On the paramodularity of typical abelian surfaces

Brumer, Armand; Pacetti, Ariel; Poor, Cris; Tornaría, Gonzalo; Voight, John; Yuen, David

doi:10.2140/ant.2019.13.1145

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On the paramodularity of typical abelian surfaces

Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight and David S. Yuen

Vol. 13 (2019), No. 5, 1145–1195

DOI: 10.2140/ant.2019.13.1145

Abstract

Generalizing the method of Faltings–Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves.

An errata, with an appendix by J.-P. Serre, was submitted on 9 November 2020 and posted online on 11 November 2020.

Keywords

abelian surfaces, Siegel modular forms, computation

Mathematical Subject Classification 2010

Primary: 11F46

Secondary: 11Y40

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Received: 6 July 2018

Revised: 24 January 2019

Accepted: 2 April 2019

Published: 12 July 2019

Correction posted: 11 November 2020

Authors

Armand Brumer
Department of Mathematics Fordham University Bronx, NY United States

Ariel Pacetti
FAMAF-CIEM Universidad Nacional de Córdoba Argentina

Cris Poor
Department of Mathematics Fordham University Bronx, NY United States

Gonzalo Tornaría
Centro de Matemática Universidad de la República Montevideo Uruguay

John Voight
Department of Mathematics Dartmouth College Hanover, NH United States

David S. Yuen
Department of Mathematics University of Hawaii Honolulu, HI United States

Vol. 13, No. 5, 2019