An explicit correspondence of modular curves

Imin Chen; Parinaz Salari Sharif

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arXiv · 1801.04020

An explicit correspondence of modular curves

Imin Chen·Parinaz Salari Sharif

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Abstract

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 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 expressed as a certain linear combination of double coset operators whose coefficients we are able to make explicit.

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Imin Chen, Parinaz Salari Sharif. 2018-01-11. An explicit correspondence of modular curves. https://arxiv.org/abs/1801.04020

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