Arithmetic and intermediate Jacobians of some rigid Calabi-Yau threefolds

TL;DR

This paper constructs specific rigid Calabi-Yau threefolds over the rationals, verifies their modularity through L-series computations, and explores the relationship between their L-series and intermediate Jacobians, providing new examples and counterexamples.

Contribution

It introduces a novel construction of rigid Calabi-Yau threefolds over  with explicit L-series verification and examines their intermediate Jacobians, challenging existing conjectures.

Findings

Verification of modularity without previous methods

Construction of -models for intermediate Jacobians

Counterexamples to Yui's conjecture on L-series relations

Abstract

We construct Calabi-Yau threefolds defined over $\mathbb{Q}$ via quotients of abelian threefolds, and re-verify the rigid Calabi-Yau threefolds in this construction are modular by computing their L-series, without \cite{Dieulefait} or \cite{GouveaYui}. We compute the intermediate Jacobians of the rigid Calabi-Yau threefolds as complex tori, then compute a $\mathbb{Q}$-model for the 1-torus given a $\mathbb{Q}$-structure on the rigid Calabi-Yau threefolds, and find infinitely many examples and counterexamples for a conjecture of Yui about the relation between the $L$-series of the rigid Calabi-Yau threefolds and the $L$-series of their intermediate Jacobians.