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Mathematics of Computation

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Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields

Authors: Jose Ignacio Burgos Gil and Ariel Pacetti
Journal: Math. Comp. 86 (2017), 1949-1978
MSC (2010): Primary 11F41
Published electronically: November 8, 2016
MathSciNet review: 3626544
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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

Ariel Pacetti
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina
MR Author ID: 759256

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
Article copyright: © Copyright 2016 American Mathematical Society