{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:11:41Z","timestamp":1776802301628,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"306","license":[{"start":{"date-parts":[[2017,11,8]],"date-time":"2017-11-08T00:00:00Z","timestamp":1510099200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["MTM2013- 42135-P."],"award-info":[{"award-number":["MTM2013- 42135-P."]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a real quadratic field and\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:msub>\n \\mathscr {O}_K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n its ring of integers. Let\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a congruence subgroup of\n \n \n \n \n \n \n S<\/mml:mi>\n L<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n \n \n O<\/mml:mi>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {SL}_2(\\mathscr {O}_K)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n M<\/mml:mi>\n \n (<\/mml:mo>\n \n k<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u0393\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n M_{(k_1,k_2)}(\\Gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the finite dimensional space of Hilbert modular forms of weight\n \n \n \n \n (<\/mml:mo>\n \n k<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n (k_1,k_2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Given a form\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n M<\/mml:mi>\n \n (<\/mml:mo>\n \n k<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u0393\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f(z) \\in M_{(k_1,k_2)}(\\Gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , 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\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) such that the Fourier coefficients of any form in such set determines it uniquely.\n <\/p>","DOI":"10.1090\/mcom\/3187","type":"journal-article","created":{"date-parts":[[2016,6,17]],"date-time":"2016-06-17T07:07:43Z","timestamp":1466147263000},"page":"1949-1978","source":"Crossref","is-referenced-by-count":1,"title":["Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields"],"prefix":"10.1090","volume":"86","author":[{"given":"Jose Ignacio","family":"Burgos Gil","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ariel","family":"Pacetti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,11,8]]},"reference":[{"issue":"9","key":"1","doi-asserted-by":"publisher","first-page":"2497","DOI":"10.1090\/S0002-9939-02-06609-1","article-title":"On the number of Fourier coefficients that determine a Hilbert modular form","volume":"130","author":"Baba, Srinath","year":"2002","journal-title":"Proc. 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