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  Non-paritious Hilbert modular forms

Dembélé, L., Loeffler, D., & Pacetti, A. (2019). Non-paritious Hilbert modular forms. Mathematische Zeitschrift, 292(1-2), 361-385. doi:10.1007/s00209-019-02229-5.

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Item Permalink: http://hdl.handle.net/21.11116/0000-0004-85C8-4 Version Permalink: http://hdl.handle.net/21.11116/0000-0004-85CC-0
Genre: Journal Article

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Dembélé-Loeffler-Pacetti_Non-paritious Hilbert modular forms_ 2019.pdf (Publisher version), 447KB
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© The Author(s) 2019 OpenAccess This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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https://doi.org/10.1007/s00209-019-02229-5 (Publisher version)
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 Creators:
Dembélé, Lassina1, Author              
Loeffler, David, Author
Pacetti, Ariel, Author
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 Abstract: The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious" - all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective l-adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis' result should hold, giving Galois representations into certain groups intermediate between GL(2) and PGL(2), and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with a selection of examples.

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Language(s): eng - English
 Dates: 2019
 Publication Status: Published in print
 Pages: 25
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 Table of Contents: -
 Rev. Method: Peer
 Identifiers: arXiv: 1612.06625
URI: http://arxiv.org/abs/1612.06625
DOI: 10.1007/s00209-019-02229-5
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Title: Mathematische Zeitschrift
  Abbreviation : Math. Z.
Source Genre: Journal
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Publ. Info: Springer
Pages: - Volume / Issue: 292 (1-2) Sequence Number: - Start / End Page: 361 - 385 Identifier: -