Leading terms of anticyclotomic Stickelberger elements and p-adic periods

TL;DR

This paper constructs and analyzes anticyclotomic Stickelberger elements for Hilbert modular forms over quadratic extensions, relating their leading terms to automorphic L-invariants and extending known results to broader number fields.

Contribution

It extends the theory of Stickelberger elements and L-invariants from rational fields to arbitrary totally real fields, providing bounds and explicit descriptions of leading terms.

Findings

Bound the order of vanishing of Stickelberger elements.

Describe leading terms via automorphic L-invariants.

Show equality of automorphic and arithmetic L-invariants for totally imaginary fields.

Abstract

Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spie{\ss} from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic L-invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.