Periods, supercongruences, and their motivic lifts

Julian Rosen

Published 2016-08-24Version 1

A period is a complex number arising as the integral of a rational function, with rational coefficients, over a rationally-defined region. Though periods are typically transcendental, the theory of motives predicts a version of Galois theory should hold for periods. The motivic Galois theory has an explicit description for a class of periods called multiple zeta values. This paper explores an analogy between multiple zeta values and supercongruences, which are congruences modulo prime powers. We prove many new supercongruences by expressing terms as $p$-adic series involving combinatorial defined rational numbers called multiple harmonic sums. We use the series expansions to express certain quantities as infinite series involving $p$-adic analogues of multiple zeta values. We construct a period mapping, and use it to define motivic lifts of supercongruences. The lifts are formal Laurent series over the ring of motivic multiple zeta values. Via the motivic lifts, we construct a Galois theory for supercongruences. We also provide a software package for proving supercongruences, and for computing the motivic lifts.