Residual paramodularity of a certain Calabi-Yau threefold

TL;DR

This paper establishes congruences between Hecke eigenvalues of certain Hilbert modular forms and a Siegel modular form associated with a Calabi-Yau threefold, providing evidence for a paramodularity conjecture.

Contribution

It proves a specific case of paramodularity for a Calabi-Yau threefold via congruences of automorphic forms, linking Galois representations to Siegel modular forms.

Findings

Proves congruences of Hecke eigenvalues between Hilbert modular forms.

Shows the 4-dimensional Galois representation arises from a Siegel modular form.

Establishes a congruence involving a lift of a Hilbert modular form.

Abstract

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms $f_{79}$ and $h_{79}$ over $F=\mathbb Q(\sqrt{5})$, 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 $\mathrm{Gal}(\overline{\mathbb Q} / \mathbb Q)$ on the 3rd cohomology of a certain Calabi-Yau threefold comes from a Siegel modular form $F_{79}$ 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 $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$, which has weight 3 and paramodular level $79\times 5^2$.