New realizations of modular forms in Calabi-Yau threefolds arising from ϕ4 theory

Brown and Schnetz found that the number of points over Fp 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).