Zeros of $p$-adic $L$-functions

COMPUTATION OF THE ZEROS OF p-ADIC ¿FUNCTIONS. II

The authors have carried out a computational study of the zeros of Kubota-Leopoldt p-adic L-functions. Results of this study have appeared recently in a previous article. The present paper is a sequel to that article, dealing with the computation of the zeros under certain conditions that complicate the original situation.

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES

Zeros of p-adic forms

A variant of Brauer’s induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.

L-FUNCTIONS OF p-ADIC CHARACTERS

We define a p-adic character to be a continuous homomorphism from 1 + tFq [[t]] to Zp. 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-f...

T-ADIC L-FUNCTIONS OF p-ADIC EXPONENTIAL SUMS

The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...