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2021 | OriginalPaper | Chapter

# A Prym Variety with Everywhere Good Reduction over $$\mathbb {Q}(\sqrt {61})$$

Authors: Nicolas Mascot, Jeroen Sijsling, John Voight

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## Abstract

We compute an equation for a modular abelian surface A that has everywhere good reduction over the quadratic field $$K = \mathbb {Q}(\sqrt {61})$$ and that does not admit a principal polarization over K.
Literature
1.
V. A. Abrashkin, Galois modules of group schemes of period p over the ring of Witt vectors, translated from Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 4, 691–736, 910, Math. USSR-Izv. 31 (1988), no. 1, 1–46.
2.
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 (Special Issue A) (2016), 235–254.
3.
Armand Brumer and Kenneth Kramer, Certain abelian varieties bad at only one prime, Algebra Number Theory 12 (2018), no. 5, 1027–1071.
4.
Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight, and David S. Yuen, On the paramodularity of typical abelian surfaces, Algebra & Number Theory 13 (2019), no. 5, 1145–1195.
5.
Christina Birkenhake and Herbert Lange, Complex abelian varieties, volume 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 2004.
6.
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), vol. 3–4, 235–265.
7.
Nils Bruin, Jeroen Sijsling, and Alexandre Zotine, Numerical computation of endomorphism rings of Jacobians, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Series 2 (2019), 155–171. MathSciNet
8.
Henri Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, 213–237.
9.
W. Casselman, On abelian varieties with many endomorphisms and a conjecture of Shimura’s, Invent. Math. 12 (1971), 225–236.
10.
Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight, Rigorous computation of the endomorphism ring of a Jacobian, Math. Comp. 88 (2019), 1303–1339.
11.
Lassina Dembélé, Abelian varieties with everywhere good reduction over certain real quadratic fields with small discriminant, 2019, preprint.
12.
Lassina Dembélé and Abhinav Kumar, Examples of abelian surfaces with everywhere good reduction, Math. Ann. 364 (2016), no. 3–4, 1365–1392.
13.
Luis Dieulefait, Lucio Guerberoff, and Ariel Pacetti, Proving modularity for a given elliptic curve over an imaginary quadratic field, Math. Comp. 79 (2010), no. 270, 1145–1170.
14.
15.
Gerd Faltings, Finiteness theorems for abelian varieties over number fields, translated from Invent. Math. 73 (1983), no. 3, 349–366; ibid. 75 (1984), no. 2, 381, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, 9–27.
16.
17.
Jean-Marc Fontaine, Il n’y a pas de variété abélienne sur $$\mathbb {Z}$$, Invent. Math. 81 (1985), no. 3, 515–538.
18.
Josep González, Jordi Guàrdia, and Victor Rotger, Abelian surfaces of GL 2- type as Jacobians of curves, Acta Arith. 116 (2005), no. 3, 263–287.
19.
Jordi Guàrdia, Jacobian nullwerte and algebraic equations, J. Algebra 253 (2002), no. 1, 112–132.
20.
Jeroen Hanselman, Gluing curves along their torsion, Ph.D. thesis, Universität Ulm, in progress, (2021) available at https://​oparu.​uni-ulm.​de/​xmlui/​bitstream/​handle/​123456789/​33412/​Hanselman_​Thesis.​pdf
21.
Jeroen Hanselman and Jeroen Sijsling, gluing : Gluing curves of low genus along their torsion, https://​github.​com/​JRSijsling/​gluing.
22.
Ivanyos, Gábor, Lajos Rónyai, and Josef Schicho, Splitting full matrix algebras over algebraic number fields, J. Algebra 354 (2012), 211–223.
23.
Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals Math. Studies, vol. 108, Princeton University Press, 1985.
24.
Pınar Kılıçer, Hugo Labrande, Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling, and Marco Streng, Plane quartics over $$\mathbb {Q}$$ with complex multiplication, Acta Arith. 185 (2018), no. 2, 127–156.
25.
Reynald Lercier, Christophe Ritzenthaler, and Jeroen Sijsling, Reconstructing plane quartics from their invariants, Discrete Comput. Geom. (2018), https://​doi.​org/​10.​1007/​s00454-018-0047-4.
26.
The LMFDB Collaboration, The L- functions and Modular Forms Database, http://​www.​lmfdb.​org, 2019, [Online; accessed 30 July 2019].
27.
Pascal Molin and Christian Neurohr, Computing period matrices and the Abel-Jacobi map of superelliptic curves, Math. Comp. 88 (2019), no. 316, 847–888.
28.
Christian Neurohr, Efficient integration on Riemann surfaces & applications, Ph.D. thesis, Carl-von-Ossietzky-Universität Oldenburg, 2018.
29.
Nicolas Mascot, Computing modular Galois representations, Rendiconti Circ. Mat. Palermo, vol. 62 no 3 (2013), 451–476.
30.
Nicolas Mascot, Companion forms and explicit computation of $$\operatorname {PGL}_2$$- number fields with very little ramification, Journal of Algebra, vol. 509 (2018) 476–506.
31.
Nicolas Mascot, Jeroen Sijsling, and John Voight, Prym calculation code, https://​github.​com/​JRSijsling/​prym.
32.
Elisabeth E. Pyle, Abelian varieties over $$\mathbb Q$$ with large endomorphism algebras and their simple components over $$\overline {\mathbb Q}$$, Modular curves and abelian varieties, Progr. Math., vol. 224, Birkhäuser, Basel, 2004, 189–239.
33.
Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 143–162.
34.
René Schoof, Abelian varieties over cyclotomic fields with good reduction everywhere, Math. Ann. 325 (2003), no. 3, 413–448.
35.
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, no. 1, Princeton University Press, Princeton, 1971.
36.
Jeroen Sijsling, curve_reconstruction : Geometric and arithmetic reconstruction of curves from their period matrices, https://​github.​com/​JRSijsling/​curve_​reconstruction.
37.
Richard Taylor, Representations of Galois groups associated to modular forms, Proc. ICM (Zürich, 1994), vol. 2, Birkhäuser, Basel, 1995, 435–442.