On the existence of abelian surfaces with everywhere good reduction

Dembélé, Lassina

doi:10.1090/mcom/3692

On the existence of abelian surfaces with everywhere good reduction
HTML articles powered by AMS MathViewer

by Lassina Dembélé HTML | PDF

Math. Comp. 91 (2022), 1381-1403 Request permission

Abstract:

Let $D \le 2000$ be a positive discriminant such that $F = \mathbf {Q}(\sqrt {D})$ has narrow class number one, and $A/F$ an abelian surface of $\operatorname {GL}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is either a $\mathbf {Q}$-surface or is a base change from $\mathbf {Q}$ of an abelian surface $B$ such that $\operatorname {End}_\mathbf {Q}(B) = \mathbf {Z}$, except for $D = 353, 421, 1321, 1597$ and $1997$. In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over $F$ for $D = 353, 421$ and $1597$, which are non-isogenous to their Galois conjugates. These are the first known such examples.

References

Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Nils Bruin and Kevin Doerksen, The arithmetic of genus two curves with $(4,4)$-split Jacobians, Canad. J. Math. 63 (2011), no. 5, 992–1024. MR 2866068, DOI 10.4153/CJM-2011-039-3
R. P. Bending, Curves of genus $2$ with $\sqrt {2}$-multiplication, Unpublished, 1999.
Clifton Cunningham and Lassina Dembélé, Lifts of hilbert modular forms and application of modularity of abelian varieties, Preprint, 2017.
Gaëtan Chenevier, An automorphic generalization of the Hermite-Minkowski theorem, Duke Math. J. 169 (2020), no. 6, 1039–1075. MR 4085077, DOI 10.1215/00127094-2019-0049
J. E. Cremona, Modular symbols for $\Gamma _1(N)$ and elliptic curves with everywhere good reduction, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 199–218. MR 1142740, DOI 10.1017/S0305004100075307
John R. Doyle and David Krumm, Computing algebraic numbers of bounded height, Math. Comp. 84 (2015), no. 296, 2867–2891. MR 3378851, DOI 10.1090/mcom/2954
Lassina Dembélé and Abhinav Kumar, Examples of abelian surfaces with everywhere good reduction, Math. Ann. 364 (2016), no. 3-4, 1365–1392. MR 3466871, DOI 10.1007/s00208-015-1252-6
Lassina Dembélé and John Voight, Explicit methods for Hilbert modular forms, Elliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2013, pp. 135–198. MR 3184337, DOI 10.1007/978-3-0348-0618-3_{4}
Noam Elkies and Abhinav Kumar, K3 surfaces and equations for Hilbert modular surfaces, Algebra Number Theory 8 (2014), no. 10, 2297–2411. MR 3298543, DOI 10.2140/ant.2014.8.2297
N. Elkies, Elliptic curves of unit discriminant over real quadratic number fields, Unpublished, 2014.
Jordan S. Ellenberg, Serre’s conjecture over $\Bbb F_9$, Ann. of Math. (2) 161 (2005), no. 3, 1111–1142. MR 2180399, DOI 10.4007/annals.2005.161.1111
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
G. Faltings, Erratum: “Finiteness theorems for abelian varieties over number fields”, Invent. Math. 75 (1984), no. 2, 381 (German). MR 732554, DOI 10.1007/BF01388572
Nuno Freitas, Bao V. Le Hung, and Samir Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math. 201 (2015), no. 1, 159–206. MR 3359051, DOI 10.1007/s00222-014-0550-z
Jean-Marc Fontaine, Il n’y a pas de variété abélienne sur $\textbf {Z}$, Invent. Math. 81 (1985), no. 3, 515–538 (French). MR 807070, DOI 10.1007/BF01388584
Josep González, Jordi Guàrdia, and Victor Rotger, Abelian surfaces of $\textrm {GL}_2$-type as Jacobians of curves, Acta Arith. 116 (2005), no. 3, 263–287. MR 2114780, DOI 10.4064/aa116-3-3
Haruzo Hida, On $p$-adic Hecke algebras for $\textrm {GL}_2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384. MR 960949, DOI 10.2307/1971444
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
John W. Jones and David P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), no. 1, 595–618. MR 3356048, DOI 10.1112/S1461157014000424
Takaaki Kagawa, Determination of elliptic curves with everywhere good reduction over certain real quadratic fields, Sūrikaisekikenkyūsho K\B{o}kyūroku 998 (1997), 67–77 (Japanese). Algebraic number theory and related topics (Japanese) (Kyoto, 1996). MR 1622162
Takaaki Kagawa, Determination of elliptic curves with everywhere good reduction over real quadratic fields $\Bbb Q(\sqrt {3p})$, Acta Arith. 96 (2001), no. 3, 231–245. MR 1814279, DOI 10.4064/aa96-3-4
Masanari Kida and Takaaki Kagawa, Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields, J. Number Theory 66 (1997), no. 2, 201–210. MR 1473878, DOI 10.1006/jnth.1997.2177
Chandrashekhar B. Khare and Jack A. Thorne, Automorphy of some residually $S_5$ Galois representations, Math. Z. 286 (2017), no. 1-2, 399–429. MR 3648503, DOI 10.1007/s00209-016-1766-y
Qing Liu, Courbes stables de genre $2$ et leur schéma de modules, Math. Ann. 295 (1993), no. 2, 201–222 (French). MR 1202389, DOI 10.1007/BF01444884
Laurent Moret-Bailly, Problèmes de Skolem sur les champs algébriques, Compositio Math. 125 (2001), no. 1, 1–30 (French, with English summary). MR 1818054, DOI 10.1023/A:1002686625404
R. G. E. Pinch, Elliptic curves over number fields, Oxford University thesis, 1982.
N. I. Shepherd-Barron and R. Taylor, $\textrm {mod}\ 2$ and $\textrm {mod}\ 5$ icosahedral representations, J. Amer. Math. Soc. 10 (1997), no. 2, 283–298. MR 1415322, DOI 10.1090/S0894-0347-97-00226-9
René Schoof, Abelian varieties over cyclotomic fields with good reduction everywhere, Math. Ann. 325 (2003), no. 3, 413–448. MR 1968602, DOI 10.1007/s00208-002-0368-7
Bennett Setzer, Elliptic curves with good reduction everywhere over quadratic fields and having rational $j$-invariant, Illinois J. Math. 25 (1981), no. 2, 233–245. MR 607025
Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
R. J. Stroeker, Reduction of elliptic curves over imaginary quadratic number fields, Pacific J. Math. 108 (1983), no. 2, 451–463. MR 713747, DOI 10.2140/pjm.1983.108.451
John Wilson, Explicit moduli for curves of genus 2 with real multiplication by $\textbf {Q}(\sqrt 5)$, Acta Arith. 93 (2000), no. 2, 121–138. MR 1757185, DOI 10.4064/aa-93-2-121-138