L-functions of p-adic characters

Abstract

We define a $p$-adic character to be a continuous homomorphism from $1+t{\mathbb{F}}_q\left[\right[t\left]\right]$ to ${\mathbb{Z}}_p^{\ast }$. For $p>2$, we use the ring of big Witt vectors over ${\mathbb{F}}_q$ to exhibit a bijection between $p$-adic characters and sequences $(c_i{)}_{(i,p)=1}$ of elements in ${\mathbb{Z}}_q$, indexed by natural numbers relatively prime to $p$, and for which ${lim\hspace{0.17em}}_{i\to \infty }{c}_i=0$. 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 $\left(c_i\right)$ is overconvergent. If more generally the sequence is $Clog$-convergent, we show that the associated $L$-function is meromorphic in the open disk of radius $q^C$. Finally, we exhibit examples of $Clog$-convergent sequences with associated $L$-functions which are not meromorphic in the disk of radius $q^{C+ϵ}$ for any $ϵ>0$.