Classification of p-adic functions satisfying Kummer type congruences

TL;DR

This paper introduces and classifies p-adic Kummer spaces of continuous functions satisfying specific congruences, exploring their properties, zeros, fixed points, and applications to number theory including Bernoulli numbers and Fermat quotients.

Contribution

It provides a classification of p-adic Kummer spaces, analyzes their algebraic properties, and connects these functions to Dirichlet L-values, Bernoulli numbers, and Fermat quotients.

Findings

Functions in these spaces have fixed points and zeros that are effectively computable.

The classification reveals ring properties and decompositions of these spaces.

Applications include insights into Bernoulli and Euler numbers and conjectures on L-functions.

Abstract

We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions. As a result, these functions have always a fixed point, functions of certain subclasses have always a unique simple zero in $\mathbb{Z}_p$. The fixed points and the zeros are effectively computable by given algorithms. This theory can be transferred to values of Dirichlet $L$-functions at negative integer arguments. That leads to a conjecture about their structure supported by several computations. In particular we give an application to the classical Bernoulli and Euler numbers. Finally, we present a link to $p$-adic functions that are related to Fermat quotients.