Elliptic curves with full 2-torsion and maximal adelic Galois representations
HTML articles powered by AMS MathViewer
- by David Corwin, Tony Feng, Zane Kun Li and Sarah Trebat-Leder PDF
- Math. Comp. 83 (2014), 2925-2951 Request permission
Abstract:
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in $\mathrm {GL}_2(\widehat {\mathbb {Z}})$. In (2010), Greicius developed necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius’s methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over $\mathbb {Q}(\alpha )$ with maximal image, where $\alpha$ is the real root of $x^3 + x + 1$. Next, we extend Greicius’s tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups.References
- Ahmed Abbes, Réduction semi-stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, $\ldots$, Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998) Progr. Math., vol. 187, Birkhäuser, Basel, 2000, pp. 59–110 (French). MR 1768094
- Clemens Adelmann, The decomposition of primes in torsion point fields, Lecture Notes in Mathematics, vol. 1761, Springer-Verlag, Berlin, 2001. MR 1836119, DOI 10.1007/b80624
- David Corwin, Tony Feng, Zane Kun Li, and Sarah Trebat-Leder, Transcript of computations, Available at http://code.google.com/p/maximal-adelic-image/.
- William Duke, Elliptic curves with no exceptional primes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 8, 813–818 (English, with English and French summaries). MR 1485897, DOI 10.1016/S0764-4442(97)80118-8
- Aaron Greicius, Elliptic curves with surjective adelic Galois representations, Experiment. Math. 19 (2010), no. 4, 495–507. MR 2778661, DOI 10.1080/10586458.2010.10390639
- Nathan Jones, Almost all elliptic curves are Serre curves, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1547–1570. MR 2563740, DOI 10.1090/S0002-9947-09-04804-1
- Serge Lang and Hale Trotter, Frobenius distributions in $\textrm {GL}_{2}$-extensions, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin-New York, 1976. Distribution of Frobenius automorphisms in $\textrm {GL}_{2}$-extensions of the rational numbers. MR 0568299, DOI 10.1007/BFb0082087
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
- 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
- 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
- David Zywina, Elliptic curves with maximal Galois action on their torsion points, Bull. Lond. Math. Soc. 42 (2010), no. 5, 811–826. MR 2721742, DOI 10.1112/blms/bdq039
- David Zywina, Hilbert’s irreducibility theorem and the larger sieve, Preprint (2010), arXiv:1011.6465v1.
Additional Information
- David Corwin
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1073361
- Email: corwind@mit.edu
- Tony Feng
- Affiliation: 479 Quincy Mail Center, 58 Plympton Street, Cambridge, Massachusetts 02138
- Email: tfeng@college.harvard.edu
- Zane Kun Li
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Address at time of publication: Department of Mathematics, UCLA, Los Angeles, California 90095
- MR Author ID: 869116
- Email: zkli@math.ucla.edu
- Sarah Trebat-Leder
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- Email: strebat@emory.edu
- Received by editor(s): July 20, 2012
- Received by editor(s) in revised form: February 4, 2013
- Published electronically: January 30, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 2925-2951
- MSC (2010): Primary 11F80, 11G05
- DOI: https://doi.org/10.1090/S0025-5718-2014-02804-4
- MathSciNet review: 3246816