2018/01/11 by Imin Chen, Chen, Imin, Parinaz Salari Sharif +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1801.04020
openalex publication_date 2018/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ. 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.