A moduli interpretation for the non-split Cartan modular curve
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...
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| Format: | Article |
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Cambridge University Press
2018
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| Online Access: | https://eprints.nottingham.ac.uk/43660/ |
| _version_ | 1848796738389803008 |
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| author | Rebolledo, Marusia Wuthrich, Christian |
| author_facet | Rebolledo, Marusia Wuthrich, Christian |
| author_sort | Rebolledo, Marusia |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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]. |
| first_indexed | 2025-11-14T19:52:45Z |
| format | Article |
| id | nottingham-43660 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:52:45Z |
| publishDate | 2018 |
| publisher | Cambridge University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-436602020-05-04T19:35:24Z https://eprints.nottingham.ac.uk/43660/ A moduli interpretation for the non-split Cartan modular curve Rebolledo, Marusia Wuthrich, Christian 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]. Cambridge University Press 2018-05-01 Article PeerReviewed Rebolledo, Marusia and Wuthrich, Christian (2018) A moduli interpretation for the non-split Cartan modular curve. Glasgow Mathematical Journal, 60 (2). pp. 411-434. ISSN 1469-509X https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/moduli-interpretation-for-the-nonsplit-cartan-modular-curve/B5C148F18FDB5B33E2470293ECD94D3A doi:10.1017/S0017089517000180 doi:10.1017/S0017089517000180 |
| spellingShingle | Rebolledo, Marusia Wuthrich, Christian A moduli interpretation for the non-split Cartan modular curve |
| title | A moduli interpretation for the non-split Cartan modular curve |
| title_full | A moduli interpretation for the non-split Cartan modular curve |
| title_fullStr | A moduli interpretation for the non-split Cartan modular curve |
| title_full_unstemmed | A moduli interpretation for the non-split Cartan modular curve |
| title_short | A moduli interpretation for the non-split Cartan modular curve |
| title_sort | moduli interpretation for the non-split cartan modular curve |
| url | https://eprints.nottingham.ac.uk/43660/ https://eprints.nottingham.ac.uk/43660/ https://eprints.nottingham.ac.uk/43660/ |