On the order of vanishing of Stickelberger elements of Hilbert modular forms

TL;DR

This paper constructs Stickelberger elements for Hilbert modular cusp forms of weight 2 and uses recent results to establish lower bounds on their vanishing order, relating to conjectures on elliptic curves.

Contribution

It introduces a method to bound the order of vanishing of Stickelberger elements for Hilbert modular forms, connecting to conjectures on elliptic curves of rank zero.

Findings

Bound on the order of vanishing of Stickelberger elements.

Connection to the Birch and Swinnerton-Dyer conjecture for rank 0 elliptic curves.

Application of recent results by Dasgupta and Spiess.

Abstract

We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spiess to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's refined "Birch and Swinnerton-Dyer type"-conjecture for elliptic curves of rank 0 follows.