the GP property, and by Theorem 447, its Galois group Ɣ Mr is full Further, for each 1 r d any diagonal of M r has canonical rigid analytic trivialization given by r n for some n U r(s), and hence its Galois group is m because r is transcendental over k(t) Since by Lemma 421 the functions 1,, d are algebraically independent over k(t), particularly the Galois group of N i N j is a two-dimensional torus over p l(t) for any diagonal N i (resp N j ) of M i (resp M j ) with i = j, 1 i, j d Put M := d r=1 M r, and note that M has the GP property by Corollary 224 Applying Theorem 322 to this situation, we obtain the explicit dimension of Ɣ M :

19 128 Chieh-Yu Chang, Matthew A Papanikolas and Jing Yu Theorem 451 Given any positive integers s and d with d 2, let M be defined as above Then the Galois group Ɣ M is full, that is, dim Ɣ M = d + d (m rn + 1) As a consequence, we completely determine all the algebraic relations among the families of Carlitz zeta values: Corollary 452 Given any positive integers d and s, the transcendence degree of the field ( d ) k { π r, ζ r (1),, ζ r (s)} r=1 over k is d ( s r=1 s p s p r 1 r=1 + s p(p r 1) ) + 1 Proof We may assume d 2, since the case d = 1 is already given in [Chang and Yu 2007, Corollary 46] For 1 r d, let { V1 (s) := if p = 2; V r (s) := U r (s) otherwise Since M has the GP property, by Theorem 451 we see that the elements of the set ( d { 1 (θ),, d (θ)} r=1 n V r (s) ) { 0,rn (θ),, mrn,rn(θ)} are algebraically independent over k In particular, by Definition 445, we have that ( d ) { π 1,, π d } {ζ r (n)} r=1 n V r (s) (13) is an algebraically independent set over k Counting the cardinality of V r (s) for each 1 r d, we complete the proof Acknowledgements We thank D Thakur for many helpful discussions during the preparation of this manuscript The first author thanks the NSC and National Center for Theoretical Sciences for financial support, and Texas A&M University for its hospitality References [Anderson 1986] G W Anderson, t-motives, Duke Math J 53:2 (1986), MR 87j:11042 Zbl

20 Difference equations and algebraic independence of zeta values 129 [Anderson and Thakur 1990] G W Anderson and D S Thakur, Tensor powers of the Carlitz module and zeta values, Ann of Math (2) 132:1 (1990), MR 91h:11046 Zbl [Anderson et al 2004] G W Anderson, W D Brownawell, and M A Papanikolas, Determination of the algebraic relations among special Ɣ-values in positive characteristic, Ann of Math (2) 160:1 (2004), MR 2005m:11140 Zbl [Carlitz 1935] L Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math J 1:2 (1935), MR Zbl [Chang 2009] C-Y Chang, A note on a refined version of Anderson Brownawell Papanikolas criterion, J Number Theory 129:3 (2009), MR 2010a:11144 Zbl [Chang and Yu 2007] C-Y Chang and J Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv Math 216:1 (2007), MR 2008j:11114 Zbl [Chang et al 2010] C-Y Chang, M A Papanikolas, D S Thakur, and J Yu, Algebraic independence of arithmetic gamma values and Carlitz zeta values, Adv Math 223:4 (2010), MR 2011d:11180 Zbl [Denis 1998] L Denis, Indépendance algébrique de différents π, C R Acad Sci Paris Sér I Math 327:8 (1998), MR 99i:11059 Zbl [Goss 1996] D Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 35, Springer, Berlin, 1996 MR 97i:11062 Zbl [Papanikolas 2008] M A Papanikolas, Tannakian duality for Anderson Drinfeld motives and algebraic independence of Carlitz logarithms, Invent Math 171:1 (2008), MR 2009b:11127 Zbl [Thakur 2004] D S Thakur, Function field arithmetic, World Scientific, River Edge, NJ, 2004 MR 2005h:11115 Zbl [Wade 1941] L I Wade, Certain quantities transcendental over GF(p n, x), Duke Math J 8 (1941), MR 3,263f Zbl [Yu 1991] J Yu, Transcendence and special zeta values in characteristic p, Ann of Math (2) 134:1 (1991), 1 23 MR 92g:11075 Zbl [Yu 1997] J Yu, Analytic homomorphisms into Drinfeld modules, Ann of Math (2) 145:2 (1997), MR 98c:11054 Zbl Communicated by Brian Conrad Received Revised Accepted cychang@mathctsnthuedutw map@mathtamuedu yu@mathntuedutw National Center for Theoretical Sciences, Mathematics Division, National Tsing Hua University, Hsinchu City 30042, Taiwan ~ cychang/ Department of Mathematics, Texas A&M University, College Station, TX , United States ~ map/ Department of Mathematics, National Taiwan University, Taipei City 106, Taiwan ~ yu/ mathematical sciences publishers msp

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22 Algebra & Number Theory Volume 5 No Formules pour l invariant de Rost P HILIPPE G ILLE and A NNE Q UÉGUINER -M ATHIEU Modular abelian varieties of odd modular degree 1 37 S OROOSH YAZDANI Group algebra extensions of depth one ROBERT B OLTJE and B URKHARD K ÜLSHAMMER Set-theoretic defining equations of the variety of principal minors of symmetric matrices L UKE O EDING Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic C HIEH -Y U C HANG, M ATTHEW A PAPANIKOLAS and J ING Y U (2011)5:1;1-G