Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Elliptic curves with $2$-torsion contained in the $3$-torsion field
HTML articles powered by AMS MathViewer

by Julio Brau and Nathan Jones PDF
Proc. Amer. Math. Soc. 144 (2016), 925-936 Request permission


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 $\mathbb {Q}$.
  • 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) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (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:
  • 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:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 925-936
  • MSC (2010): Primary 11G05
  • DOI:
  • MathSciNet review: 3447646