Explicit Formulae for $L$-values in Positive Characteristic

Abstract

We focus on the generating series for the rational special values of Pellarin's $L$-series in $1\le s\le 2(q-1)$ indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson-Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin's $L$-series in the same range of $s$, and divisibility results on the numerators of the Bernoulli-Carlitz numbers by monic irreducibles of degrees one and two.

Some helpful clarifications in the introduction. Final edit, as appears online in Math. Z

References in corpus (3)

• Values of certain L-series in positive characteristic
• Arithmetic of positive characteristic L-series values in Tate algebras
• A Dirichlet unit theorem for Drinfeld modules

Cited by in corpus (3)

• An exact degree for multivariate special polynomials
• Effective rigid analytic trivializations for Drinfeld modules
• On a Conjecture of Gezmis and Pellarin