Quantitative level lowering for weight two Hilbert modular forms

TL;DR

This paper extends the understanding of Shimura curves and modular abelian varieties over totally real fields, showing that cohomological and ring theoretic congruence modules coincide even without multiplicity one.

Contribution

It generalizes Ribet and Takahashi's results to the Hilbert modular setting, analyzing parametrizations as quaternion algebras vary.

Findings

Cohomological and ring theoretic congruence modules are equal on Shimura curves.

Extension of results to cases lacking multiplicity one.

Provides new insights into the parametrization of modular abelian varieties.

Abstract

We generalize a result of Ribet and Takahashi on the parametrization of elliptic curves by Shimura curves to the Hilbert modular setting. In particular, we study the behaviour of the parametrization of modular abelian varieties by Shimura curves associated to quaternion algebras $D$ over a totally real field $F$, as we vary $D$. As a consequence, we obtain that on these Shimura curves, the cohomological congruence module is equal to the ring theoretic congruence module even in cases where we do not have multiplicity one, thereby extending results of Manning and B\"ockle-Khare-Manning.