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Level raising mod 2 and arbitrary 2-Selmer ranks

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2016

Bao V. Le Hung  and
Chao Li
Bao V. Le Hung
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email lhvietbao@googlemail.com
Chao Li
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA email chaoli@math.columbia.edu
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Abstract

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We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$ . It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$ . We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

Keywords

modular formsSelmer groups

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1576 - 1608
Copyright
© The Authors 2016 

References

Allen, P. B., Modularity of nearly ordinary 2-adic residually dihedral Galois representations , Compos. Math. 150 (2014), 1235–1346; MR 3252020.CrossRefGoogle Scholar
Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., Potential automorphy and change of weight , Ann. of Math. (2) 179 (2014), 501–609; MR 3152941.CrossRefGoogle Scholar
Bertolini, M. and Darmon, H., Euler systems and Jochnowitz congruences , Amer. J. Math. 121 (1999), 259–281; MR 1680333 (2001d:11060).CrossRefGoogle Scholar
Boeckle, G., Deformations of Galois representations,http://www.iwr.uni-heidelberg.de/groups/arith-geom/boeckle/Deformations-Barca.pdf.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), 235–265; Computational algebra and number theory (London, 1993); MR 1484478.CrossRefGoogle Scholar
Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over Q : wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), 843–939, (electronic); MR 1839918 (2002d:11058).CrossRefGoogle Scholar
Brinon, O. and Conrad, B., CMI summer school notes on p-adic Hodge theory, http://math.stanford.edu/∼conrad/papers/notes.pdf.Google Scholar
Conrad, B., The flat deformation functor , in Modular forms and Fermat’s last theorem (Boston, MA, 1995) (Springer, New York, 1997), 373–420; MR 1638486.CrossRefGoogle Scholar
Darmon, H., Diamond, F. and Taylor, R., Fermat’s last theorem , in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) (International Press, Cambridge, MA, 1997), 2–140; MR 1605752 (99d:11067b).Google Scholar
Diamond, F. and Taylor, R., Lifting modular mod l representations , Duke Math. J. 74 (1994), 253–269; MR 1272977 (95e:11052).CrossRefGoogle Scholar
Diamond, F. and Taylor, R., Nonoptimal levels of mod l modular representations , Invent. Math. 115 (1994), 435–462; MR 1262939 (95c:11060).CrossRefGoogle Scholar
Gee, T., Automorphic lifts of prescribed types , Math. Ann. 350 (2011), 107–144; MR 2785764 (2012c:11118).CrossRefGoogle Scholar
Gross, B. H. and Parson, J. A., On the local divisibility of Heegner points , in Number theory, analysis and geometry (Springer, New York, 2012), 215–241; MR 2867919.CrossRefGoogle Scholar
Kassaei, P. L., p-adic modular forms over Shimura curves over Q, PhD thesis, Massachusetts Institute of Technology, ProQuest LLC, Ann Arbor, MI (1999); MR 2716881.Google Scholar
Kisin, M., Modularity of 2-adic Barsotti–Tate representations , Invent. Math. 178 (2009), 587–634; MR 2551765 (2010k:11089).CrossRefGoogle Scholar
Li, C., 2-Selmer groups and Heegner points on elliptic curves, PhD thesis, Harvard University (2015).Google Scholar
Mazur, B. and Rubin, K., Ranks of twists of elliptic curves and Hilbert’s tenth problem , Invent. Math. 181 (2010), 541–575; MR 2660452 (2012a:11069).CrossRefGoogle Scholar
Milne, J. S., Arithmetic duality theorems, vol. 1 of Perspectives in mathematics (Academic Press, Boston, MA, 1986); MR 881804 (88e:14028).Google Scholar
O’Neil, C., The period-index obstruction for elliptic curves , J. Number Theory 95 (2002), 329–339; MR 1924106 (2003f:11079).CrossRefGoogle Scholar
Pilloni, V., The study of 2-dimensional $p$ -adic Galois deformations in the $\ell$ not $p$ case,http://perso.ens-lyon.fr/vincent.pilloni/Defo.pdf.Google Scholar
Poonen, B. and Rains, E., Random maximal isotropic subspaces and Selmer groups , J. Amer. Math. Soc. 25 (2012), 245–269; MR 2833483.CrossRefGoogle Scholar
Raynaud, M., Schémas en groupes de type (p, …, p) , Bull. Soc. Math. France 102 (1974), 241–280; MR 0419467 (54 #7488).CrossRefGoogle Scholar
Ribet, K. A., Raising the levels of modular representations , in Séminaire de théorie des nombres, Paris 1987–88, Progress in Mathematics, vol. 81 (Birkhäuser, Boston, MA, 1990), 259–271; MR 1042773 (91g:11055).Google Scholar
Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Invent. Math. 15 (1972), 259–331; MR 0387283 (52 #8126).CrossRefGoogle Scholar
Skinner, C. and Urban, E., The Iwasawa main conjectures for GL2 , Invent. Math. 195 (2014), 1–277; MR 3148103.CrossRefGoogle Scholar
Snowden, A., Singularities of ordinary deformation rings, Preprint (2011), arXiv:1111.3654 [math.NT].Google Scholar
Stein, W. A. et al. , Sage mathematics software (ver. 5.11), The Sage Development Team, 2013, http://www.sagemath.org.Google Scholar
Tate, J., Finite flat group schemes , in Modular forms and Fermat’s last theorem (Boston, MA, 1995) (Springer, New York, 1997), 121–154; MR 1638478.CrossRefGoogle Scholar
Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras , Ann. of Math. (2) 141 (1995), 553–572; MR 1333036 (96d:11072).CrossRefGoogle Scholar
Wiles, A., Modular elliptic curves and Fermat’s last theorem , Ann. of Math. (2) 141 (1995), 443–551; MR 1333035 (96d:11071).CrossRefGoogle Scholar
Zarhin, J. G., Noncommutative cohomology and Mumford groups , Math. Z. 15 (1974), 415–419; MR 0354612 (50 #7090).Google Scholar
Zhang, W., Selmer groups and the indivisibility of Heegner points , Cambridge J. Math. 2 (2014), 191–253.CrossRefGoogle Scholar