Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Elliptic curves with $2$-torsion contained in the $3$-torsion field


Authors: Julio Brau and Nathan Jones
Journal: Proc. Amer. Math. Soc. 144 (2016), 925-936
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/proc/12786
Published electronically: July 8, 2015
MathSciNet review: 3447646
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There is a modular curve $X’(6)$ of level $6$ defined over $\mathbb {Q}$ whose $\mathbb {Q}$-rational points correspond to $j$-invariants of elliptic curves $E$ over $\mathbb {Q}$ that satisfy $\mathbb {Q}(E[2]) \subseteq \mathbb {Q}(E[3])$. In this note we characterize the $j$-invariants of elliptic curves with this property by exhibiting an explicit model of $X’(6)$. Our motivation is two-fold: on the one hand, $X’(6)$ belongs to the list of modular curves which parametrize non-Serre curves (and is not well known), and on the other hand, $X’(6)(\mathbb {Q})$ gives an infinite family of examples of elliptic curves with non-abelian “entanglement fields”, which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over $\mathbb {Q}$.


References [Enhancements On Off] (What's this?)

References
  • J. Brau, Selmer groups of elliptic curves and Galois representations, Ph.D. Thesis, University of Cambridge (2014).
  • Alina-Carmen Cojocaru, David Grant, and Nathan Jones, One-parameter families of elliptic curves over $\Bbb Q$ with maximal Galois representations, Proc. Lond. Math. Soc. (3) 103 (2011), no. 4, 654–675. MR 2837018, DOI 10.1112/plms/pdr001
  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
  • Tim Dokchitser and Vladimir Dokchitser, Surjectivity of mod $2^n$ representations of elliptic curves, Math. Z. 272 (2012), no. 3-4, 961–964. MR 2995149, DOI 10.1007/s00209-011-0967-7
  • Noam D. Elkies, Points of low height on elliptic curves and surfaces. I. Elliptic surfaces over $\Bbb P^1$ with small $d$, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 287–301. MR 2282931, DOI 10.1007/11792086_{2}1
  • 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
  • N. Jones, $\operatorname {GL}_2$-representations with maximal image, Math. Res. Lett., to appear.
  • S. Lang and H. Trotter, Frobenius distribution in $\operatorname {GL}_2$ extensions, Lecture Notes in Math. 504, Springer (1976).
  • V. Radhakrishnan, Asymptotic formula for the number of non-Serre curves in a two-parameter family, Ph.D. Thesis, University of Colorado at Boulder (2008).
  • Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 457455, DOI 10.2307/2373815
  • 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, Cours d’arithmétique, Le Mathématicien, No. 2, Presses Universitaires de France, Paris, 1977 (French). Deuxième édition revue et corrigée. MR 0498338
  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11G05

Retrieve articles in all journals with MSC (2010): 11G05


Additional Information

Julio Brau
Affiliation: Faculty of Mathematics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Email: jb711@cam.ac.uk

Nathan Jones
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
MR Author ID: 842244
Email: ncjones@uic.edu

Received by editor(s): June 8, 2014
Received by editor(s) in revised form: February 4, 2015
Published electronically: July 8, 2015
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society