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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Felix Bergunde and Lennart Gehrmann
Journal: Trans. Amer. Math. Soc. 370 (2018), 6297-6329
MSC (2010): Primary 11F67; Secondary 11F75, 11G18, 11G40
DOI: https://doi.org/10.1090/tran/7120
Published electronically: February 21, 2018
MathSciNet review: 3814331
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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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Additional Information

Felix Bergunde
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany
Email: fbergund@math.uni-bielefeld.de

Lennart Gehrmann
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
MR Author ID: 1209875
Email: lennart.gehrmann@uni-due.de

Received by editor(s): August 2, 2016
Received by editor(s) in revised form: November 7, 2016
Published electronically: February 21, 2018
Additional Notes: The first-named author was financially supported by the DFG within the CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.
Article copyright: © Copyright 2018 American Mathematical Society