Congruences for Convolutions of Hilbert Modular Forms

TL;DR

This paper explores congruences related to Rankin convolution L-values of Hilbert modular forms, providing evidence for conjectured non-commutative Iwasawa theory relations.

Contribution

It proves weak forms of Kato's false Tate curve congruences for Rankin convolution L-values of Hilbert modular forms, advancing understanding in non-commutative Iwasawa theory.

Findings

Proved weak forms of Kato's congruences

Connected L-values with non-commutative Iwasawa theory

Extended congruence results to Hilbert modular forms

Abstract

Let $\f$ be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution $L$-values $L(\f,\g,s)$, where $\g$ is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's `false Tate curve' congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.