Regular poles for the p-adic group $GSp_4$

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...

Group Rings over the p-Adic Integers

Lower bound for the poles of Igusa’s p-adic zeta functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P . Let f be a polynomial over R in n > 1 variables and let χ be a character of R×. Let Mi(u) be the number of solutions of f = u in (R/P i)n for i ∈ Z≥0 and u ∈ R/P i. These numbers are related with Igusa’s p-adic zeta function Zf,χ(s) of f . We explain the connection betwe...