L-functions of p-adic characters

TL;DR

This paper introduces a new framework for defining p-adic characters using Witt vectors, and studies the meromorphic properties of their associated L-functions based on convergence conditions.

Contribution

It establishes a bijection between p-adic characters and sequences in Z_q, and analyzes the meromorphicity of their L-functions under various convergence assumptions.

Findings

L-functions are p-adic meromorphic for overconvergent sequences.

Meromorphicity extends to sequences with c ext{-}log-convergence within a radius q^c.

Examples show some L-functions are not meromorphic beyond certain disks.

Abstract

We define a p-adic character to be a continuous homomorphism from 1 + t\Fq[[t]] to \Zp^*. We use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences (c_i) of elements in Zq, indexed by natural numbers relatively prime to p, and which converge to zero p-adically. To such a p-adic character we associate an L-function, and we prove that this L-function is p-adic meromorphic if the corresponding sequence (c_i) is overconvergent. If more generally the sequence is c\log-convergent, we show that the associated L-function is meromorphic in the open disk of radius q^c. Finally, we exhibit examples of c\log-convergent sequences with associated L-functions which are not meromorphic in any disk of radius greater than q^c.