Algebra & Number Theory

Modular curves of prime-power level with infinitely many rational points

Andrew Sutherland and David Zywina

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For each open subgroup G of GL2( ̂) containing I with full determinant, let XG denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in G. Up to conjugacy, we determine a complete list of the 248 such groups G of prime power level for which XG() is infinite. For each G, we also construct explicit maps from each XG to the j-line. This list consists of 220 modular curves of genus 0 and 28 modular curves of genus 1. For each prime , these results provide an explicit classification of the possible images of -adic Galois representations arising from elliptic curves over that is complete except for a finite set of exceptional j-invariants.

Article information

Source
Algebra Number Theory, Volume 11, Number 5 (2017), 1199-1229.

Dates
Received: 20 May 2016
Revised: 10 February 2017
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513090725

Digital Object Identifier
doi:10.2140/ant.2017.11.1199

Mathematical Reviews number (MathSciNet)
MR3671434

Zentralblatt MATH identifier
1374.14022

Subjects
Primary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Secondary: 11F80: Galois representations 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
modular curves elliptic curves Galois representations

Citation

Sutherland, Andrew; Zywina, David. Modular curves of prime-power level with infinitely many rational points. Algebra Number Theory 11 (2017), no. 5, 1199--1229. doi:10.2140/ant.2017.11.1199. https://projecteuclid.org/euclid.ant/1513090725


Export citation

References

  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • K. S. Chua, M. L. Lang, and Y. Yang, “On Rademacher's conjecture: congruence subgroups of genus zero of the modular group”, J. Algebra 277:1 (2004), 408–428.
  • J. E. Cremona, “Elliptic curve data”, electronic reference, University of Warwick, 2016, http://johncremona.github.io/ecdata/.
  • C. J. Cummins and S. Pauli, “Congruence subgroups of ${\rm PSL}(2,{\mathbb Z})$ of genus less than or equal to 24”, Experiment. Math. 12:2 (2003), 243–255. A database containing the tables is available at {inbibhttp://www.uncg.edu/mat/faculty/pauli/congruence/.
  • P. Deligne and M. Rapoport, “Les schémas de modules de courbes elliptiques”, pp. 143–316 in Modular functions of one variable, II (Antwerp, 1972), edited by P. Deligne and W. Kuyk, Lecture Notes in Math. 349, Springer, Berlin, 1973.
  • J. B. Dennin, Jr., “The genus of subfields of $K(p^{n})$”, Illinois J. Math. 18 (1974), 246–264.
  • G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math. 73:3 (1983), 349–366. Correction in 75: 2 (1984), 381.
  • E. Halberstadt, “Sur la courbe modulaire $X_{\text{nd\'ep}}(11)$”, Experiment. Math. 7:2 (1998), 163–174.
  • D. S. Kubert and S. Lang, Modular units, Grundlehren Math. Wissenschaften 244, Springer, Berlin, 1981.
  • LMFDB Collaboration, “The $L$-functions and modular forms database”, electronic reference, 2013, http://www.lmfdb.org.
  • J. Rouse and D. Zureick-Brown, “Elliptic curves over $\mathbb Q$ and 2-adic images of Galois”, Res. Number Theory 1 (2015), art. id. 12.
  • J.-P. Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, New York, 1968.
  • J.-P. Serre, “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”, Invent. Math. 15:4 (1972), 259–331.
  • J.-P. Serre, “Quelques applications du théorème de densité de Chebotarev”, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401.
  • J.-P. Serre, Lectures on the Mordell–Weil theorem, 3rd ed., Friedr. Vieweg & Sohn, Braunschweig, 1997.
  • G. Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures 1, Iwanami Shoten, Tokyo, 1971.
  • A. V. Sutherland, “Computing images of Galois representations attached to elliptic curves”, Forum Math. Sigma 4 (2016), art. id. e4.
  • A. V. Sutherland and D. Zywina, Magma scripts associated to “Modular curves of prime-power level with infinitely many rational points”, 2016, http://math.mit.edu/~drew/SZ16.
  • D. Zywina, “On the possible images of the mod $l$ representations associated to elliptic curves over ${\mathbb Q}$”, preprint, 2015.
  • D. Zywina, “Possible indices for the Galois image of elliptic curves over ${\mathbb Q}$”, preprint, 2015.

Supplemental materials

  • Group Tables.