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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: [email protected]
CHRISTIAN WUTHRICH
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK e-mail: [email protected]
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Abstract

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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).

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

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