A moduli interpretation for the non-split Cartan modular curve

Rebolledo, Marusia; Wuthrich, Christian

Abstract

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 X+ nsp(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 [Che98, Che00].

Citation

Rebolledo, M., & Wuthrich, C. (2018). A moduli interpretation for the non-split Cartan modular curve. Glasgow Mathematical Journal, 60(2), https://doi.org/10.1017/S0017089517000180

• Journal Article Type: Article
• Acceptance Date: Jun 16, 2017
• Online Publication Date: Oct 30, 2017
• Publication Date: May 1, 2018
• Deposit Date: Jun 20, 2017
• Publicly Available Date: Oct 30, 2017
• Journal: Glasgow Mathematical Journal
• Print ISSN: 0017-0895
• Electronic ISSN: 1469-509X
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 60
• Issue: 2
• DOI: https://doi.org/10.1017/S0017089517000180
• Public URL: https://nottingham-repository.worktribe.com/output/930843
• Publisher URL: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/moduli-interpretation-for-the-nonsplit-cartan-modular-curve/B5C148F18FDB5B33E2470293ECD94D3A