Proving modularity for a given elliptic curve over an imaginary quadratic field

Dieulefait, Luis; Guerberoff, Lucio; Pacetti, Ariel

doi:10.1090/S0025-5718-09-02291-1

Proving modularity for a given elliptic curve over an imaginary quadratic field
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by Luis Dieulefait, Lucio Guerberoff and Ariel Pacetti PDF

Math. Comp. 79 (2010), 1145-1170 Request permission

Abstract:

We present an algorithm to determine if the $L$-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible $\ell$-adic representations. Our algorithm is based on Faltings-Serre’s method to prove that $\ell$-adic Galois representations are isomorphic. Using the algorithm we provide the first examples of modular elliptic curves over imaginary quadratic fields with residual $2$-adic image isomorphic to $S_3$ and $C_3$.

References

Tobias Berger and Gergely Harcos, $l$-adic representations associated to modular forms over imaginary quadratic fields, Int. Math. Res. Not. IMRN 23 (2007), Art. ID rnm113, 16. MR 2380006
Computational number theory. http://www.ma.utexas.edu/users/villegas/cnt/.
Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), no. 3, 275–324. MR 743014
J. E. Cremona, Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields, J. London Math. Soc. (2) 45 (1992), no. 3, 404–416. MR 1180252, DOI 10.1112/jlms/s2-45.3.404
Gary Cornell, Joseph H. Silverman, and Glenn Stevens (eds.), Modular forms and Fermat’s last theorem, Springer-Verlag, New York, 1997. Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995. MR 1638473, DOI 10.1007/978-1-4612-1974-3
J. E. Cremona and E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp. 62 (1994), no. 205, 407–429. MR 1185241, DOI 10.1090/S0025-5718-1994-1185241-6
Michael Harris, David Soudry, and Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to $\textrm {GSp}_4(\textbf {Q})$, Invent. Math. 112 (1993), no. 2, 377–411. MR 1213108, DOI 10.1007/BF01232440
Mark Lingham. Modular forms and elliptic curves over imaginary quadratic fields. Ph.D. thesis, University of Nottingham, October 2005. Available from http://www.warwick.ac.uk/staff/J.E.Cremona/theses/index.html.
Ron Livné, Cubic exponential sums and Galois representations, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 247–261. MR 902596, DOI 10.1090/conm/067/902596
The PARI Group, Bordeaux. PARI/GP, version 2.4.3, 2008. available from http://pari.math.u-bordeaux.fr/.
Matthias Schütt, On the modularity of three Calabi-Yau threefolds with bad reduction at 11, Canad. Math. Bull. 49 (2006), no. 2, 296–312. MR 2226253, DOI 10.4153/CMB-2006-031-9
Jean-Pierre Serre, Groupes de Lie $l$-adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algèbre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1966, pp. 239–256 (French). MR 0218366
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
Jean-Pierre Serre. Résumé des cours de 1984-1985. Annuaire du Collège de France, pages 85–90, 1985.
Jean-Pierre Serre. Représentations linéaires sur des anneaux locaux, d’apres carayol. Publ. Inst. Math. Jussieu, 49, 1995.
Richard Taylor. $l$-adic representations associated to modular forms over imaginary quadratic fields. II. Invent. Math., 116(1-3):619–643, 1994.