An explicit correspondence between certain modular curves

Date created: 
2017-05-03
Identifier: 
etd10163
Keywords: 
Modular curves, explicit correspondences, representation theory, split and non-split Cartan subgroups.
Abstract: 

In this thesis, we recall an alternative proof of Merel's conjecture, which asserts that a certain explicit correspondence gives the isogeny relation between the Jacobians associated to the normalizer of split and non-split Cartan subgroups. This alternative proof does not require extensive representation theory and can be formulated in terms of certain finite geometries modulo $\ell$. Secondly, we generalize these arguments to exhibit an explicit correspondence which gives the isogeny relation between the Jacobians associated to split and non-split Cartan subgroups. An interesting feature is that the required explicit correspondence is considerably more complicated but can be expressed as a certain linear combination of double coset operators whose coefficients we are able to make explicit.

Document type: 
Thesis
Rights: 
This thesis may be printed or downloaded for non-commercial research and scholarly purposes. Copyright remains with the author.
File(s): 
Senior supervisor: 
Imin Chen
Department: 
Science: Department of Mathematics
Thesis type: 
(Thesis) M.Sc.
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