L-functions of p-ADIC characters

We define a p-adic character to be a continuous homomorphism from 1 + tFq[[t]] to ℤ*p. For p > 2, we use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences (ci)(i,p)=1 of elements in ℤq, indexed by natural numbers relatively prime to p, and for which limi→ci = 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 (ci) 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 qC. Finally, we exhibit examples of C log-convergent sequences with associated L-functions which are not meromorphic in the disk of radius qC+ε for any ε > 0. © 2014 by The Editorial