Skip to main content

Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois

Abstract

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over \(\mathbb {Q}\). To this end, we compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

References

  1. 1

    Arai, K: On uniform lower bound of the Galois images associated to elliptic curves. J. Théor. Nombres Bordeaux. 20(1), 23–43 (2008). http://jtnb.cedram.org/item?id=JTNB_2008__20_1_23_0.

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Baran, B: A modular curve of level 9 and the class number one problem. J. Number Theory. 129(3), 715–728 (2009). doi:10.1016/j.jnt.2008.09.013.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Baran, B: Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem. J. Number Theory. 130(12), 2753–2772 (2010). doi:10.1016/j.jnt.2010.06.005.

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Baran, B: An exceptional isomorphism between modular curves of level 13. J. Number Theory. 145, 273–300 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Bröker, R, Lauter, K, Sutherland, AV: Modular polynomials via isogeny volcanoes. Math. Comp. 81(278), 1201–1231 (2012). doi:10.1090/S0025-5718-2011-02508-1.

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Bilu, Y, Parent, P, Rebolledo, M: Rational points on \(X_{0}^{+} (p^{r})\) (2011). arXiv:1104.4641, Preprint.

  7. 7

    Bruin, N, Poonen, Bj, Stoll, M: Generalized explicit descent and its application to curves of genus, 3 (2012). Preprint.

  8. 8

    Bruin, N: Chabauty methods using elliptic curves. J. Reine Angew. Math. 562, 27–49 (2003).

    MathSciNet  MATH  Google Scholar 

  9. 9

    Bruin, N: Some ternary Diophantine equations of signature (n,n,2). In: Discovering mathematics with Magma, Algorithms Comput. Math. vol. 19, pp. 63–91. Springer, Berlin (2006). doi:10.1007/978-3-540-37634-7_3.

    Google Scholar 

  10. 10

    Bruin, N: The arithmetic of Prym varieties in genus 3. Compos. Math. 144(2), 317–338 (2008). doi:10.1112/S0010437X07003314.

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Bruin, N, Stoll, M: Deciding existence of rational points on curves: an experiment, Experiment. Math. 17(2), 181–189 (2008). http://projecteuclid.org/getRecord?id=euclid.em/1227118970.

    MathSciNet  MATH  Google Scholar 

  12. 12

    Coombes, KR, Grant, DR: On heterogeneous spaces. J. London Math. Soc. (2). 40(3), 385–397 (1989). doi:10.1112/jlms/s2-40.3.385.

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Chen, I: On Siegel’s modular curve of level 5 and the class number one problem. J. Number Theory. 74(2), 278–297 (1999). doi:10.1006/jnth.1998.2320.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Cohen, H: Number theory. Vol. I. Tools and Diophantine equations. Graduate Texts in Mathematics, Vol. 239. Springer, New York (2007).

    Google Scholar 

  15. 15

    Cummins, CJ, Pauli, S: Congruence subgroups of PSL(2,) of genus less than or equal to 24. Experiment. Math. 12(2), 243–255 (2003). http://projecteuclid.org/getRecord?id=euclid.em/1067634734.

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Dokchitser, T, Dokchitser, V: Surjectivity of mod 2n representations of elliptic curves. Math. Z. 272(3-4), 961–964 (2012). doi:10.1007/s00209-011-0967-7.

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Dokchitser, T, Dokchitser, V: Identifying Frobenius elements in Galois groups. Algebra Number Theory. 7(6), 1325–1352 (2013). doi:10.2140/ant.2013.7.1325.

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Deligne, P, Rapoport, M: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math, pp. 143–316. Springer, Berlin (1973).

    Google Scholar 

  19. 19

    Diamond, F, Shurman, J: A first course in modular forms: Graduate Texts in Mathematics, Vol. 228. Springer-Verlag, New York (2005).

    MATH  Google Scholar 

  20. 20

    Elkies, ND: Elliptic curves with 3-adic galois representation surjective mod 3 but not mod 9 (2006). Preprint.

  21. 21

    Flynn, EV, Wetherell, JL: Covering collections and a challenge problem of Serre. Acta Arith. 98(2), 197–205 (2001). doi:10.4064/aa98-2-9.

    MathSciNet  Article  MATH  Google Scholar 

  22. 22

    González-Jiménez, E, González, J: Modular curves of genus 2, Vol. 72 (2003). doi:10.1090/S0025-5718-02-01458-8.

  23. 23

    González-Jiménez, E, Lozano-Robledo, Á: Elliptic curves with abelian division fields. Preprint.

  24. 24

    Hecke, E: Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Sem. Univ. Hamburg. 5(1), 199–224 (1927). doi:10.1007/BF02952521.

    MathSciNet  Article  MATH  Google Scholar 

  25. 25

    Heegner, K: Diophantische Analysis und Modulfunktionen. Math. Z. 56, 227–253 (1952).

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Jones, JW: Number fields unramified away from 2. J. Number Theory. 130(6), 1282–1291 (2010). doi:10.1016/j.jnt.2010.02.005.

    MathSciNet  Article  MATH  Google Scholar 

  27. 27

    Jones, R, Rouse, J: Galois theory of iterated endomorphisms. Proc. Lond. Math. Soc. (3). 100(3), 763–794 (2010). doi:10.1112/plms/pdp051. Appendix A by Jeffrey D. Achter.

    MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Kenku, MA: A note on the integral points of a modular curve of level 7. Mathematika. 32(1), 45–48 (1985). doi:10.1112/S0025579300010846.

    MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Knapp, AW: Elliptic curves. Mathematical Notes, Vol. 40. Princeton University Press, Princeton, NJ (1992).

    Google Scholar 

  30. 30

    Mazur, B: Rational points on modular curves. In: Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pp. 107–148. Springer, Berlin (1977).

    Google Scholar 

  31. 31

    Mazur, B: Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44(2), 129–162 (1978).

    MathSciNet  Article  MATH  Google Scholar 

  32. 32

    McMurdy, K: Explicit equations for X0(N). http://phobos.ramapo.edu/~kmcmurdy/research/Models/index.html.

  33. 33

    Momose, F: Rational points on the modular curves Xsplit(p). Compositio Math. 52(1), 115–137 (1984). http://www.numdam.org/item?id=CM_1984__52_1_115_0.

    MathSciNet  MATH  Google Scholar 

  34. 34

    McCallum, W, Poonen, B: The method of Chabauty and Coleman. Soc. Math., France, Paris (2012).

    MATH  Google Scholar 

  35. 35

    Nishioka, K: The 2-adic representations attached to elliptic curves defined over Q whose points of order 2 are all Q-rational. J. Math. Soc. Japan. 35(2), 191–219 (1983). doi:10.2969/jmsj/03520191.

    MathSciNet  Article  MATH  Google Scholar 

  36. 36

    Poonen, B: Computing rational points on curves. In: Number theory for the millennium, III (Urbana, IL, 2000), pp. 149–172. A K Peters, Natick, MA (2002).

    Google Scholar 

  37. 37

    Poonen, B, Schaefer, EF: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997).

    MathSciNet  MATH  Google Scholar 

  38. 38

    Rouse, J, Zureick-Brown, D: Electronic transcript of computations for the paper ‘Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois’. Available at, http://users.wfu.edu/rouseja/2adic/. (Data files and scripts will also be posted on arXiv).

  39. 39

    Serre, J-P: Propriétés Galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972).

    MathSciNet  Article  Google Scholar 

  40. 40

    Serre, J-P: Lectures on the Mordell-Weil theorem. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. With a foreword by Brown and Serre. 3rd ed. Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1997). x+218 pp. ISBN: 3-528-28968-6 MR1757192(2000m:11049).

    MATH  Google Scholar 

  41. 41

    Shimura, G: Introduction to the arithmetic theory of automorphic functions. In: Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo (1971). Kanô Memorial Lectures, No. 1.

    Google Scholar 

  42. 42

    Shimura, M: Defining equations of modular curves X0(N). Tokyo J. Math. 18(2), 443–456 (1995). doi:10.3836/tjm/1270043475.

    MathSciNet  Article  MATH  Google Scholar 

  43. 43

    Silverman, JH: The arithmetic of elliptic curves, Second edition, Graduate Texts in Mathematics, Vol. 106. Springer, Dordrecht (2009).

    Book  Google Scholar 

  44. 44

    Skorobogatov, A: Torsors and rational points, Cambridge Tracts in Mathematics, Vol. 144. Cambridge University Press, Cambridge (2001). http://dx.doi.org.ezproxy.library.wisc.edu/10.1017/CBO9780511549588.

    Book  MATH  Google Scholar 

  45. 45

    Schoof, R, Tzanakis, N: Integral points of a modular curve of level 11. Acta Arith. 152(1), 39–49 (2012). doi:10.4064/aa152-1-4.

    MathSciNet  Article  MATH  Google Scholar 

  46. 46

    Stoll, M: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98(3), 245–277 (2001). doi:10.4064/aa98-3-4.

    MathSciNet  Article  MATH  Google Scholar 

  47. 47

    Sutherland, AV: Constructing elliptic curves over finite fields with prescribed torsion. Math. Comp. 81(278), 1131–1147 (2012). doi:10.1090/S0025-5718-2011-02538-X.

    MathSciNet  Article  MATH  Google Scholar 

  48. 48

    Sutherland, AV: Computing images of Galois representations attached to elliptic curves (2015). arXiv:1504.07618, Preprint.

  49. 49

    Sutherland, A: Defining equations for X1(N). http://math.mit.edu/~drew/X1_altcurves.html.

  50. 50

    Stoll, M, van Luijk, R: Explicit Selmer groups for cyclic covers of 1. Acta Arith. 159(2), 133–148 (2013). doi:10.4064/aa159-2-4.

    MathSciNet  Article  MATH  Google Scholar 

  51. 51

    Sutherland, A, Zywina, D: Modular curves of genus zero and prime-power level. in preparation.

  52. 52

    Wetherell, JL: Bounding the number of rational points on certain curves of high rank. ProQuest LLC, Ann Arbor, MI (1997). http://search.proquest.com/docview/304343505. Thesis (Ph.D.)--University of California, Berkeley

  53. 53

    Wilson, JS: Profinite groups. London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, New York (1998).

    Google Scholar 

  54. 54

    Zywina, D: On the surjectivity of mod l representations associated to elliptic curves (2011). Preprint.

Download references

Acknowledgments

We thank Jeff Achter, Nils Bruin, Tim Dokchitser, Bjorn Poonen, William Stein, Michael Stoll, Drew Sutherland, and David Zywina for useful conversations and University of Wisconsin-Madison’s Spring 2011 CURL (Collaborative undergraduate research Labs) students (Eugene Yoong, Collin Smith, Dylan Blanchard) for doing initial group theoretical computations. The second author is supported by an NSA Young Investigator grant. We would also like to thank anonymous referees for helpful comments and suggestions that have improved the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to David Zureick-Brown.

Rights and permissions

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rouse, J., Zureick-Brown, D. Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois. Res. Number Theory 1, 12 (2015). https://doi.org/10.1007/s40993-015-0013-7

Download citation

Keywords

  • Conjugacy Class
  • Elliptic Curve
  • Rational Point
  • Maximal Subgroup
  • Elliptic Curf