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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields
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by Jose Ignacio Burgos Gil and Ariel Pacetti PDF
Math. Comp. 86 (2017), 1949-1978 Request permission

Abstract:

Let $K$ be a real quadratic field and $\mathscr {O}_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\mathrm {SL}_2(\mathscr {O}_K)$ and $M_{(k_1,k_2)}(\Gamma )$ be the finite dimensional space of Hilbert modular forms of weight $(k_1,k_2)$ for $\Gamma$. Given a form $f(z) \in M_{(k_1,k_2)}(\Gamma )$, how many Fourier coefficients determine it uniquely in such space? This problem was solved by Hecke for classical forms, and Sturm proved its analogue for congruences modulo a prime ideal. The present article solves the same problem for Hilbert modular forms over $K$. We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to $\Gamma$) such that the Fourier coefficients of any form in such set determines it uniquely.
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Additional Information
  • Jose Ignacio Burgos Gil
  • Affiliation: ICMAT (CSIC-UAM-UCM-UC3), C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain
  • MR Author ID: 349969
  • Email: jiburgosgil@gmail.com
  • Ariel Pacetti
  • Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina
  • MR Author ID: 759256
  • Email: apacetti@dm.uba.ar
  • Received by editor(s): July 1, 2015
  • Received by editor(s) in revised form: August 9, 2015, and January 19, 2016
  • Published electronically: November 8, 2016
  • Additional Notes: The first author was partially supported by grant MTM2013-42135-P
    The second author was partially supported by CONICET PIP 2010-2012 11220090100801, ANPCyT PICT-2013-0294 and UBACyT 2014-2017-20020130100143BA
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1949-1978
  • MSC (2010): Primary 11F41
  • DOI: https://doi.org/10.1090/mcom/3187
  • MathSciNet review: 3626544