Regular poles for the p-adic group $GSp_4$-II

On the smallest poles of Igusa’s p-adic zeta functions

Let K be a p-adic field. We explore Igusa’s p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kn. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n = 2 and less than −1 for n = 3 which occur as ...

P-adic Valuations and K-regular Sequences

Abstract A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x) ∈ Qp[x], we consider the sequence {vp(f(n))}n=0, where vp is the p-adic valuation. We show that this sequence...

Complexity of regular invertible p-adic motions.

We consider issues of computational complexity that arise in the study of quasi-periodic motions (Siegel discs) over the p-adic integers, where p is a prime number. These systems generate regular invertible dynamics over the integers modulo p(k), for all k, and the main questions concern the computation of periods and orbit structure. For a specific family of polynomial maps, we identify condit...

SIMPLE p - ADIC GROUPS , II

0.1. For any finite group Γ, a “nonabelian Fourier transform matrix” was introduced in [L1]. This is a square matrix whose rows and columns are indexed by pairs formed by an element of Γ and an irreducible representation of the centralizer of that element (both defined up to conjugation). As shown in [L2], this matrix, which is unitary with square 1, enters (for suitable Γ) in the character for...

Group Rings over the p-Adic Integers