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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
DOI: https://doi.org/10.1090/mcom/3187
Published electronically: November 8, 2016
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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
Email: jiburgosgil@gmail.com

Ariel Pacetti
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina
Email: apacetti@dm.uba.ar

DOI: https://doi.org/10.1090/mcom/3187
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