Shuffle Relations for Function Field Multizeta Values

Abstract

Despite the failure of naive analogs of the sum shuffle or the integral shuffle relations, we prove the existence of ‘shuffle’ relations for the multizeta values (for a general A, with a rational place at infinity) introduced [T04] by the author in the function field context. This makes the Fp-span of the multizeta values into an algebra. We effectively determine and prove all the Fp-coefficients identities (but not the Fp(t)-coefficients identities). 0. Introduction Multizeta values introduced and studied originally by Euler have been pursued recently again with renewed interest because of their emergence in studies in mathematics and mathematical physics connecting diverse viewpoints. See eg. introduction to [T09b] and references there. This paper is sequel to [T09b]. The author defined and studied two types of multizeta [T04, Sec 5.10] for function fields, one complex valued (generalizing the Artin-Weil zeta function) and the other with values in Laurent series ring over finite fields (generalizing the Carlitz zeta values). (For general background on function field arithmetic, we refer to [G96, T04]. ) For the Fq[t] case, the first type was completely evaluated in [T04] (see [M06] for more detailed study in the higher genus case). For the second type, the failure of sum and integral shuffle identities was noted, but different combinatorially involved identities were established or conjectured in [T04, T09b] as well as in the Masters thesis work [L09, L?] of Jose Alejandro Lara Rodriguez done at the University of Arizona. Also, period interpretation for these multizeta values was given in [AT09] in terms of explicit iterated extensions of the Carlitz-Tate t-motives. In contrast to the classical division between the convergent versus the divergent (normalized) values, all the values are convergent in our case. In place of the sum or the integral shuffle relations, we have different kinds of relations: the shuffle type relations with Fp-coefficients and the relations with Fp(t)-coefficients. (Classically, of course, there is no such distinction, the rational number field being the prime field in that case). In this paper, we show the existence of shuffle type relations proving that the product of multizeta values can also be expressed as a sum of some multizeta values, so that the Fp-span of all multizeta values is an algebra. While [T09b, L09, L?] conjectured and proved many such interesting relations (in the special case A = Fq[t]), which are combinatorially quite involved to describe unlike the classical case, here we prove the existence directly (for general A, defined below) rather than proving those conjectures. Date: October 16, 2009. The author was supported by NSA grant H98230-08-1-0049.