Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T19:34:41.208Z Has data issue: false hasContentIssue false
  • English
  • Français

A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

Part of: Arithmetic problems. Diophantine geometry Abelian varieties and schemes Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  30 October 2017

MARUSIA REBOLLEDO  and
CHRISTIAN WUTHRICH
MARUSIA REBOLLEDO
Affiliation:
Université Clermont Auvergne, Laboratoire de Mathématiques, UMR 6620 CNRS, Campus universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière, France e-mail: marusia.rebolledo@uca.fr
CHRISTIAN WUTHRICH
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK e-mail: christian.wuthrich@nottingham.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc. (3) 77(1) (1998), 1–38; J. Algebra231(1) (2000), 414–448).

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11F80: Galois representations 14G35: Modular and Shimura varieties 14K10: Algebraic moduli, classification
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 411 - 434
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Baran, B., Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem, J. Number Theory 130 (12) (2010), 2753–2772.CrossRefGoogle Scholar
2. Coates, J., Greenberg, R., Ribet, K. A. and Rubin, K., Arithmetic theory of elliptic curves, Lecture Notes in Mathematics, vol. 1716, (Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 1999). Lectures from the 3rd C.I.M.E. Session held in Cetraro, July 12–19, 1997 (Viola C., Editor).CrossRefGoogle Scholar
3. Chen, I., The Jacobians of non-split Cartan modular curves, Proc. London Math. Soc. (3) 77 (1) (1998), 1–38.CrossRefGoogle Scholar
4. Chen, I., On relations between Jacobians of certain modular curves, J. Algebra 231 (1) (2000), 414–448.CrossRefGoogle Scholar
5. Darmon, H. and Merel, L., Winding quotients and some variants of Fermat's last theorem, J. Reine Angew. Math. 490 (1997), 81–100.Google Scholar
6. Deligne, P. and Rapoport, M., Les schémas de modules de courbes elliptiques, in Modular functions of one variable, II, Volume 349 of Lecture Notes in Mathematics (Springer, Berlin, 1973), 143–316. Proceedings of the International Summer School, University of Antwerp, Antwerp, 1972.CrossRefGoogle Scholar
7. Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics, vol. 228 (Springer-Verlag, New York, 2005).Google Scholar
8. de Smit, B. and Edixhoven, B., Sur un résultat d'Imin Chen, Math. Res. Lett. 7 (2–3) (2000), 147–153.CrossRefGoogle Scholar
9. Edixhoven, B., On a result of Imin Chen, Available at http://arxiv.org/abs/alg-geom/9604008, 1996, unpublished.Google Scholar
10. Halberstadt, E., Sur la courbe modulaire X ndép(11), Exp. Math. 7 (2) (1998), 163–174.Google Scholar
11. Katz, N. M. and Mazur, B., Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985).CrossRefGoogle Scholar
12. Kohen, D. and Pacetti, A., Heegner points on Cartan non-split curves, Canad. J. Math. 68 (2016), 422–444.CrossRefGoogle Scholar
13. Ligozat, G., Courbes modulaires de niveau 11, in Modular functions of one variable, V: Proceedings of the 2nd International Conference, University of Bonn, Bonn, 1976, Lecture Notes in Math., vol. 601 (Springer, Berlin, 1977), 149–237.CrossRefGoogle Scholar
14. Merel, L., Arithmetic of elliptic curves and Diophantine equations, J. Théor. Nombres Bordeaux 11 (1) (1999), 173–200, Les XXèmes Journées Arithmétiques (Limoges, 1997).CrossRefGoogle Scholar
15. Momose, F., Rational points on the modular curves X split(p), Compositio Math. 52 (1) (1984), 115–137.Google Scholar
16. Serre, J.-P., Lectures on the Mordell-Weil theorem, 3rd ed., Aspects of Mathematics (Friedr. Vieweg & Sohn, Braunschweig, 1997), Translated from the French and edited by Brown, Martin from notes by Michel Waldschmidt, With a foreword by Brown and Serre.CrossRefGoogle Scholar
17. Shimura, G., Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11 (Princeton University Press, Princeton, NJ, 1994). Reprint of the 1971 original, Kanô Memorial Lectures, 1.Google Scholar