Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Bergunde, Felix; Gehrmann, Lennart

doi:10.1090/tran/7120

Leading terms of anticyclotomic Stickelberger elements and $p$-adic periods
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by Felix Bergunde and Lennart Gehrmann PDF

Trans. Amer. Math. Soc. 370 (2018), 6297-6329 Request permission

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 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.

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