# The completed finite period map and Galois theory of supercongruences

## TL;DR

This paper develops a Galois theory for supercongruences by constructing an analogue of the motivic period map, enabling the systematic discovery and proof of supercongruences with an accompanying algorithm and software.

## Contribution

It introduces a novel Galois theory for supercongruences by formalizing an analogy with periods and constructing a corresponding period map and algorithm.

## Key findings

- Constructed a motivic period map analogue for supercongruences
- Developed an algorithm to find and prove supercongruences
- Provided software implementation of the algorithm

## Abstract

A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a rationally-defined region. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values, and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analogue of the motivic period map in the setting of supercongruences, and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.