On a conjecture of Furusho over function fields

Taylor coefficients of Anderson-Thakur series and explicit formulae

“…In [CM19b], the first and third authors of the present paper verify a function field analogue of Furusho's conjecture asserting that the p-adic multiple zeta values (abbreviated as MZV's) satisfy the same Q-linear relations that the corresponding real-valued MZV's satisfy. The key technique in that paper is to show that any ∞-adic MZV can be realized as a coordinate of the logarithm of an explicitly constructed t-module at a specific algebraic point.…”

“…We begin by reviewing fiber coproducts of dual t-motives and the construction of G s and Z s mentioned in the introduction. With the results from Theorem 3.3.7 and Lemma 5.2.3, we use techniques developed in [CM19b] to derive an explicit formula for Z s , stated in our Main Theorem 6.2.1. Finally, in Theorem 6.3.4 we further relate any monomial of MZV's to a coordinate of the logarithm of a certain t-module evaluated at a special point, and give a description of the other coordinates of this logarithmic vector explicitly in terms of Taylor coefficients of t-motivic CMSPL's.…”

“…Carlitz multiple polylogarithms (CMPL's) were introduced by the first author in [C14] and are generalizations of the Carlitz polylogarithm of Anderson and Thakur from [AT90]. Carlitz multiple star polylogarithms (CMSPL's) were introduced by the first and third authors in [CM19b] in order to connect MZV's with logarithms of t-modules. The primary goal of this section is to derive the formula stated in Theorem 1.2.1, which connects the logarithmic vector (1.2.2) with these CMSPL's.…”

“…The key idea of the proof is that formula (5.2.4) is the deformation of (5.2.1). In the proof of Theorem 6.2.1 we do not need to know the precise coefficients b ℓ , so we avoid presenting the repetitive details given in [CM19b,p. 23].…”

“…6.1.3 for the detailed definitions). In [CM19b], the authors gave a logarithmic interpretation of ζ A (s) in the sense that there exists a vector Z s ∈ Lie G s (C ∞ ) for which Γ s ζ A (s) occurs as the wt(s)-th coordinate of Z s . The primary purpose of this paper is to give explicit formulae for all the coordinates of Z s .…”