Cohomology and congruences

TL;DR

This paper provides an elementary introduction to $p$-adic methods in de Rham cohomology of algebraic hypersurfaces, illustrating their applications in number theory, combinatorics, physics, and arithmetic geometry through explicit examples.

Contribution

It offers an accessible account of $p$-adic techniques in cohomology, based on joint work with Beukers, with applications across multiple mathematical disciplines.

Findings

Explicit examples of $p$-adic de Rham cohomology

Applications to number theory and combinatorics

Connections to mathematical physics and arithmetic geometry

Abstract

These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in S\~ao Paulo. We give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theory and combinatorics. These lectures are based on the series of our joint papers with Frits Beukers entitled \emph{Dwork crystals} (\cite{DCI,DCII,DCIII}). These methods also have applications in mathematical physics and arithmetic geometry (\cite{IN,Cartier0}), which we overview here towards the end. I am grateful to the organisers of both schools and to the participants of my courses whose questions stimulated writing these notes.