Abstract. In this paper, we give a simultaneous vanishing principle for the v-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where v is a finite place of the rational function field over a finite field. This principle establishes the fact that the v-adic vanishing of CMPLs at algebraic points is equivalent to its ∞-adic counterpart being Eulerian. This reveals a nontrivial connection between the v-adic and ∞-adic worlds in positive characteristic.
In this paper, we define finite Carlitz multiple polylogarithms and show that every finite multiple zeta value over the rational function field F q (θ) is an F q (θ)-linear combination of finite Carlitz multiple polylogarithms at integral points. It is completely compatible with the formula for Thakur MZV's established in [C14].
In this paper, we study multizeta values over function fields in characteristic p. For each d ≥ 2, we show that when the constant field has cardinality > 2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values. As a special case, this gives an affirmative answer to the function field analogue of a question of Y.
Let k be the rational function field over the finite field of q elements and k its fixed algebraic closure. In this paper, we study algebraic relations over k among the fundamental period π of the Carlitz module and the positive characteristic multizeta values ζ(n) and ζ(n, n) for an "odd" integer n, where we say that n is "odd" if q − 1 does not divide n. We prove that these three elements are either algebraically independent over k or satisfy some simple relation over k. We also prove that if 2n is "odd", then these three elements are algebraically independent over k.
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