Modular forms and $p$-adic numbers (in Russian)

TL;DR

This paper explores the $p$-adic properties of arithmetical functions linked to modular form coefficients, illustrating congruences and computational methods to analyze their arithmetic behavior.

Contribution

It introduces new $p$-adic analysis techniques for modular forms and demonstrates their application to solving arithmetical problems with computational support.

Findings

Examples of congruences modulo $p$ and $p^n$ for modular forms.

Use of computer algorithms to study modular forms.

Insights into $p$-adic properties of arithmetical functions.

Abstract

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated.