On Zagier-Hoffman's conjectures in positive characteristic

Abstract

We study Todd-Thakur's analogues of Zagier-Hoffman's conjectures in positive characteristic. These conjectures predict the dimension and an explicit basis Tw of the span of characteristic p multiple zeta values of fixed weight w which were introduced by Thakur as analogues of classical multiple zeta values of Euler.In the present paper we first establish the algebraic part of these conjectures which states that the span of characteristic p multiple zeta values of weight w is generated by the set Tw. As a consequence, we obtain upper bounds for the dimension. This is the analogue of Brown's theorem and also those of Deligne-Goncharov and Terasoma.We then prove two results towards the transcendental part of these conjectures. First, we establish the linear independence for a large subset of Tw and yield lower bounds for the dimension. Second, for small weights we prove the linear independence for the whole set Tw and completely solve Zagier-Hoffman's conjectures in positive characteristic. Our key tool is the Anderson-Brownawell-Papanikolas criterion for linear independence in positive characteristic.

On Thakur’s basis conjecture for multiple zeta values in positive characteristic

“…We show in Corollary 3.7 that the two sets and are generating sets of the k -vector space for every positive integer w (the latter one was known by Ngo Dac [26]) and further show in Theorem 3.8 that the set is a generating set for .…”

“…In [26], Ngo Dac showed that for each , is a generating set for the k -vector space . As a consequence of Ngo Dac’s result, one has the upper bound result : Theoretically, to prove Thakur’s basis conjecture, it suffices to show that his conjectural basis is linearly independent over k .…”

“…Given a positive integer w , we let be the set consisting of all tuples of positive integers with and for all , and note that was used and studied in [26]. The overall arguments in the proof of Theorem 1.5 are divided into the following two parts: We show in Corollary 3.7 that the two sets and are generating sets of the k -vector space for every positive integer w (the latter one was known by Ngo Dac [26]) and further show in Theorem 3.8 that the set is a generating set for . We prove in Theorem 5.2 that is a linearly independent set over k . Since (see Proposition 2.1), it follows that is a k -basis of . …”

“…Then we adopt the ideas and methods rooted in [30, 26] to achieve Theorem 3.6, which enables us to show the results mentioned in (I). Since some essential arguments are from those in [26], we leave the detailed proofs in the appendix. However, under such abstraction, the maps given in Definition 3.4 are explicit.…”

Algebraic relations among Goss’s zeta values on elliptic curves

“…The special values of this type of L -function, called Goss’s zeta values, have been at the heart of function field arithmetic for the last forty years. Various works have revealed the importance of these zeta values for both their independent interest and for their applications to a wide variety of arithmetic problems, including multiple zeta values (see the excellent articles [54, 53] for an overview and also [46] for some recent results), Anderson’s log-algebraicity identities (see [2, 3, 6, 36, 51]) and Taelman’s units and the class formula à la Taelman (see [11, 24, 25, 26, 28, 43, 50] for recent progress and [10] for an overview).…”