scholarly journals Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

2010 ◽  
Vol 130 (12) ◽  
pp. 2753-2772 ◽  
Author(s):  
Burcu Baran
Keyword(s):  
Class Number ◽  
Modular Curves ◽  
Cartan Subgroups

scholarly journals A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

2017 ◽  
Vol 60 (2) ◽  
pp. 411-434 ◽  
Author(s):  
MARUSIA REBOLLEDO ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Elliptic Curves ◽  
Half Plane ◽  
London Math ◽  
Number Fields ◽  
Arithmetic Geometry ◽  
Modular Curves ◽  
Torsion Points ◽  
Modular Curve ◽  
Upper Half Plane ◽  
Cartan Subgroups

AbstractModular curves likeX0(N) andX1(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 byXnsp(p) andXnsp+(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 ofp-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).

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