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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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:
  • 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:
  • 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:
  • MathSciNet review: 3626544