Belyi parametrisations of elliptic curves and congruence defects

2002/10/24 by Chandrashekhar Khare, Khare, Chandrashekhar
Mathematics · #11F #11R #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11F #msc:11R

paper · pdf · doi:10.48550/arxiv.math/0210391

arxiv created 2002/10/24 · openalex publication_date 2002/10/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL₂(Z). We ask if we can choose such a parametrisation of an elliptic curve to reflect the field over which it is defined: this can be viewed as a generalisation of the Shimura-Taniyama-Weil conjecture.

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