Some modular abelian surfaces

Calegari, Frank; Chidambaram, Shiva; Ghitza, Alexandru

doi:10.1090/mcom/3434

Some modular abelian surfaces
HTML articles powered by AMS MathViewer

by Frank Calegari, Shiva Chidambaram and Alexandru Ghitza;

Math. Comp. 89 (2020), 387-394

DOI: https://doi.org/10.1090/mcom/3434

Published electronically: April 1, 2019

HTML | PDF | Request permission

Abstract:

In this note, we use the main theorem of Boxer, Calegari, Gee, and Pilloni in Abelian surfaces over totally real fields are potentially modular ( arXiv:1812.09269, 2018) to give explicit examples of modular abelian surfaces $A$ with $\operatorname {End}_{\mathbf {C}} A = \mathbf {Z}$ and $A$ smooth outside $2$, $3$, $5$, and $7$.

References

Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
G. Boxer, F. Calegari, T. Gee, and V. Pilloni, Abelian surfaces over totally real fields are potentially modular, arXiv:1812.09269 [math.NT], 2018.
Hans Ulrich Besche, Bettina Eick, and E. A. O’Brien, A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644. MR 1935567, DOI 10.1142/S0218196702001115
T. Berger and K. Klosin, Deformations of Saito-Kurokawa type and the Paramodular Conjecture (with an appendix by Cris Poor, Jerry Shurman, and David S. Yuen), arXiv:1710.10228 [math.NT], 2017.
A. Brumer, A. Pacetti, C. Poor, G. Tornaría, J. Voight, and D. S. Yuen, On the paramodularity of typical abelian surfaces, arXiv:1805.10873 [math.NT], 2018.
Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235–254. MR 3540958, DOI 10.1112/S146115701600019X
J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090, DOI 10.1017/CBO9780511526084
Francesc Fité, Kiran S. Kedlaya, Víctor Rotger, and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math. 148 (2012), no. 5, 1390–1442. MR 2982436, DOI 10.1112/S0010437X12000279
Cris Poor, Jerry Shurman, and David S. Yuen, Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory 13 (2017), no. 10, 2627–2652. MR 3713095, DOI 10.1142/S1793042117501469
Cris Poor and David S. Yuen, Paramodular cusp forms, Math. Comp. 84 (2015), no. 293, 1401–1438. MR 3315514, DOI 10.1090/S0025-5718-2014-02870-6
B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, 1859, pp. 671–680.
Jean-Pierre Serre, Facteurs locaux des fonctions zêta des varietés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou. 11e année: 1969/70. Théorie des nombres. Fasc. 1: Exposés 1 à 15; Fasc. 2: Exposés 16 à 24, Secrétariat Math., Paris, 1970, pp. 15 (French). MR 3618526
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559