Leading terms of anticyclotomic Stickelberger elements and $p$-adic periods

摘要

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 Spieß  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 $ mathcal {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 $ mathcal {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.