New realizations of modular forms in Calabi-Yau threefolds arising from $\phi^4$ theory

TL;DR

This paper establishes a connection between modular forms and Calabi-Yau threefolds derived from graph hypersurfaces, providing new geometric realizations of modular forms of weights 3 and 4.

Contribution

It proves a relation between point counts over finite fields and modular forms for a specific example, refines this relation, and constructs new Calabi-Yau threefolds realizing certain modular forms.

Findings

Proved the relation for one modular form of weight 4 and two of weight 3.

Constructed two new rigid Calabi-Yau threefolds realizing Hecke eigenforms.

Suggested methods to prove similar relations for additional modular forms.

Abstract

It has been found experimentally by Brown and Schnetz that the number of points over ${\mathbb F}_p$ of a graph hypersurface is often related to the coefficients of a modular form. In this paper I prove this relation for one example of a modular form of weight $4$ and two of weight $3$, refine the statement and suggest a method of proving it for four more of weight $4$, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight $4$ (one provably and one conjecturally).