Trivial multiple zeta values in Tate algebras

Abstract

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras introduced by the second author. If the number of variables involved is “not large” in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over ${\mathbb{F}}_q$. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of linear relations between Thakur’s multiple zeta values.

Universal families of Eulerian multiple zeta values in positive characteristic

“…Harmonic products. We are ready to prove our main result, by variants of the harmonic products discussed in [24]. Proposition 3.5.…”

“…It is in fact the analysis of the interesting phenomenology described in [28] that brought us to the path that gave rise to the present paper. The reader will also notice similarities in the processes behind the proofs of, on one side, [27, Theorem 3.2] and, on the other side, the proof of [24,Lemma 6.11].…”

“…The proof of Theorem 0.3 is based on the key notion of L-series in Tate algebras introduced by the third author in [31]. This notion has turned out to be very fruitful in function field arithmetic, see for example [5,6,7,9,10,11,12,24,26]. In our setting, these functions can be identified with rigid analytic functions over the analytification (X \ {∞}) an C∞ of the affine curve Spec(A ⊗ Fq C ∞ ) over C ∞ .…”

“…The entire function Z k can be viewed as an example of multiple zeta value in the Tate algebra T(K). These functions have been investigated in [24] in the case of X = P 1 and ∞ an F q -rational point. We will not pursue the general theory of such functions because in the present paper we only need to work with the entire functions…”

“…We also set Z −1 := 0. Also, Lemma 1.1 can be easily generalized to match with the formalism of [24] and the reader can see that all the functions that naturally arise in this picture define entire functions over products of the curve X(C ∞ ) \ {∞} in an appropriate sense.…”