Algebra structure of multiple zeta values in positive characteristic
aa r X i v : . [ m a t h . N T ] J u l ALGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVECHARACTERISTIC
CHIEH-YU CHANG, YEN-TSUNG CHEN, AND YOSHINORI MISHIBA
Abstract.
This paper is a culmination of [CM20] on the study of multiple zeta values(MZV’s) over function fields in positive characteristic. For any finite place v of the rationalfunction field k over a finite field, we prove that the v -adic MZV’s satisfy the same ¯ k -algebraicrelations that their corresponding ∞ -adic MZV’s satisfy. Equivalently, we show that the v -adic MZV’s form an algebra with multiplication law given by the q -shuffle product whichcomes from the ∞ -adic MZV’s, and there is a well-defined ¯ k -algebra homomorphism fromthe ∞ -adic MZV’s to the v -adic MZV’s. Introduction
Classical conjecture.
Let N be the set of positive integers. For a positive integer r ,an r -tuple s = ( s , . . . , s r ) ∈ N r is called an index, and called admissible if s >
1. We putwt( s ) := P ri =1 s i and dep( s ) := r . Classical real-valued multiple zeta values (abbreviated asMZV’s) are generalizations of special values of the Riemann zeta function at positive integersat least 2. The MZV at an admissible index s = ( s , . . . , s r ) is defined by the following series ζ ( s ) := X n >n > ··· >n r ≥ n s · · · n s r r ∈ R × . The weight and depth of the presentation ζ ( s ) are defined by wt( s ) and dep( s ) respectively.MZV’s have deep properties and have appeared in recent decades in connection with vari-ous topics including Grothendieck-Teichm¨uller groups, Drinfeld associators and KZ equationsand periods of mixed Tate motives etc (see [An04, Br12, BGF19, DG05, Dr90, F11, Gon02,R02, Te02, Zh16]). One of the core problems on the topic of MZV’s is to study their algebraicrelations, and how to generate Q -linear relations among them has been well-developed. Forinstance, the machinery of regularized double shuffle relations [IKZ06] produces rich Q -linearrelations among MZV’s of the same weight.Let p be a prime number. In the parallel but extremely different world, namely the p -adicfield, Furusho [F04] defined p -adic MZV’s. The starting point is that for an admissible index s = ( s , . . . , s r ) ∈ N r , the MZV ζ ( s ) is the limit of the one-variable multiple polylogarithmLi s ( z ) := X n >n > ··· >n r ≥ z n n s · · · n s r r for | z | < , z →
1. Furusho considered the one-variable p -adic multiple polylogarithm Li s ( z ) p ,which is the same power series as Li s ( z ), but treated p -adically. He then made an analyticcontinuation of Li s ( z ) p by Coleman’s p-adic iterated integration theory and then definedthe p-adic MZV ζ p ( s ) to be certain limit value at 1 of analytically continued function of Date : July 17, 2020.2010
Mathematics Subject Classification.
Primary 11R58, 11J93.
Key words and phrases.
Multiple zeta values, v -adic multiple zeta values, Carlitz multiple star polyloga-rithms, logarithms of t -modules, t -motives. Li s ( z ) p . Related details are referred to Furusho’s paper [F04]. The weight and depth of thepresentation of the p -adic MZV ζ p ( s ) are defined to be wt( s ) and dep( s ) respectively.Note that in the case of depth one, Furusho’s p-adic zeta value ζ p ( s ) equals the Kubota-Leopoldt p -adic zeta value at s up to a scalar multiplication by (1 − p − s ) − . In particular, wehave ζ p (2 n ) = 0 for n ∈ N . As Kubota-Leopoldt p -adic zeta function p -adically interpolates the special values of Riemann zeta function at negative inetgers, one can ask the naturalquestion: what kind of spark can these two seemingly similar values, real-valued MZV’s and p -adic MZV’s, but living in completely different worlds have? The following fundamentalconjecture gives an explicit connection between these two kinds of MZV’s. Conjecture 1.1.1.
For any prime number p , the p -adic MZV’s satisfy the same Q -algebraicrelations that their corresponding real-valued MZV’s satisfy. That is, if f ( ζ ( s ) , . . . , ζ ( s m )) = 0 for f ∈ Q [ X , . . . , X m ] , then we have f ( ζ p ( s ) , . . . , ζ p ( s m )) = 0 . Let Z (resp. Z p ) be the Q -vector space spanned by 1 and all real-valued MZV’s (resp. by1 and all p -adic MZV’s). It is well-known that Z forms a Q -algebra with two multiplicationlaws given by shuffle product and stuffle product [R02, IKZ06, BGF19]. By [F04, BF06], onealso knows that Z p forms a Q -algebra with two multiplication laws given by shuffle productand stuffle product such as the case of real-valued MZV’s. Therefore, the conjecture aboveis equivalent to the following one. Conjecture 1.1.2.
For any prime number p , the following map φ p := ( ζ ( s ) ζ p ( s )) : Z ։ Z p is a well-defined Q -algebra homomorphism. There are several ways to illustrate the conjectures above.(1) Ihara, Kaneko and Zagier [IKZ06] gave a conjecture asserting that the regular-ized double shuffle relations generate all Q -algebraic relations among the real-valuedMZV’s. Furusho-Jafari [FJ07] showed that the p -adic MZV’s satisfy the regularizeddouble shuffle relations. It follows that combining Ihara-Kaneko-Zagier conjectureand Furusho-Jafari’s result would imply Conjecture 1.1.1.(2) Clues of the formulation of the conjecture above also come from [F06, F07]. Foran integer n ≥
2, we let Z n (resp. Z n,p ) be the Q -vector space spanned by real-valued MZV’s of weight n (resp. p -adic MZV’s of weight n ). Considering the gradedalgebra Z := Q ⊕ L n ≥ Z n (resp. Z p := Q ⊕ L n ≥ Z n , p ), Furusho [F06, Conj. A]conjectured that O ( GRT ) is isomorphic to Z/ ( π ) and in [F07, Sec. 3.1] he explainedthat there is a surjection from O ( GRT ) to Z p . Here GRT is the unipotent partof the graded Grothendieck-Teichm¨uller group GRT , which is a pro-algebraic groupover Q . For more details, see [F06, F07]. On the other hand, Goncharov’s direct sumconjecture [Gon97] for MZV’s asserts that Z = Z , and on the p -adic side we have anatural surjective Q -algebra homomorphism Z p ։ Z p . So, conjecturally the composite map Z = Z ։ Z/ ( π ) ≃ O ( GRT ) ։ Z p ։ Z p gives rise to a surjective Q -algebra homomorphism from Z to Z p . LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 3
Moreover, there is a motivic illustration for a conjectural surjective homomorphism Z ։ Z p , which the authors learned from F. Brown’s talk on “motivic periods and applications”in Hausdorff Research Institute for Mathematics in 2018. First, Deligne also defined p -adicMZV’s and Furusho showed in [F07] that Deligne’s p -adic MZV’s generate the same space Z p . Let Z mot be the Q -algebra of motivic MZV’s, and one knows that there is a Q -algebrahomomorphism (see [Gon02] and [Br12]) addressed as the period map Z mot ։ Z , and the Grothendick periods conjecture for MZV’s predicts that this is an isomorphism. Onthe other hand, from the p -adic period map one has a Q -algebra homomorphism (cf. [F07,(3.11)]) Z mot / ( ζ mot (2)) ։ Z p , and so conjecturally there is a Q -algebra homomorphism Z ։ Z p . The aim of this paper is to prove the precise analogue of Conjecture 1.1.1 in the settingof function fields in positive characteristic. Note that our methods of proof are throughlogarithms of t -modules, which are entirely different from the above points of view in thecharacteristic zero case.1.2. The main result.
Let q be a power of a prime number p , and let F q be a finite fieldof q elements. Let A := F q [ θ ] be the polynomial ring with quotient field k := F q ( θ ). We let k ∞ be the completion of k at the infinite place ∞ , and C ∞ be the ∞ -adic completion of afixed algebraic closure of k ∞ . We let ¯ k be the algebraic closure of k inside C ∞ .The ∞ -adic multiple zeta values are defined by Thakur [T04]: for any index s = ( s , . . . , s r ) ∈ N r , we define ζ A ( s ) := X a s · · · a s r r ∈ k ∞ , where the sum is over all monic polynomials a , . . . , a r in A with the restriction deg θ a > deg θ a > · · · > deg θ a r . For r = 1, the values above were introduced by Carlitz [Ca35] andcalled Carlitz zeta values. We call wt( s ) := P ri =1 s i the weight and r := dep( s ) the depthof the presentation ζ A ( s ). In [T10], Thakur showed that for any two indices s ∈ N r and s ′ ∈ N r ′ , one has(1.2.1) ζ A ( s ) · ζ A ( s ′ ) = X j f j ζ A ( s j )for some finitely many f j ∈ F p and s j ∈ N dep( s j ) depending on q with wt( s j ) = wt( s ) + wt( s ′ )and dep( s j ) ≤ dep( s ) + dep( s ′ ), where F p is the prime field of k . We simply call (1.2.1) the q -shuffle relations (or q -shuffle product), which Thakur called sum-shuffle relations. Notethat in our positive characteristic setting, the q -shuffle product is neither the classical shuffleproduct nor stuffle product (see H.-J. Chen’s explicit formula (5.3.1)). Because of the q -shuffle product, the ∞ -adic MZV’s form an F p -algebra.Given a monic irreducible polynomial v of A , we let k v be the completion of k at v and let C v be the v -adic completion of a fixed algebraic closure of k v . Throughout thisarticle, we always fix an embedding ¯ k ֒ → C v over k once a finite place v is given. Based onthe formula of ∞ -adic MZV’s in terms of Carlitz multiple polylogarithms (abbreviated asCMPL’s) established in [C14], the first and third authors of the present paper introduced theCarlitz multiple star polylogarithms (abbreviated as CMSPL’s) given in (2.2.2) and derivedthe formula of ∞ -adic MZV’s as k -linear combinations of CMSPL’s at integral points in CHIEH-YU CHANG, YEN-TSUNG CHEN, AND YOSHINORI MISHIBA (2.3.4). In the depth one case, CMSPL’s are reduced to Carlitz polylogarithms and suchformula was established by Anderson-Thakur [AT90].Inspired by Furusho’s strategy for defining p -adic MZV’s [F04], for any index s ∈ N r thefirst and third authors treated CMSPL’s for v -adic convergence in [CM19] and used actionof certain t -modules for which v -adic CMSPL’s can be extended to be defined at integralpoints. Then they used the same formula of ∞ -adic MZV’s (2.3.4) to define the v -adic MZV ζ A ( s ) v in (2.3.5) for any index s . As same as the case of ∞ -adic MZV’s, the weight and thedepth of the presentation ζ A ( s ) v are defined to be wt( s ) and dep( s ) respectively. Note thatThakur [T04] also defined v -adic MZV’s but his definition is different from ours.In [Go79], Goss defined a v -adic zeta function that interpolates Carlitz zeta values at non-positive integers and obtained v -adic zeta values at positive integers, which are simply called Goss’ v -adic zeta values , which are equal to Thakur’s v -adic MZV’s of depth one. In thedepth one case, our v -adic zeta value ζ A ( s ) v is identical to Goss’ v -adic zeta value [Go79] at s multiplied by (1 − v − s ) − (see [AT90, Thm. 3.8.3. (II)]), and so ζ A ( n ) v = 0 for all positiveintegers n divisible by q − p -adiccase mentioned above.Let Z ⊂ C ∞ (resp. Z v ⊂ C v ) be the k -vector space spanned by 1 and all ∞ -adic MZV’s(resp. 1 and all v -adic MZV’s). It is shown in [CM20, Cor. 6.4.3] that the map Z ։ Z v given by ζ A ( s ) ζ A ( s ) v is a well-defined k -linear map. Our main theorem stated below is afunction field analogue of Conjecture 1.1.1 but it is in stronger form as it is over algebraiccoefficients. Theorem 1.2.2.
For any finite place v of k , the v -adic MZV’s satisfy the same ¯ k -algebraicrelations that their corresponding ∞ -adic MZV’s satisfy. That is, if g ( ζ A ( s ) , . . . , ζ A ( s m )) = 0 for g ∈ ¯ k [ X , . . . , X m ] , then we have g ( ζ A ( s ) v , . . . , ζ A ( s m ) v ) = 0 . Note that Z forms a k -algebra because of (1.2.1). The theorem above is equivalent to thefollowing. Theorem 1.2.3.
Let v be a finite place of k . Then the following hold. (1) v -adic MZV’s satisfy the q -shuffle relations in the sense that ζ A ( s ) v · ζ A ( s ′ ) v = X j f j ζ A ( s j ) v with notation given in (1.2.1) . (2) Z v forms a k -algebra and the following map Z ։ Z v given by ζ A ( s ) ζ A ( s ) v is awell-defined ¯ k -algebra homomorphism. In particular, the kernel contains the principalideal generated by ζ A ( q − . The theorem above gives an affirmative answer of part of the questions in [CM20, Rem. 6.4.4],which arose from numerical evidence using Sage. In Section 5.3 we give an example for com-puting the product of the v -adic single zeta value ζ v (1) with itself for a very special q . Ascan be seen from that example, a direct calculation proof of Theorem 1.2.3 is impractica-ble because the definition of ζ A ( s ) v is through the logarithm of a concrete t -module, whosedimension is huge when wt( s ) and dep( s ) are large. In this paper, we aim to prove The-orem 1.2.3 in a more robust way via the logarithmic points of view. Since we have shown LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 5 in [CM20] that the map Z ։ Z v is well-defined and k -linear, the statement (1) of Theo-rem 1.2.3 is equivalent to the statement (2). It is natural to ask about the kernel of the k -algebra homomorphism Z ։ Z v given in Theorem 1.2.3. We conjecture that the kernel inquestion is the principal ideal generated by the single zeta value ζ A ( q −
1) and discuss someapplications in Sec. 5.4.1.3.
Ideas of the proof.
To sketch the key ideas of our proof, we first set up the followingnotation. Fix any finite place v of k . For any index s = ( s , . . . , s r ) ∈ N r , let Li ⋆ s ( z , . . . , z r )be the Carlitz multiple star polylogarithm defined in (2.2.2). We define(1.3.1) D s , ∞ := { ( z , . . . , z r ) ∈ C r ∞ | | z | ∞ < q s qq − and | z i | ∞ ≤ q siqq − (2 ≤ i ≤ r ) } ⊂ C r ∞ and note that by [CM20, Rem. 4.1.3.] Li ⋆ s converges ∞ -adically on D s , ∞ . Concerning the v -adic convergence, Li ⋆ s converges on(1.3.2) D Conv s ,v := { ( z , . . . , z r ) ∈ C rv | | z | v < | z i | v ≤ ≤ i ≤ r ) } ⊂ C rv , but by [CM19] it can be extended to be defined on the closed polydisc(1.3.3) D Def s ,v := { ( z , . . . , z r ) ∈ C rv | | z i | v ≤ ≤ i ≤ r ) } ⊂ C rv . We then define the following common sets of algebraic points(1.3.4) D Conv s ,k := D s , ∞ ∩ k dep s ∩ D Conv s ,v ⊂ k dep s , at which Li ⋆ s converges both ∞ -adically and v -adically, and(1.3.5) D Def s ,k := D s , ∞ ∩ k dep s ∩ D Def s ,v ⊂ k dep s , at which Li ⋆ s is defined both ∞ -adically and v -adically. Definition 1.3.6.
We define the following k -vector spaces. • L Conv ∞ := the k -vector space spanned by 1 and all Li ⋆ s ( u ) for s ∈ ∪ r> N r and u ∈ D Conv s ,k . • L Conv v := the k -vector space spanned by 1 and all Li ⋆ s ( u ) v for s ∈ ∪ r> N r and u ∈ D Conv s ,k . • L Def ∞ := the k -vector space spanned by 1 and all Li ⋆ s ( u ) for s ∈ ∪ r> N r and u ∈ D Def s ,k . • L Def v := the k -vector space spanned by 1 and all Li ⋆ s ( u ) v for s ∈ ∪ r> N r and u ∈ D Def s ,k .As D Conv s ,k ⊂ D Def s ,k we have the natural inclusions: L Conv ∞ ⊂ L Def ∞ and L Conv v ⊂ L Def v . We illustrate our strategy via the following commutative diagram(1.3.7) Z [CM20] (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) [C14] / / L Def ∞ Thm. 5.1.5 φ v (cid:15) (cid:15) (cid:15) (cid:15) Thm. 4.2.1 L Conv ∞ (cid:15) (cid:15) (cid:15) (cid:15) Z v (cid:31) (cid:127) Def. / / L Def v L Conv v ? _ = o o with the following descriptions:(1) The inclusion Z ֒ → L Def ∞ follows from [CM20, Thm. 5.2.5] (see [C14, Thm. 5.5.2] also).(2) The map Z ։ Z v given by ζ A ( s ) ζ A ( s ) v is a well-defined k -linear map by [CM20,Cor. 6.4.3].(3) We prove in Theorem 4.2.1 that the inclusion L Conv ∞ ֒ → L Def ∞ is in fact an equality. CHIEH-YU CHANG, YEN-TSUNG CHEN, AND YOSHINORI MISHIBA (4) We prove in Theorem 5.1.5 that the map φ v : L Def ∞ ։ L Def v given by Li ⋆ s ( u ) Li ⋆ s ( u ) v is a well-defined k -linear map.(5) By definition, the restriction of φ v to L Conv ∞ is a well-defined k -linear map onto L Conv v .With the above properties established, we mention that since the map φ v is surjectiveand L Def ∞ = L Conv ∞ , it implies the equality L Def v = L Conv v . Furthermore, because the CMSPL’sconverge ∞ -adically and v -adically on D Conv s ,k , the values in L Conv ∞ and L Conv v satisfy the stufflerelations respectively (see (2.2.5) and (2.2.6)) and hence the map φ v is a k -algebra homo-morphism. As the restriction of φ v to Z is given by ζ A ( s ) ζ A ( s ) v (see (2.3.4) and (2.3.5)),we derive the desried k -algebra homomorphism Z ։ Z v .1.4. Organization of this paper.
In Sec. 2, we first review the theory of Anderson onhis t -modules. We then review CMSPL’s and describe stuffle relations. We further recallfrom [CM19, CM20] how we relate CMSPL’s to coordinates of logarithms of certain t -modulesat specific points as is used to define v -adic MZV’s.Sections 3 and 4 are the most technical parts, which are devoted to prove Theorem 4.2.1.Given any Li ⋆ s ( u ) ∈ L Def ∞ , ie., u ∈ D Def s ,k , we mention that Li ⋆ s ( u ) is realized as the wt( s )-thcoordinate of the logarithm of an explicitly constructed t -module G defined over k at analgebraic point v ∈ G ( k ). To show that Li ⋆ s ( u ) ∈ L Conv ∞ , the key of our strategy is to find asuitable algebraic point v ′ ∈ G ( k ), at which the logarithm log G converges both ∞ -adicallyand v -adically and we use techniques of division points to do the trick. From the functionalequation of log G , the wt( s )-th coordinate of log G ( v ′ ) is related to the value Li ⋆ s ( u ).Moreover, we have to control the v -adic size of the point v ′ as is needed when definingthe v -adic MZV ζ A ( s ) v . However, Papanikolas’ computation [Pp] concerning the leadingcoefficient matrices of the t m -action of the s -th tensor power of the Carlitz module C ⊗ s enables us to ensure that the point v ′ satisfies the desired properties. The crucial result isstated as Theorem 3.1.1.Section 4 is devoted to establish a kind of algebraic functional equations for certain co-ordinate of the logarithm of certain explicit t -module at any algebraic point whenever it isdefined. The primary result is given as Theorem 4.1.4, which is mainly used to prove theidentity L Def ∞ = L Conv ∞ . In the final section, we use Yu’s sub- t -module theorem [Yu97] to provein Theorem 5.1.5 that the map φ v is a well-defined k -linear map, and then give a proof forTheorem 1.2.3. 2. Preliminaries
Notations.Table of Symbols 2.1.1.
We use the following symbols throughout this paper. N = the set of positive integers. q = a power of a prime number p . F q = a finite field of q elements. A = F q [ θ ], the polynomial ring in the variable θ over F q . k = F q ( θ ), the quotient field of A . |·| ∞ = the normalized absolute value on k for which | θ | ∞ = q . LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 7 k ∞ = F q ((1 /θ )), the completion of k at the infinite place. C ∞ = c k ∞ , the ∞ -adic completion of an algebraic closure of k ∞ . v = a monic irreducible polynomial in A . |·| v = the normalized absolute value on k for which | v | v = q − v , where q v := q deg θ v . k v = the completion of k at v . C v = b k v , the v -adic completion of an algebraic closure of k v . k = an algebraic closure of k with fixed embeddings into C ∞ and C v over k .˜Λ = ( λ r , . . . , λ ) for any r -tuple Λ = ( λ , . . . , λ r ) of symbols. k M k w = max i,j {| M ij | w } for M = ( M ij ) ∈ Mat ℓ × m ( C w ) where w = ∞ or w = v .wt( s ) = P ri =1 s i for an index s = ( s , . . . , s r ) ∈ N r .dep( s ) = r for an index s = ( s , . . . , s r ) ∈ N r .2.2. t -modules associated to CMSPL’s. Review of Anderson’s theory on t -modules. For any F q -algebra R , any matrix M =( M ij ) ∈ Mat ℓ × m ( R ) and any non-negative integer n , we define the n -th fold Frobenius twistby M ( n ) := ( M q n ij ). We then define the non-commutative ring Mat d ( R )[ τ ], whose elementsare of the form P i ≥ a i τ i with a i ∈ Mat d ( R ) and a i = 0 for i ≫
0, and whose multiplicationlaw is given by ( X i ≥ a i τ i )( X j ≥ b j τ j ) = X i X j a i b ( i ) j τ i + j . We put R [ τ ] := Mat ( R )[ τ ] then we have natural identificationsMat d ( R [ τ ]) ≃ Mat d ( R )[ τ ] ≃ End F q ( G da/R ) , where End F q ( G da/R ) is the ring of F q -linear endomorphisms over R of the d -dimensionaladditive group scheme G da/R and τ corresponds to the Frobenius operator ( x , . . . , x d ) tr ( x q , . . . , x qd ) tr .Let t be a new variable. Given an A -subalgebra R ⊂ k and a positive integer d , a t -moduleof dimension d defined over R is an F q -linear ring homomorphism[ − ] : F q [ t ] → Mat d ( R [ τ ])so that ∂ [ t ] − θ · I d is a nilpotent matrix. Here, for a ∈ F q [ t ] we define ∂ [ a ] := a whenever[ a ] = P mi =0 a i τ i for a i ∈ Mat d ( R ). We denote by G = ( G da/R , [ − ]), whose underlying space isthe group scheme G da over R and whose F q [ t ]-module structure is through the F q -linear ringhomomorphism [ − ], ie, for any R -algebra R ′ , the F q [ t ]-module structure on G da/R ( R ′ ) = ( R ′ ) d is given by a · x := [ a ]( x )for a ∈ F q [ t ] and x ∈ ( R ′ ) d .Fix a d -dimensional t -module G defined over R as above, and let K be the fraction fieldof R . Anderson [A86] showed that there is a d -variable F q -linear formal power seriesexp G ∈ K [[ z , . . . , z d ]] d satisfying that exp G ( z ) ≡ z (mod deg q ) for z = ( z , . . . , z d ) tr , and as formal power seriesidentity we have(2.2.1) exp G ◦ ∂ [ a ] = [ a ] ◦ exp G CHIEH-YU CHANG, YEN-TSUNG CHEN, AND YOSHINORI MISHIBA for all a ∈ F q [ t ]. The power series exp G is called the exponential map of G , and it is shownby Anderson that as ∞ -adic convergence, it is an entire function from Lie G ( C ∞ ) = C d ∞ to G ( C ∞ ) = C d ∞ . The formal inverse of exp G is denoted by log G and is called the logarithmmap of G . So as formal power series one has the following propertities: • log G ( z ) ≡ z (mod deg q ). • log G ◦ [ a ] = ∂ [ a ] ◦ log G for a ∈ F q [ t ].From the point of view of transcendence theory [Yu91, Yu97], the values of logarithms of t -modules at algebraic points provide rich resources of interesting transcendental numbers.In this paper, we deepen this logarithmic perspective and such logarithmic interpretationsfor our special values studied here provide the key approaches to prove Theorem 1.2.3.2.2.2. CMSPL’s and stuffle relations.
Let L := 1 and L i := Q ij =1 ( θ − θ q j ) for i ≥
1. Forany index s = ( s , . . . , s r ) ∈ N r , we define the s -th Carlitz multiple polylogarithm (CMPL)as follows (see [C14]):Li s ( z , . . . , z r ) := X i > ··· >i r ≥ z q i . . . z q ir r L s i · · · L s r i r ∈ k [[ z , · · · , z r ]] . We also define the s -th Carlitz multiple star polylogarithm (CMSPL) as follows (see [CM19]):(2.2.2) Li ⋆ s ( z , . . . , z r ) := X i ≥···≥ i r ≥ z q i · · · z q ir r L s i · · · L s r i r ∈ k [[ z , · · · , z r ]] . We denote by Li s ( z , · · · , z r ) v and Li ⋆ s ( z , · · · , z r ) v when we consider the v -adic convergenceof those two infinite series.In what follows, we describe the stuffle relations arising from CMSPL’s. Let X := { ( s, u ) | s ∈ N , u ∈ k, | u | ∞ ≤ q sqq − , | u | v ≤ } ⊂ N × k and X := { ( s, u ) | s ∈ N , u ∈ k, | u | ∞ < q sqq − , | u | v < } ⊂ X. Let h := k h z s,u | ( s, u ) ∈ X i be the non-commutative polynomial algebra over k generated by the variables { z s,u } ( s,u ) ∈ X and h := k ⊕ M ( s,u ) ∈ X z s,u h ⊂ h . For any s = ( s , . . . , s r ) ∈ N r and u = ( u , . . . , u r ) ∈ k r with ( s i , u i ) ∈ X for i = 1 , . . . , r , weshall call z s ,r · · · z s r ,u r the monomial associated to the pair ( s , u ), and vice versa.We define the k -bilinear stuffle product ⋆ on h by1 ⋆ w = w ⋆ w and z s,u w ⋆ z s ′ ,u ′ w ′ = z s,u ( w ⋆ z s ′ ,u ′ w ′ ) + z s ′ ,u ′ ( z s,u w ⋆ w ′ ) − z s + s ′ ,uu ′ ( w ⋆ w ′ )for each ( s, u ) , ( s ′ , u ′ ) ∈ X and w, w ′ ∈ h (cf. [IKOO11, Sec. 2.1]). Clearly, h isclosed under ⋆ . LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 9
Now we define L i ⋆ ( − ) and L i ⋆ ( − ) v to be the k -linear maps given by L i ⋆ (1) := 1, L i ⋆ (1) v :=1, and L i ⋆ ( − ) := (cid:0) z s ,u · · · z s r ,u r Li ⋆ ( s ,...,s r ) ( u , . . . , u r ) (cid:1) : h ։ L Conv ∞ ⊂ C ∞ , and L i ⋆ ( − ) v := (cid:0) z s ,u · · · z s r ,u r Li ⋆ ( s ,...,s r ) ( u , . . . , u r ) v (cid:1) : h ։ L Conv v ⊂ C v . The following describes the stuffle relations for the convergent values of CMSPL’s, and itmay be well-known for experts but to be self-contained we provide the detailed but shortarguments here.
Proposition 2.2.3.
The k -linear maps L i ⋆ ( − ) and L i ⋆ ( − ) v are multiplicative in the sensethat L i ⋆ ( w ⋆ w ′ ) = L i ⋆ ( w ) · L i ⋆ ( w ′ ) and L i ⋆ ( w ⋆ w ′ ) v = L i ⋆ ( w ) v · L i ⋆ ( w ′ ) v for each w, w ′ ∈ h . In particular, L Conv ∞ and L Conv v form k -algebras and their genera-tors satisfy the same stuffle relations in the sense that for monomials w, w ′ ∈ h withexpression (2.2.4) w ⋆ w ′ = X i α i w i , α i ∈ F p , we have (2.2.5) L i ⋆ ( w ) · L i ⋆ ( w ′ ) = L i ⋆ ( w ⋆ w ′ ) = X i α i L i ⋆ ( w i ) in C ∞ and (2.2.6) L i ⋆ ( w ) v · L i ⋆ ( w ′ ) v = L i ⋆ ( w ⋆ w ′ ) v = X i α i L i ⋆ ( w i ) v in C v . Proof.
For each non-negative integer n , we define the (truncated) k -linear map L i ⋆ ≤ n givenby L i ⋆ ≤ n (1) := 1 and L i ⋆ ≤ n := z s ,u · · · z s r ,u r X n ≥ i ≥···≥ i r ≥ u q i · · · u q ir r L s i · · · L s r i r ! : h → k. Since lim n →∞ L i ⋆ ≤ n ( w ) = L i ⋆ ( w ) in C ∞ and lim n →∞ L i ⋆ ≤ n ( w ) = L i ⋆ ( w ) v in C v , it suffices to showthat L i ⋆ ≤ n ( w ⋆ w ′ ) = L i ⋆ ≤ n ( w ) · L i ⋆ ≤ n ( w ′ )for all w, w ′ ∈ h and n ∈ Z ≥ .Because of linearity, we may assume that w and w ′ are monomials. We prove the desiredclaim by induction on the sum of total degrees of w and w ′ . If w = 1 or w ′ = 1, then theequality is clearly valid. Let w = 1 , w ′ = 1 and suppose that the equality holds for all n andfor monomials whose total degree is less than deg( w ) + deg( w ′ ). If we write w = z s,u w and w ′ = z s ′ ,u ′ w ′ , then we have L i ⋆ ≤ n ( z s,u w ) · L i ⋆ ≤ n ( z s ′ ,u ′ w ′ )= X n ≥ i ≥ u q i L si L i ⋆ ≤ i ( w ) · L i ⋆ ≤ i ( z s ′ ,u ′ w ′ ) + X n ≥ i ≥ ( u ′ ) q i L s ′ i L i ⋆ ≤ i ( z s,u w ) · L i ⋆ ≤ i ( w ′ ) − X n ≥ i ≥ ( uu ′ ) q i L s + s ′ i L i ⋆ ≤ i ( w ) · L i ⋆ ≤ i ( w ′ )= X n ≥ i ≥ u q i L si L i ⋆ ≤ i ( w ⋆ z s ′ ,u ′ w ′ ) + X n ≥ i ≥ ( u ′ ) q i L s ′ i L i ⋆ ≤ i ( z s,u w ⋆ w ′ ) − X n ≥ i ≥ ( uu ′ ) q i L s + s ′ i L i ⋆ ≤ i ( w ⋆ w ′ )= L i ⋆ ≤ n ( z s,u ( w ⋆ z s ′ ,u ′ w ′ )) + L i ⋆ ≤ n ( z s ′ ,u ′ ( z s,u w ⋆ w ′ )) − L i ⋆ ≤ n ( z s + s ′ ,uu ′ ( w ⋆ w ′ ))= L i ⋆ ≤ n ( z s,u w ⋆ z s ′ ,u ′ w ′ ) , where the second identity comes from the induction hypothesis and the last two identitiesfollow from definitions. (cid:3) The construction of G s , u . Throughout this section, we fix s = ( s , . . . , s r ) ∈ N r and u = ( u , . . . , u r ) ∈ k r . For 1 ≤ i ≤ r , we set(2.2.7) d i := s i + · · · + s r and(2.2.8) d := d + · · · + d r . Let B be a d × d -matrix of the form B [11] · · · B [1 r ]... ... B [ r · · · B [ rr ] where B [ ℓm ] is a d ℓ × d m -matrix for each ℓ and m . We call B [ ℓm ] the ( ℓ, m )-th blocksub-matrix of B .For 1 ≤ ℓ ≤ m ≤ r , we set N ℓ := · · ·
00 1 . . . .... . . . . . 0. . . 10 ∈ Mat d ℓ ( k ) ,N := N N . . . N r ∈ Mat d ( k ) , LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 11 E [ ℓm ] := · · · · · · · · · ∈ Mat d ℓ × d m ( k ) (if ℓ = m ) ,E [ ℓm ] := · · · · · · − m − ℓ Q m − e = ℓ u e · · · ∈ Mat d ℓ × d m ( k ) (if ℓ < m ) ,E := E [11] E [12] · · · E [1 r ] E [22] . . . .... . . E [ r − , r ] E [ rr ] ∈ Mat d ( k ) . Also, we define the t -module G s , u := ( G da , [ − ]) by(2.2.9) [ t ] = θI d + N + Eτ ∈ Mat d ( k [ τ ]) . Note that G s , u depends only on u , . . . , u r − . Finally, we define(2.2.10) v s , u := d ...0( − r − u · · · u r d ...0( − r − u · · · u r ... ...0 d r ...0 u r ∈ G s , u ( k ) . v -adic MZV’s. To review the definition of v -adic MZV’s, we need to recall how weextend the defining domain of the v -adic CMSPL Li ⋆ s to D Def s ,v . Fix an r -tuple s = ( s , . . . , s r ) ∈ N r and note that Li ⋆ s converges v -adically on D Conv s ,v given in (1.3.2) (see [CM19, Sec. 2.2]).The following result gives logarithmic interpretation for CMSPL’s at algebraic points. Theorem 2.3.1 ([CM19, Thm. 3.3.3] and [CM20, Thm. 4.2.3]) . Fixing any r -tuples s =( s , . . . , s r ) ∈ N r and u = ( u , . . . , u r ) ∈ k r , let G s , u and v s , u be defined as above. (1) If ˜ u = ( u r , . . . , u ) ∈ k r ∩ D ˜ s , ∞ ⊂ C r ∞ defined in (1.3.1) and x = ( x i ) ∈ G s , u ( C ∞ ) with | x d + ··· d m − + j | ∞ < q − ( d m − j )+ dmqq − for each ≤ m ≤ r and ≤ j ≤ d m , then log G s , u ( x ) converges ∞ -adically. In particular, log G s , u ( v s , u ) converges ∞ -adically. Moreover,its wt( s ) -th coordinate is equal to ( − r − · Li ⋆ ˜ s (˜ u ) = ( − r − · Li ⋆ ( s r ,...,s ) ( u r , . . . , u ) . (2) If ˜ u = ( u r , . . . , u ) ∈ k r ∩ D Conv ˜ s ,v ⊂ C rv defined in (1.3.2) and x ∈ G s , u ( C v ) with k x k v < , then log G s , u ( x ) v converges v -adically. In particular, log G s , u ( v s , u ) v converges v -adically. Moreover, its wt( s ) -th coordinate is equal to ( − r − · Li ⋆ ˜ s (˜ u ) v = ( − r − · Li ⋆ ( s r ,...,s ) ( u r , . . . , u ) v . Remark . In fact, all coordinates of log G s , u ( v s , u ) can be written explicitly in [CGM19],and the tractable coordinates (see Definition 5.1.4) of log G s , u ( v s , u ) v are given explicitlyin [CM19].We put v ( t ) := v | θ = t ∈ F q [ t ]. Define the local ring O C v := { α ∈ C v ; | α | v ≤ } and denoteby m v the maximal ideal of O C v . The purpose of constructing G s , u for given s and u is for thepurpose of connecting Li ⋆ ˜ s (˜ u ) as well as Li ⋆ ˜ s (˜ u ) v with a coordinate logarithm of the specialalgebraic point v s , u , where the depth one case was established in [AT90]. The followingprovides an approach to extend the v -adic defining domains of CMSPL’s. Proposition 2.3.3 ([CM19, Prop. 4.1.1]) . Let s = ( s , . . . , s r ) ∈ N r and u := ( u , . . . , u r ) ∈ k r ∩ D Def s ,v defined in (1.3.3) . Let G s , u be the t -module defined in (2.2.9) and v s , u ∈ G s , u ( k ) bedefined in (2.2.10) . Let ℓ ≥ be an integer such that each image of u i in O C v / m v ∼ = F q v iscontained in F q ℓv . Let d , . . . , d r be defined in (2.2.7) . Then (cid:12)(cid:12)(cid:12)(cid:12) [ v ( t ) d ℓ − v ( t ) d ℓ − · · · [ v ( t ) d r ℓ − v s , u (cid:12)(cid:12)(cid:12)(cid:12) v < . In particular, log G s , u (cid:0) [ v ( t ) d ℓ − v ( t ) d ℓ − · · · [ v ( t ) d r ℓ − v s , u (cid:1) v converges in Lie G s , u ( C v ) . For any u ∈ k r ∩ D Def s ,v (this is equivalent to ˜ u ∈ k r ∩ D Def ˜ s ,v ), let a ∈ F q [ t ] be nonzero suchthat (cid:12)(cid:12)(cid:12)(cid:12) [ a ] v s , u (cid:12)(cid:12)(cid:12)(cid:12) v <
1. The v -adic CMSPL Li ⋆ ˜ s at ˜ u is defined to beLi ⋆ ( s r ,...,s ) ( u r , . . . , u ) v := ( − r − a ( θ ) × (cid:16) the wt( s )-th coordinate of log G s , u ([ a ] v s , u ) v (cid:17) . It is shown in [CM19] that the v -adic value Li ⋆ ˜ s (˜ u ) for u ∈ k r ∩ D Def s ,v is independent of thechoices of a ( t ) and the existence of such a ( t ) ∈ F q [ t ] is guaranteed by Proposition 2.3.3.We now recall the formula expressing MZV’s as linear combinations of CMSPL’s at alge-braic points from [C14, Thm. 5.5.2] or [CM20, Thm. 5.2.5]. For any index s ∈ N r , there aresome explicit tuples s ℓ ∈ N dep( s ℓ ) with dep( s ℓ ) ≤ dep( s ) and wt( s ℓ ) = wt( s ), some explicitelements b ℓ ∈ k and some explicit integral points u ℓ ∈ A dep( s ℓ ) ∩ D s ℓ , ∞ so that(2.3.4) ζ A ( s ) = X ℓ b ℓ · ( − dep( s ℓ ) − Li ⋆ s ℓ ( u ℓ ) ∈ k ∞ . So based on Proposition 2.3.3 Li ⋆ s ℓ ( u ℓ ) v is defined. The v -adic MZV ζ A ( s ) v in [CM20] isdefined by LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 13 (2.3.5) ζ A ( s ) v := X ℓ b ℓ · ( − dep( s ℓ ) − Li ⋆ s ℓ ( u ℓ ) v ∈ k v , which can be thought of an analogue of Furusho’s p -adic MZV’s. Such as the ∞ -adic case, wedefine the weight and depth of the presentation ζ A ( s ) v to be wt( s ) and dep( s ) respectively.These ∞ -adic MZV’s and v -adic MZV’s are expressed in terms of CMSPL’s at algebraicpoints and have nice logarithmic interpretations in [CM20], extending Anderson-Thakur’swork [AT90] from depth one to arbitrary depth.3. A trick on division points
The main theme in this section is to demonstrate that we are able to find a specificalgebraic point at which the logarithm of the t -module in question converges both ∞ -adicallyand v -adically. This trick is achieved using the uniformization of the t -module as well as itsproperty of iterated extensions of Carlitz tensor powers.3.1. The crucial result.
Recall the notation D Def s ,k = D s , ∞ ∩ k dep s ∩ D Def s ,v ⊂ k dep s given inSec. 1.3.5. Note that exp G is locally isometric for any t -module G defined over k , so we canfind a small domain D G ⊂ Lie G ( C ∞ ) on which exp G is an isometry (see [HJ16, Lem. 5.3]).For each x ∈ O C v , we denote by x the image of x in the residue field O C v / m v ∼ = F q v . Foreach ℓ ∈ N , we define a local ring A ( v ) ,ℓ by A ( v ) ,ℓ := { x ∈ k ∩ O C v | x ∈ F q ℓv } . For each n ∈ N and x = ( x i ) ∈ Mat n × ( O C v ), we define x := ( x i ) ∈ Mat n × ( F q v ). Theorem 3.1.1.
Let s = ( s , . . . , s r ) ∈ N r , u ∈ k r with ˜ u ∈ D Def ˜ s ,k defined in (1.3.5) , and G s , u = ( G da , [ − ]) be the t -module defined in (2.2.9) . Given any v ∈ G s , u ( k ∩ O C v ) suchthat log G s , u ( v ) ∈ Lie G s , u ( C ∞ ) converges ∞ -adically. Pick a positive integer ℓ for which v ∈ G s , u ( A ( v ) ,ℓ ) and define a ( t ) := ( v ( t ) d ℓ − · · · ( v ( t ) d r ℓ − . For each n ∈ Z ≥ , put Z n := ∂ [ v ( t ) n ] − log G s , u ( v ) ∈ Lie G s , u ( C ∞ ) and v n := exp G s , u ( Z n ) ∈ G s , u ( C ∞ ) . Then the following properties hold. (1) [ v ( t ) n ] v n = v and v n ∈ G s , u ( k ) . (2) (cid:12)(cid:12)(cid:12)(cid:12) [ a ( t )] v n (cid:12)(cid:12)(cid:12)(cid:12) v < if n is divisible by lcm( d , . . . , d r ) . In particular, log G s , u ([ a ( t )] v n ) v converges v -adically (see Theorem 2.3.1). (3) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) Z n (cid:12)(cid:12)(cid:12)(cid:12) ∞ = 0 . In particular, log G s , u ([ a ( t )] v n ) , ∂ [ a ( t )] log G s , u ( v n ) and Z n = ∂ [ v ( t ) n ] − log G s , u ( v ) converge ∞ -adically, and they are contained in D G s , u for suffi-ciently large n . In order to prove the theorem above, we need to establish some lemmas. For any positiveinteger s , we denote by C ⊗ s = ( G sa , [ − ] s ) the s -th tensor power of the Carlitz module, which is defined over A . Precisely,[ t ] s := θ · · · θ . . . .... . . 1 θ + · · ·
00 0 · · · · · · τ ∈ Mat s ( A [ τ ]) . (3.1.2)By [Pp, Cor. 3.5.4] for each m ∈ Z ≥ with m ≡ ℓ (mod s ) (0 < ℓ ≤ s ), the leading coefficientmatrix of [ t m ] s is of the form · · · · · · · · · · · · ∗ . . . . . . . . . ... ∗ ∗ · · · where the ( i, j )-th component is 1 (resp. 0) if i = j + s − ℓ (resp. i < j + s − ℓ ). Furthermore,we have deg τ [ t m ] s = l ms m . So the above properties directly lead to the following.
Lemma 3.1.3.
Let s be a positive integer and let b ( t ) ∈ F q [ t ] be a monic polynomial such that deg t b ( t ) is divisible by s . Then there exist polynomials f ij ( X ) ∈ XA [ X ] with deg X f ij ( X ) ≤ q (deg t b ( t )) /s and deg X f ij ( X ) < q (deg t b ( t )) /s if i ≤ j so that for any x = ( x , . . . , x s ) tr ∈ C ⊗ s ( k ) = k s and ( y , . . . , y s ) tr := [ b ( t )] s x , we have x q (deg t b ( t )) /s + s X j =1 f j ( x j ) = y x q (deg t b ( t )) /s + s X j =1 f j ( x j ) = y ... x q (deg t b ( t )) /s s + s X j =1 f sj ( x j ) = y s . (3.1.4)Then Lemma 3.1.3 enables us to ensure the v -adic integrality of certain v ( t ) n -divisionpoints in G s , u . Lemma 3.1.5.
Given s = ( s , . . . , s r ) ∈ N r and u = ( b , . . . , u r ) ∈ k r , let G s , u = ( G da , [ − ]) be the t -module defined in (2.2.9) . Let n ∈ N be a positive integer divisible by lcm( d , . . . , d r ) .Fix any algebraic point v ∈ G s , u ( k ) and suppose that v ′ ∈ G s , u ( k ) satisfies [ v ( t ) n ] v ′ = v . If u ∈ A r ( v ) ,ℓ and v ∈ G s , u ( A ( v ) ,ℓ ) for some ℓ ∈ N , then v ′ also lies in G s , u ( A ( v ) ,ℓ ) . LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 15
Proof.
We write[ v ( t ) n ] = [ v ( t ) n ] d M · · · M r [ v ( t ) n ] d · · · M r . . . ...[ v ( t ) n ] d r ∈ Mat d ( k [ τ ]) , and note that since u ∈ A r ( v ) ,ℓ , we have M ij ∈ Mat d i × d j ( A ( v ) ,ℓ [ τ ]) for all i, j . Hence we have[ v ( t ) n ] ∈ Mat d ( A ( v ) ,ℓ [ τ ]) as [ v ( t ) n ] d i ∈ Mat d i ( A [ τ ]) for i = 1 , . . . , r .We further write v = v ... v r ( v i ∈ Mat d i × ( A ( v ) ,ℓ )) and v ′ = v ′ ... v ′ r ( v ′ i ∈ Mat d i × ( k )) , then the equation [ v ( t ) n ] v ′ = v is expressed as(3.1.6) [ v ( t ) n ] d i v ′ i = v i − X i
1. Recall that q v := q deg θ v = q deg t b ( t ) s . Then we have | f ij ( x j ) | v ≤ max { , | x j | v } deg X f ij ( X ) ≤ (cid:26) | x i | deg X f ij ( X ) v ( j ≥ i )max { , | x j | v } q v ( j < i ) < | x i | q v v for each 1 ≤ j ≤ s . Therefore1 < | x i | q v v = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q v i + s X j =1 f ij ( x j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v = | y i | v ≤ v ′ ∈ C ⊗ s ( k ∩ O C v ).By [AT90, Proposition 1.6.1], we have[ v ( t ) s ] s v ′ ≡ v ′ (deg θ v ) (mod m v ) . So the relation [ v ( t ) s ] s v ′ ≡ v (mod m v ) implies v ′ (deg θ v ) = v ∈ C ⊗ s ( F q ℓv ) . Since F q ℓv is perfect, we have v ′ ∈ C ⊗ s ( F q ℓv ), and hence v ′ ∈ C ⊗ s ( A ( v ) ,ℓ ). (cid:3) In what follows, we need the notion of regular t -modules introduced by Yu in [Yu97,p. 218]. Definition 3.1.7.
Let G be a t -module defined over k . We say that G is regular if there is apositive integer ν for which the a -torsion submodule of G ( k ) is free of rank ν over F q [ t ] / ( a )for every nonzero polynomial a ∈ F q [ t ]. The following proposition is needed in the following subsection, where we prove Theo-rem 3.1.1. Moreover, it also enables us to apply Yu’s sub- t -module theorem for the t -module G s , u when proving Theorem 1.2.3 in Section 5. Proposition 3.1.8.
For any s ∈ N r and u ∈ k r with ˜ u ∈ D ˜ s , ∞ defined in (1.3.1) , the t -module G s , u is regular.Proof. The t -module G s , u is uniformizable by [CM19, Rem. 3.3.5] and [CM20, Rem. 2.5.2].Then by the same arguments of [CM20, Prop. 6.3.2], the desired result follows. (cid:3) Proof of Theorem 3.1.1.
Now we give a proof of Theorem 3.1.1. For each n ∈ Z ≥ ,we have[ v ( t ) n ] v n := [ v ( t ) n ] exp G s , u (cid:16) ∂ [ v ( t ) n ] − log G s , u ( v ) (cid:17) = exp G s , u (cid:16) ∂ [ v ( t ) n ] ∂ [ v ( t ) n ] − log G s , u ( v ) (cid:17) = exp G s , u (cid:16) log G s , u ( v ) (cid:17) = v . Denote by G s , u [ v ( t ) n ] the algebraic subgroup of v ( t ) n -torsion points of G s , u . As the alge-braic variety defined by [ v ( t ) n ] X = v is a G s , u [ v ( t ) n ]-torsor, it is a zero-dimensional varietydefined over k since G s , u is a regular t -module by Proposition 3.1.8. Therefore we have v n ∈ G s , u ( k ) , whence proving the property (1).To prove the property (2), we mention that by Lemma 3.1.5, if n is divisible by lcm( d , . . . , d r )then we have v n ∈ G s , u ( A ( v ) ,ℓ ) . Furthermore, from [CM19, Prop. 4.1.1] the choice of a ( t ) implies that (cid:12)(cid:12)(cid:12)(cid:12) [ a ( t )] v n (cid:12)(cid:12)(cid:12)(cid:12) v < n ∈ N with lcm( d , . . . , d r ) | n .Now we aim to show that (cid:12)(cid:12)(cid:12)(cid:12) Z n (cid:12)(cid:12)(cid:12)(cid:12) ∞ → n → ∞ . Let N b := ∂ [ b ( t )] − b ( θ ) I d for eachnonzero polynomial b ( t ) ∈ F q [ t ]. Then N b is a nilpotent matrix. Recall that N is thenilpotent matrix given in [ t ] in (2.2.9). Since N ∈ Mat d ( F p ) and ∂ [ t i ] = ( θI d + N ) i = θ i I d + X ≤ j
The main purpose of this section is to show Theorem 4.2.1.
LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 17
Formula for the weight coordinate.
In this subsection, we aim to give an formulafor the wt( s )-th coordinate of log G s , u at algebraic points in terms of CMSPL’s for each s ∈ N r and u ∈ k r with ˜ u ∈ D ˜ s , ∞ . We first mention that the ∞ -adic convergence domain of log G s , u is given in Theorem 2.3.1. We then recall that the wt( s )-th row of the coefficient matricesof log G s , u is explicitly given as follows. Proposition 4.1.1 ([CM19, Prop. 3.2.1], [Chen20, Prop. 3.2.2]) . Fix any s = ( s , . . . , s r ) ∈ N r and u = ( u , . . . , u r ) ∈ k r , let G s , u be defined in (2.2.9) , d i be given in (2.2.7) for i = 1 , . . . , r and d be given in (2.2.8) . We put log G s , u := X i ≥ P i τ i , P := I d , P i ∈ Mat d ( k ) , and write the wt( s ) -th row of P i as (cid:0) y , , . . . , y ,d , y , , . . . , y ,d , . . . , y r, , . . . , y r,d r (cid:1) . Then for each i ≥ , we have (4.1.2) y ,j = ( θ − θ q i ) d − j L d i for ≤ j ≤ d , and for each ≤ m ≤ r and ≤ j ≤ d m we have (4.1.3) y m,j = ( − m − ( θ − θ q i ) d m − j X ≤ i ≤···≤ i m −
Theorem 4.1.4 (cf. [Chen20, Thm. 3.2.9]) . Fix any s = ( s , . . . , s r ) ∈ N r and u =( u , . . . , u r ) ∈ k r with ˜ u ∈ D ˜ s , ∞ defined in (1.3.1) . Let G s , u be defined in (2.2.9) and d i be given in (2.2.7) for i = 1 , . . . , r . If we set x := ( x , , . . . , x ,d , x , , . . . , x ,d , . . . , x r, , . . . , x r,d r ) tr ∈ G s , u ( C ∞ ) , and assume | x m,j | ∞ < q − ( d m − j )+ dmqq − for each ≤ m ≤ r and ≤ j ≤ d m , then | θ ℓ x m,j | ∞ < q dmqq − (1 ≤ m ≤ r, ≤ j ≤ d m , ≤ ℓ ≤ d m − j ) and | θ ℓ x m,j u m − | ∞ < q dm − qq − (2 ≤ m ≤ r, ≤ j ≤ d m , ≤ ℓ ≤ d m − j ) , and the wt( s ) -th coordinate of log G s , u ( x ) is given by d X j =1 d − j X ℓ =0 ( − ℓ (cid:18) d − jℓ (cid:19) θ d − j − ℓ Li ⋆d ( θ ℓ x ,j )+ X ≤ m ≤ r ( − m − d m X j =1 d m − j X ℓ =0 ( − ℓ (cid:18) d m − jℓ (cid:19) θ d m − j − ℓ × (cid:8) Li ⋆ ( d m ,s m − ,...,s ) ( θ ℓ x m,j , u m − , . . . , u ) − Li ⋆ ( d m − ,s m − ,...,s ) ( θ ℓ x m,j u m − , u m − , . . . , u ) (cid:9) , where in the case of m = 2 , we denote by ( d m − , s m − , . . . , s ) := ( d ) and ( θ ℓ x m,j u m − , u m − , . . . , u ) := ( θ ℓ x ,j u ) . Proof.
We use the same notations as in Proposition 4.1.1. First of all, we note that since | θ ℓ x m,j | ∞ < q ℓ · q − ( d m − j )+ dmqq − ≤ q d m − j · q − ( d m − j )+ dmqq − = q dmqq − for each 1 ≤ m ≤ r , 1 ≤ j ≤ d m and 0 ≤ ℓ ≤ d m − j , and | θ ℓ x m,j u m − | ∞ < q ℓ · q − ( d m − j )+ dmqq − · q sm − qq − ≤ q d m − j · q − ( d m − j )+ dmqq − · q sm − qq − = q dm − qq − , for each 2 ≤ m ≤ r , 1 ≤ j ≤ d m and 0 ≤ ℓ ≤ d m − j , each CMSPL in Theorem 4.1.4converges ∞ -adically.According to Theorem 2.3.1, log G s , u ( x ) converges ∞ -adically. Since we write log G s , u = P ∞ i =0 P i τ i , the wt( s )-th coordinate of log G s , u ( x ) is given by ∞ X i =0 r X m =1 d m X j =1 y m,j x q i m,j . We claim that the series r X m =1 d m X j =1 ∞ X i =0 y m,j x q i m,j converges ∞ -adically, and so it is equal to the wt( s )-th coordinate of log G s , u ( x ).To prove the claim above, we compute P ∞ i =0 y m,j x q i m,j for each 1 ≤ m ≤ r and 1 ≤ j ≤ d m .When m = 1, we have n X i =0 y ,j x q i ,j = n X i =0 ( θ − θ q i ) d − j x q i ,j L d i = n X i =0 d − j X ℓ =0 (cid:18) d − jℓ (cid:19) θ d − j − ℓ ( − θ q i ) ℓ x q i ,j L d i = d − j X ℓ =0 ( − ℓ (cid:18) d − jℓ (cid:19) θ d − j − ℓ n X i =0 ( θ ℓ x ,j ) q i L d i → d − j X ℓ =0 ( − ℓ (cid:18) d − jℓ (cid:19) θ d − j − ℓ Li ⋆d ( θ ℓ x ,j )as n → ∞ for each 1 ≤ j ≤ d . When 2 ≤ m ≤ r , we have n X i =0 y m,j x q i m,j = n X i =0 ( − m − ( θ − θ q i ) d m − j X ≤ i ≤···≤ i m −
Consider the following k -vector spaces L Conv ∞ ⊂ L Def ∞ ⊂ C ∞ given in Definition 1.3.6. An important application of Theorem 4.1.4 is the following equality,which is the key for us to prove Theorem 1.2.3 in the next section. Theorem 4.2.1.
Let notation be given in Definition 1.3.6. Then we have the followingequality L Conv ∞ = L Def ∞ . Proof.
Suppose n = ( n , . . . , n r ) ∈ N r and w = ( w , . . . , w r ) ∈ D Def n , k (so Li ⋆ n ( w ) ∈ L Def ∞ ), ourgoal is to show that Li ⋆ n ( w ) ∈ L Conv ∞ .Put s := e n = ( n r , . . . , n ) , u := e w = ( w r , . . . , w )and write s = ( s , . . . , s r ). Let G s , u be defined in (2.2.9) and v s , u be defined in (2.2.10). Let v := v s , u . For this v , we let a ( t ), n , Z n and v n be given as in Theorem 3.1.1 by taking n sufficiently large and divisible by lcm( d , . . . , d r ) so that all the properties of Theorem 3.1.1hold. We claim the following: • log G s , u ([ v ( t ) n ] v n ) = ∂ [ v ( t ) n ] log G s , u ( v n ). • log G s , u ([ a ( t )] v n ) = ∂ [ a ( t )] log G s , u ( v n ).We mention that although we have the functional equations of log G s , u as formal power series,one can not argue directly that the two identities above follow from the functional equationsas log G s , u is not entire.Assume the claim first. Then we have ∂ [ v ( t ) n ] log G s , u ([ a ( t )] v n ) = ∂ [ v ( t ) n ] ∂ [ a ( t )] log G s , u ( v n )= ∂ [ a ( t )] log G s , u ([ v ( t ) n ] v n )= ∂ [ a ( t )] log G s , u ( v s , u ) , where the first and second equalities follow from the claim. By Theorem 2.3.1 and comparingthe wt( s )-th coordinates of the both sides, we haveLi ⋆ n ( w ) = Li ⋆ ˜ s (˜ u ) = ( − r − a ( θ ) × (cid:16) wt( s )-th coordinate of ∂ [ a ( t )] log G s , u ( v s , u ) (cid:17) = ( − r − v n a ( θ ) × (cid:16) wt( s )-th coordinate of log G s , u ([ a ( t )] v n ) (cid:17) . Here we used the fact that the wt( s )-th component of the wt( s )-th row of ∂ [ b ( t )] is b ( θ )and the other components are zero for each b ( t ) ∈ F q [ t ]. By Theorem 3.1.1 we have that (cid:12)(cid:12)(cid:12)(cid:12) [ a ( t )] v n (cid:12)(cid:12)(cid:12)(cid:12) v <
1. Since k u k v = k w k v ≤
1, by putting x := [ a ( t )] v n into Theorem 4.1.4 wesee that the first coordinate of each CMSPL appearing in the formula of Theorem 4.1.4 hasthe v -adic absolute value strictly less than one and hence the right hand side of the equationabove is in L Conv ∞ .Now, we prove the claim above. We first recall that D G s , u is the domain on which exp G s , u is an isometry. Since Z n ∈ D G s , u and v n := exp G s , u ( Z n ), we have log G s , u ( v n ) = Z n . It follows that we obtain the first desired identitylog G s , u ([ v ( t ) n ] v n ) = log G s , u ( v ) = ∂ [ v ( t ) n ] Z n = ∂ [ v ( t ) n ] log G s , u ( v n ) , where the second equality comes from the definition Z n := ∂ [ v ( t ) n ] − log G s , u ( v ). Sincelog G s , u ([ a ( t )] v n ) , ∂ [ a ( t )] log G s , u ( v n )belong to D G s , u , on which exp G s , u is an isometry, the second desired identitylog G s , u ([ a ( t )] v n ) = ∂ [ a ( t )] log G s , u ( v n )follows from the functional equations of exp G s , u and its entireness:exp G s , u (cid:16) log G s , u ([ a ( t )] v n ) (cid:17) = [ a ( t )] v n = exp G s , u (cid:16) ∂ [ a ( t )] log G s , u ( v n ) (cid:17) . (cid:3) Main theorem and proof
The primary goal of this section is to prove Theorem 1.2.3.5.1.
Yu’s sub- t -module theorem. In our function field setting, we have the followinganalogue of W¨ustholz’s theory, called Yu’s sub- t -module theorem. Theorem 5.1.1 ([Yu97, Thm. 0.1]) . Let G be a regular t-module defined over k . Let Z be a vector in Lie G ( C ∞ ) such that exp G ( Z ) ∈ G ( k ) . Then the smallest linear subspace in Lie G ( C ∞ ) defined over k , which is invariant under ∂ [ t ] and contains Z , is the tangent spaceat the origin of a sub- t -module H of G over k . Here, a sub- t -module of G over k is a connected algebraic subgroup of G defined over k so that it is invariant under the [ t ]-action. The following lemma plays a crucial role so thatwe can apply Yu’s sub- t -module theorem appropriately to prove Theorem 5.1.5. Lemma 5.1.2.
Let s ∈ N r be an index and u ∈ k r such that ˜ u ∈ k r ∩ D Def ˜ s ,v defined in (1.3.3) .For any point x ∈ G s , u ( C v ) with k x k v < and any ǫ > , there exists n ∈ Z ≥ such that (cid:12)(cid:12)(cid:12)(cid:12) [ v ( t ) n ] x (cid:12)(cid:12)(cid:12)(cid:12) v < ǫ .Proof. For each ǫ >
0, we set a ǫ := { x ∈ O C v | | x | v < ǫ } . Since G s , u is defined over k ∩ O C v , it is clear that the F q [ t ]-action [ − ] on G s , u induces an F q [ t ]-action on ( O C v / a ǫ ) d via [ − ] where d is given in (2.2.8), and without confusion we denoteby G s , u ( O C v / a ǫ ) for the F q [ t ]-module (( O C v / a ǫ ) d , [ − ]). Note that by definition we have thefollowing equivalence (cid:12)(cid:12)(cid:12)(cid:12) [ v ( t ) n ] x (cid:12)(cid:12)(cid:12)(cid:12) v < ǫ ⇐⇒ [ v ( t ) n ]( x mod a ǫ ) = 0 in G s , u ( O C v / a ǫ ) . We prove the lemma by induction on the depth r = dep( s ). When r = 1 and s = ( s ),by [AT90, Proposition 1.6.1], we have f ij ( X ) ∈ vXA [ X ] (1 ≤ i, j ≤ r ) for b ( t ) = v ( t ) s inLemma 3.1.3. Therefore we have(5.1.3) (cid:12)(cid:12)(cid:12)(cid:12) [ v ( t ) s ] x (cid:12)(cid:12)(cid:12)(cid:12) v ≤ max {k x k q v v , k x k v /q v } = ( k x k q v v (if k x k v ≥ q − / ( q v − v ) k x k v /q v (if k x k v ≤ q − / ( q v − v ) . LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 21
We set x i := [ v ( t ) is ] x . If k x i k v > q − / ( q v − v for all i ∈ Z ≥ , then q − / ( q v − v < k x i k v ≤ k x k q iv v for all i ∈ Z ≥ , where the second inequality comes from (5.1.3). Since k x k v <
1, we have acontradiction. Therefore there exists i ∈ Z ≥ such that k x i k v ≤ q − / ( q v − v . Then we have k x i k v ≤ k x i k v /q i − i v ( i ≥ i )and hence k x i k v → i → ∞ ).Next, let r ≥ G ′ := G ( s ,...,s r ) , ( u ,...,u r ) . Let π := (cid:0) ( x , , . . . , x ,d , x , , . . . , x ,d , . . . ) tr ( x , , . . . , x ,d , . . . ) tr (cid:1) : G s , u ։ G ′ be the natural projection. Then we have the following exact sequence of F q [ t ]-modules0 / / C ⊗ wt( s ) ( O C v / a ǫ ) / / G s , u ( O C v / a ǫ ) π ǫ / / G ′ ( O C v / a ǫ ) / / , which is induced from the short exact sequence of t -modules0 / / C ⊗ wt( s ) / / G s , u π / / G ′ / / . By the induction hypothesis, there exists n ′ ∈ Z ≥ such that π ǫ (cid:16) [ v ( t ) n ′ ]( x mod a ǫ ) (cid:17) = [ v ( t ) n ′ ] ( π ǫ ( x mod a ǫ )) = 0and hence [ v ( t ) n ′ ]( x mod a ǫ ) ∈ ker π ǫ = C ⊗ wt( s ) ( O C v / a ǫ ). By the argument of the depth onecase, there exists n ∈ Z ≥ such that[ v ( t ) n ] (cid:16) [ v ( t ) n ′ ]( x mod a ǫ ) (cid:17) = 0 , whence deriving [ v ( t ) n + n ′ ]( x mod a ǫ ) = 0 . (cid:3) We recall the notion of tractable coordinates introduced by Brownawell-Papanikolas, whichis convenient for us when applying Yu’s sub- t -module theorem. Definition 5.1.4.
Let G = ( G da , [ − ]) be a t -module over k and let X = ( X , . . . , X d ) tr bethe coordinates of Lie G . The i -th coordinate X i is called tractable if the i -th coordinate of ∂ [ a ( t )] X is equal to a ( θ ) · X i for each a ( t ) ∈ F q [ t ].By the definition of the t -module G s , u in (2.2.9), the ( d + · · · + d i )-th coordinate of Lie G s , u is tractable for each 1 ≤ i ≤ r . In particular, the wt( s )-th coordinate of Lie G s , u is tractable. Theorem 5.1.5.
The map φ v := (Li ⋆ s ( u ) Li ⋆ s ( u ) v ) : L Def ∞ ։ L Def v is a well-defined k -linear map.Proof. Suppose that we have α + P mi =1 α i Li ⋆ n i ( w i ) = 0 for α , α , . . . , α m ∈ k (not all zero), n i ∈ S r> N r , w i ∈ D Def n i , ∞ . Our aim is to show that α + m X i =1 α i Li ⋆ n i ( w i ) v = 0 . We set s i := ˜ n i and u i := ˜ w i and define the t -module G := G a ⊕ m M i =1 G s i , u i with diagonal t -action, where G a is referred to the trivial t -module with exponential andlogarithm maps given by the identity map z z .Let X be the coordinate of Lie G a and X i be the coordinates of Lie G s i , u i for i = 1 , . . . , m .Put j i := wt( s i ) and define X ij i to be the j i -th coordinate of X i which is tractable inLie G s i , u i for i = 1 , . . . , m . So X := ( X , X tr1 , . . . , X tr m ) tr are coordinates of Lie G and { X , X j . . . , X mj m } are tractable coordinates.Put v := (1) ⊕ L mi =1 v s i , u i ∈ G ( k ). By Theorem 2.3.3, there exists a nonzero polynomial a ( t ) ∈ F q [ t ] such that (cid:12)(cid:12)(cid:12)(cid:12) [ a ( t )] v (cid:12)(cid:12)(cid:12)(cid:12) v <
1. By Theorem 2.3.1 we have that log G converges ∞ -adically (resp. v -adically) at v (resp. [ a ( t )] v ). Note that ∂ [ a ( t )] log G ( v ) is the column vectorwhose entries are the concatenation of a ( θ ) and the column vectors ∂ [ a ( t )] log G s , u ( v s , u ) , . . . , ∂ [ a ( t )] log G s m, u m ( v s m , u m ) , and log G ([ a ( t )] v ) v is the column vector whose entries are the concatenation of a ( θ ) and thecolumn vectors log G s , u ([ a ( t )] v s , u ) v , . . . , log G s m, u m ([ a ( t )] v s m , u m ) v . Furthermore, by Theorem 2.3.1 the value ( − dep( n i ) − a ( θ ) Li ⋆ n i ( w i ) (resp. ( − dep( n i ) − a ( θ ) Li ⋆ n i ( w i ) v )occurs as the j i -th coordinate of ∂ [ a ( t )] log G s i, u i ( v s i , u i ) (resp. log G s i, u i ([ a ( t )] v s i , u i ) v ) for i =1 , . . . , m .Let V be the smallest k -linear subvariety of Lie G for which • V ( C ∞ ) contains the vector ∂ [ a ( t )] log G ( v ). • V is invariant under the ∂ [ t ]-action.By Yu’s sub- t -module theorem, we have V = Lie H for some sub- t -module H of G definedover k . We note that the hyperplane α X + ( − dep( n ) − α X j + · · · + ( − dep( n m ) − α m X mj m = 0is a k -linear subvariety of Lie G and contains the vector ∂ [ a ( t )] log G ( v ) as a C ∞ -valued pointand is invariant under the ∂ [ t ]-action. It follows from the definition of V that V = Lie H ⊂ (cid:8) α X + ( − dep( n ) − α X j + · · · + ( − dep( n m ) − α m X mj m = 0 (cid:9) . As ∂ [ a ( t )] log G ( v ) ∈ Lie H ( C ∞ ) ⊆ Lie G ( C ∞ ), we have that[ a ( t )] v = exp G ( ∂ [ a ( t )] log G ( v )) = exp H ( ∂ [ a ( t )] log G ( v )) ∈ H ( k ) . By putting x := [ a ( t )] v in Lemma 5.1.2, there exisits n ∈ Z ≥ such that [ v ( t ) n ][ a ( t )] v ∈ H ( k )is v -adically small. Then by the same arguments of [CM20, Thm. 6.4.1], we have thatlog G ([ v ( t ) n a ( t )] v ) v = log H ([ v ( t ) n a ( t )] v ) v ∈ Lie H ( C v ) , and hence this vector is a C v -valued point of the hyperplane above. That is, the desiredlinear relation holds. (cid:3) Proof of Theorem 1.2.3.
Now we give a proof of Theorem 1.2.3. We first note thatby (2.3.4) and (2.3.5) we have Z ∞ ⊂ L Def ∞ , Z v ⊂ L Def v , φ v ( Z ) = Z v and φ v ( ζ A ( s )) = ζ A ( s ) v for each index s . We note that Theorem 4.2.1 implies L Def v = L Conv v as explained in (1.3.7).Therefore, we have the following commutative diagram: Z φ v | Z (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / L Def ∞ φ v (cid:15) (cid:15) (cid:15) (cid:15) L Conv ∞ h L i ⋆ ( − ) o o o o L i ⋆ ( − ) v w w w w ♦♦♦♦♦♦♦♦♦♦♦♦♦ Z v (cid:31) (cid:127) / / L Def v L Conv v LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 23 where the commutativity φ v ◦ L i ⋆ ( − ) = L i ⋆ ( − ) v comes from the definitions of φ v , L i ⋆ ( − )and L i ⋆ ( − ) v .Note that L Def ∞ and L Def v form k -algebras by Proposition 2.2.3. We first show that the map φ v is a k -algebra homomorphism. Indeed, for each x, x ′ ∈ L Def ∞ , let w, w ′ ∈ h such that L i ⋆ ( w ) = x and L i ⋆ ( w ′ ) = x ′ . Since L i ⋆ ( − ) and L i ⋆ ( − ) v are multiplicative in the sense ofProposition 2.2.3, we have φ v ( x · x ′ ) = φ v ( L i ⋆ ( w ) · L i ⋆ ( w ′ )) = φ v ( L i ⋆ ( w ⋆ w ′ )) = L i ⋆ ( w ⋆ w ′ ) v = L i ⋆ ( w ) v · L i ⋆ ( w ′ ) v = φ v ( L i ⋆ ( w )) · φ v ( L i ⋆ ( w ′ )) = φ v ( x ) · φ v ( x ′ ) . Next we show that v -adic MZV’s satisfy the q -shuffle relations. Let s and s ′ be two indicesand let f j ∈ F p and s j ∈ N dep( s j ) be as in (1.2.1). Then we have ζ A ( s ) v · ζ A ( s ′ ) v = φ v ( ζ A ( s )) · φ v ( ζ A ( s ′ )) = φ v ( ζ A ( s ) · ζ A ( s ′ )) = φ v X j f j ζ A ( s j ) ! = X j f j φ v ( ζ A ( s j )) = X j f j ζ A ( s j ) v . Therefore Theorem 1.2.3 (1) holds. In particular, Z v is closed under the product, and hence φ v | Z is a k -algebra homomorphism. This shows Theorem 1.2.3 (2). Remark . In the proof above, we verify the identity L Def v = L Conv v , which generalizes [Chen20, Cor. 3.2.11] for v -adic CMSPL’s at integral points. Remark . The definition of ζ A ( s ) v in (2.3.5) a priori depends on the extensions of the v -adic CMSPL’s Li ⋆ s ℓ to D Def s ℓ ,v . However, by Theorems 4.2.1 and 5.1.5, ζ A ( s ) v is the image of ζ A ( s ) via the homomorphism φ v : L Conv ∞ → L Conv v which is ‘canonical’ once we fix embeddings k ֒ → C ∞ and k ֒ → C v over k . Note that the definition of ζ A ( s ) v does not depend on thechoice of such embeddings. Indeed, if we take another pair of embeddings with φ ′ v as thecorresponding homomorphism, then the equality φ v ( ζ A ( s )) = ζ A ( s ) v = φ ′ v ( ζ A ( s )) is still validin k v .5.3. An example.
We provide an example of direct computations for Theorem 1.2.3 (1).Recall Huei-Jeng Chen’s explicit formula [Ch15] for the product of two Carlitz zeta values ζ A ( r ) ζ A ( s ) = ζ A ( r, s ) + ζ A ( s, r ) + ζ A ( r + s )(5.3.1) + X i + j = r + s, ( q − | j (cid:20) ( − s − (cid:18) j − s − (cid:19) + ( − r − (cid:18) j − r − (cid:19)(cid:21) ζ A ( i, j ) . Now, we compute the simplest case ζ A (1) θ · ζ A (1) θ for q = 2 ℓ ( ℓ ∈ N ). Note that even forthis simplest case, it still involves heavy computation on the explicit action of Carlitz tensorpowers. In fact, to verify the validity of the q -shuffle product of v -adic MZV’s by directcomputations seems impracticable.We start by specializing r = 1 and s = 1 in Huei-Jeng Chen’s formula and we get ζ A (1) · ζ A (1) = ζ A (2) as the characteristic of the base field is 2. Note that this relation also follows from thedefinition of MZV’s directly. To verify ζ A (1) θ · ζ A (1) θ = ζ A (2) θ , we recall that the Anderson-Thakur polynomials [AT90, AT09] H n − is equal to 1 if 1 ≤ n ≤ q . In this case, we have ζ A ( n ) θ = Li ⋆n (1) θ (see [CM20, Sec. 5, Sec. 6] for details). So our task is to calculate Li ⋆n (1) θ . By definition,Li ⋆n (1) θ is given by 1 θ n − × ( n -th coordinate of log C ⊗ n ([ t n − θ ) , where C ⊗ n stands for the n -th tensor power of the Carlitz module defined in (3.1.2). Onecan show that [Chen20, Ex. 3.2.12][ t n − (cid:18) n (cid:19) θ, (cid:18) n (cid:19) θ , . . . , (cid:18) nn − (cid:19) θ n − , θ n ) tr and consequentlyLi ⋆n (1) θ = 1 θ n − Li ⋆n ( θ n ) θ + X ≤ j In what follows, we conjecture that the kernel of the k -algebra homo-morphism in Theorem 1.2.3 is generated by ζ A ( q − Conjecture 5.4.1. For any finite place v of k , we have the following k -algebra isomorphism Z / ( ζ A ( q − ∼ = Z v . The conjecture above would imply the following important consequences:(i) Z v ∼ = Z v ′ for any finite places v, v ′ of k .(ii) Z v is a graded k -algebra (graded by weights) defined over k .Note that in [C14, Thm. 2.2.1], the first author of the present paper showed that Z forms agraded k -algebra (graded by weights) that is defined over k . That is: • ∞ -adic MZV’s of different weights are linearly independent over k . • k -linear independence of ∞ -adic MZV’s implies k -linear independence. LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 25 So the statement (ii) above is the v -adic analogue of [C14, Thm. 2.2.1] for ∞ -adic MZV’s.We mention that in the case of p -adic MZV’s, one has Furusho-Yamashita’s conjecture [Ya10,Conj. 5] asserting that nonzero p -adic MZV’s of different weights are linearly independentover Q , and this is the p -adic analogue of Goncharov’s direct sum conjecture [Gon97] forreal-valued MZV’s.Now, we consider the k -subalgebra Z ⊂ Z generated by the ∞ -adic single zeta values,namely the ∞ -adic MZV’s of depth one, and the k -subalgebra Z v ⊂ Z v generated by the v -adic single zeta values. When we restrict the k -algebra homomorphism given in Theo-rem 1.2.3 to Z , we obtain the surjective k -algebra homomorphism Z ։ Z v . It is shown in [CY07] that all the algebraic relations among ∞ -adic single zeta values aregenerated by Euler-Carlitz relations, namely ζ A (( q − n ) /ζ A ( q − n ∈ k for n ∈ N , andthe p -th power relations, ie., ζ A ( pn ) = ζ A ( n ) p for n ∈ N . Recall that by [Go79] we have thetrivial zeros ζ A (( q − n ) v = 0 for n ∈ N . Chang-Yu’s conjecture in [CY07, p. 323] assertsthat all the algebraic relations among Goss’ v -adic zeta values come from the trivial zerosabove and the p -th power relations, and hence it implies that the kernel of the k -algebrahomomorphism Z ։ Z v is generated by ζ A ( q − 1) in Z . So Chang-Yu’s conjecture matcheswith the phenomenon of Conjecture 5.4.1. References [A86] G. W. Anderson, t -motives , Duke Math. J. (1986), no. 2, 457–502.[ABP04] G. W. Anderson, W. D. Brownawell and M. A. Papanikolas, Determination of the algebraic relationsamong special Γ -values in positive characteristic , Ann. of Math. (2) (2004), no. 1, 237–313.[AT90] G. W. 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Yu, Transcendence and special zeta values in characteristic p , Ann. of Math. (2) (1991), no. 1,1-23.[Yu97] J. Yu, Analytic homomorphisms into Drinfeld modules , Ann. of Math. (2) (1997), no. 2, 215–233.[Zh16] J. Zhao, Multiple zeta functions, multiple polylogarithms and their special values , Series on NumberTheory and its Applications, 12. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. LGEBRA STRUCTURE OF MULTIPLE ZETA VALUES IN POSITIVE CHARACTERISTIC 27 Department of Mathematics, National Tsing Hua University, Hsinchu City 30042, TaiwanR.O.C. E-mail address : [email protected] Department of Mathematics, National Tsing Hua University, Hsinchu City 30042, TaiwanR.O.C. E-mail address : [email protected] Department of Mathematical Sciences, University of the Ryukyus, 1 Senbaru, Nishihara-cho, Okinawa 903-0213, Japan E-mail address ::