Local and global methods in arithmetic (in Russian)

TL;DR

This paper explores local and global methods for solving congruences modulo prime powers using p-adic numbers, illustrating their application in algebraic curves and computational techniques.

Contribution

It introduces a comparative analysis of local p-adic methods and global solutions, with examples linking congruences to integer solutions and demonstrating computational applications.

Findings

Local methods resemble real number computations.

Global methods connect congruences to integer solutions.

Computer illustrations for p-adic numbers and algebraic curves.

Abstract

Let $p$ be a prime. We discuss methods of solution of congruences modulo $p^n$ using $p$-adic numbers; these methods are similar to computations with real numbers (local methods). Examples of relations between local and global methods are given producing a passage from congruences to solutions in integers (global methods). The use of a computer is illistrated for the study of $p$-adic numbers and algebraic curves.