Residual paramodularity of a certain Calabi-Yau threefold

Abstract

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms f79 and h79 over F=Q(5–√), of weights (2,2) and (2,4) respectively, level of norm 79. In the main example, the modulus is a divisor of 5 in some coefficient field, in the secondary example a divisor of 2. The former allows us to prove that the 4-dimensional mod-5 representation of Gal(Q¯¯¯¯/Q) on the 3rd cohomology of a certain Calabi-Yau threefold comes from a Siegel modular form F79 of genus 2, weight 3 and paramodular level 79. This is a weak form of a conjecture of Golyshev and van Straten. In aid of this, we prove also a congruence of Hecke eigenvalues between F79 and the Johnson-Leung-Roberts lift JR(h79), which has weight 3 and paramodular level 79×52.