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Published online by Cambridge University Press:  30 October 2017

Université Clermont Auvergne, Laboratoire de Mathématiques, UMR 6620 CNRS, Campus universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière, France e-mail:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK e-mail:
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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).

Research Article
Copyright © Glasgow Mathematical Journal Trust 2017 



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