Skip to main content Accessibility help
×
Home

On families of 7- and 11-congruent elliptic curves

  • Tom Fisher (a1)

Abstract

We use an invariant-theoretic method to compute certain twists of the modular curves $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X(n)$ for $n=7,11$ . Searching for rational points on these twists enables us to find non-trivial pairs of $n$ -congruent elliptic curves over ${\mathbb{Q}}$ , that is, pairs of non-isogenous elliptic curves over ${\mathbb{Q}}$ whose $n$ -torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over ${\mathbb{Q}}(T)$ , and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves  over ${\mathbb{Q}}$ .

Supplementary materials are available with this article.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On families of 7- and 11-congruent elliptic curves
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On families of 7- and 11-congruent elliptic curves
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On families of 7- and 11-congruent elliptic curves
      Available formats
      ×

Copyright

References

Hide All
1. Adler, A., ‘Invariants of PSL2(F 11) acting on C 5 ’, Comm. Algebra 20 (1992) no. 10, 28372862.
2. Adler, A. and Ramanan, S., Moduli of abelian varieties , Lecture Notes in Mathematics 1644 (Springer, 1996).
3. Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic Comput. 24 (1997) 235265; see also the Magma home page at http://magma.maths.usyd.edu.au/magma/.
4. Cremona, J. E., Algorithms for modular elliptic curves (Cambridge University Press, Cambridge, 1997) ; see also http://www.warwick.ac.uk/∼masgaj/ftp/data/.
5. Cremona, J. E., Fisher, T. A. and Stoll, M., ‘Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves’, Algebra Number Theory 4 (2010) no. 6, 763820.
6. Cremona, J. E. and Mazur, B., ‘Visualizing elements in the Shafarevich–Tate group’, Experiment. Math. 9 (2000) no. 1, 1328.
7. Elkies, N. D., ‘The Klein quartic in number theory’, The eightfold way: The beauty of Klein’s quartic curve , Mathematical Sciences Research Institute Publications 35 (ed. Levy, S.; Cambridge University Press, Cambridge, 1999) 51101.
8. Fisher, T. A., ‘On 5 and 7 descents for elliptic curves’, PhD Thesis, University of Cambridge, 2000, http://www.dpmms.cam.ac.uk/∼taf1000/thesis.html.
9. Fisher, T. A., ‘Some examples of 5 and 7 descent for elliptic curves over Q ’, J. Eur. Math. Soc. 3 (2001) no. 2, 169201.
10. Fisher, T. A., ‘The invariants of a genus one curve’, Proc. Lond. Math. Soc. (3) 97 (2008) 753782.
11. Fisher, T. A., ‘The Hessian of a genus one curve’, Proc. Lond. Math. Soc. (3) 104 (2012) 613648.
12. Fisher, T. A., ‘Invariant theory for the elliptic normal quintic, I. Twists of X (5)’, Math. Ann. 356 (2013) no. 2, 589616.
13. Fisher, T. A., ‘On families of 7- and 11-congruent elliptic curves’, Electronic data accompanying this article, http://journals.cambridge.org/sup_S1461157014000059sup001.
14. Frey, G., ‘On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2’, Elliptic curves, modular forms & Fermat’s Last Theorem, Hong Kong, 1993 , Series on Number Theory I (eds Coates, J. and Yau, S.-T.; International Press, Cambridge, MA, 1995) 7998.
15. Halberstadt, E. and Kraus, A., ‘On the modular curves Y E (7)’, Math. Comp. 69 (2000) no. 231, 11931206.
16. Halberstadt, E. and Kraus, A., ‘Sur la courbe modulaire X E (7)’, Experiment. Math. 12 (2003) no. 1, 2740.
17. Kani, E. J. and Rizzo, O. G., ‘Mazur’s question on mod 11 representations of elliptic curves’, Preprint, http://www.mast.queensu.ca/∼kani/mdqs.htm.
18. Kani, E. and Schanz, W., ‘Modular diagonal quotient surfaces’, Math. Z. 227 (1998) no. 2, 337366.
19. Klein, F., ‘Über die Transformationen siebenter Ordnung der elliptischen Funktionen’, Math. Ann. 14 (1878) 428471; English translation in The eightfold way: The beauty of Klein’s quartic curve, Mathematical Sciences Research Institute Publications 35 (ed. S. Levy; Cambridge University Press, Cambridge 1999).
20. Klein, F., ‘Über die Transformationen elfter Ordnung der elliptischen Funktionen’, Math. Ann. 15 (1879) Reprinted in Gesammelte Mathematische Abhandlungen III (ed. R. Fricke et al.; Springer, 1923) 140–168.
21. Klein, F., ‘Über die elliptischen Normalkurven der n-ten Ordnung’ (1885); Reprinted in Gesammelte Mathematische Abhandlungen III (ed. R. Fricke et al.; Springer, 1923) 198–254.
22. Kraus, A. and Oesterlé, J., ‘Sur une question de B. Mazur’, Math. Ann. 293 (1992) no. 2, 259275.
23. Mazur, B., ‘Rational isogenies of prime degree’, Invent. Math. 44 (1978) no. 2, 129162.
24. Mumford, D., ‘Varieties defined by quadratic equations’, Questions on algebraic varieties (C.I.M.E., III Ciclo, Varenna, 1969) (Edizioni Cremonese, Rome, 1970) 29100.
25. Papadopoulos, I., ‘Courbes elliptiques ayant même 6-torsion qu’une courbe elliptique donnée’, J. Number Theory 79 (1999) no. 1, 103114.
26. Poonen, B., Schaefer, E. F. and Stoll, M., ‘Twists of X (7) and primitive solutions to x 2 + y 3 = z 7 ’, Duke Math. J. 137 (2007) no. 1, 103158.
27. Ribet, K. A., ‘Raising the levels of modular representations’, Séminaire de Théorie des Nombres, Paris, 1987–1988 , Progress in Mathematics 81 (ed. Goldstein, C.; Birkhäuser, Boston, 1990) 259271.
28. Rubin, K. and Silverberg, A., ‘Families of elliptic curves with constant mod p representations’, Elliptic curves, modular forms & Fermat’s Last Theorem, Hong Kong, 1993 , Series in Number Theory I (eds Coates, J. and Yau, S.-T.; International Press, Cambridge, MA, 1995) 148161.
29. Rubin, K. and Silverberg, A., ‘Mod 6 representations of elliptic curves’, Automorphic Forms, Automorphic representations, and arithmetic, Fort Worth, TX, 1996 , Proceedings of Symposia in Pure Mathematics, Part 1 66 (American Mathematical Society, Providence, RI, 1999) 213220.
30. Rubin, K. and Silverberg, A., ‘Mod 2 representations of elliptic curves’, Proc. Amer. Math. Soc. 129 (2001) no. 1, 5357.
31. Silverberg, A., ‘Explicit families of elliptic curves with prescribed mod N representations’, Modular forms and Fermat’s last theorem, Boston, MA, 1995 (eds Cornell, G., Silverman, J. H. and Stevens, G.; Springer-Verlag, New York, 1997) 447461.
32. Silverman, J. H., The arithmetic of elliptic curves , Graduate Text in Mathematics 106 (Springer, New York, 1986).
33. Vélu, J., ‘Isogénies entre courbes elliptiques’, C. R. Math. Acad. Sci. Paris 273 (1971) 238241.
34. Vélu, J., ‘Courbes elliptique munies d’un sous-group ℤ∕nℤ × μ n ’, Mém. Soc. Math. Fr. 57 (1978).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Type Description Title
UNKNOWN
Supplementary materials

Fisher Supplementary Material
Supplementary Material

 Unknown (51 KB)
51 KB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed