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

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).

Brief note added in separate file, indicating that a condition on which one of the results depends can now be removed, also, journal reference and DOI added. Main file unchanged

References in corpus (9)

• On Motives Associated to Graph Polynomials
• A K3 in phi4
• A database of number fields
• Modularity of rigid Calabi-Yau threefolds over Q
• Feynman integrals and motives
• Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras
• Rigid Calabi-Yau Threefolds over Q Are Modular
• Double covers of P^3 and Calabi-Yau varieties
• On the modularity of three Calabi-Yau threefolds with bad reduction at 11

Cited by in corpus (2)

• A special case of completion invariance for the $c_2$ invariant of a graph
• Modularity experiments on $S_4$-symmetric double octics