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New realizations of modular forms in Calabi-Yau threefolds arising from $$\phi^{4}$$ theory. (English) Zbl 1420.11077
Summary: F. Brown and O. Schnetz [Commun. Number Theory Phys. 7, No. 2, 293–325 (2013; Zbl 1290.81083)] found that the number of points over $$\mathbb{F}_p$$ of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relations in a general geometric context. We also prove the relation for two examples of modular forms of weight 3 and two of weight 4, refine the statement and suggest a method of proving it for three more of weight 4, and use one of the proved examples to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).

##### MSC:
 11F23 Relations with algebraic geometry and topology 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Keywords:
Calabi-Yau varieties; modular forms; $$\phi^4$$ theory
GENREG; Magma
Full Text:
##### References:
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