An explicit correspondence of modular curves

JOURNAL OF NUMBER THEORY

卷:200

Article

Chen, Imin¹  Sharif, Parinaz Salari¹ 

[1] Simon Fraser Univ, Dept Math, Burnaby, BC, Canada

关键词: Modular curves;    Elliptic curves;   

DOI  :  10.1016/j.jnt.2018.12.003

来源: Elsevier

【 摘 要 】

In this paper, 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 finite field analogues of the complex plane minus the real line. 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 expressed as a certain linear combination of double coset operators whose coefficients we are able to make explicit. (C) 2019 Published by Elsevier Inc.

【 授权许可】

Free