On the Bloch-Kato conjecture for Hilbert modular forms

TL;DR

This paper proves inequalities related to the Bloch-Kato conjecture for certain Hilbert modular forms by using an inductive Euler system approach and explicit reciprocity laws involving automorphic forms and Shimura curves.

Contribution

It introduces an inductive Euler system method combined with explicit reciprocity laws to establish inequalities towards the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two.

Findings

Established inequalities for the Bloch-Kato conjecture in specific cases

Developed explicit reciprocity laws for cohomology classes

Applied automorphic form congruences and Shimura curve points

Abstract

The aim of this paper is to prove inequalities towards instances of the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the $L$-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.