On the ramification of modular parametrizations at the cusps

François Brunault 1
1 Algèbre. Théorie des nombres
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
Abstract : We investigate the ramification of modular parametrizations of elliptic curves over Q at the cusps. We prove that if the modular form associated to the elliptic curve has minimal level among its twists by Dirichlet characters, then the modular parametrization is unramified at the cusps. The proof uses Bushnell's formula for the Godement-Jacquet local constant of a cuspidal automorphic representation of GL(2). We also report on numerical computations indicating that in general, the ramification index at a cusp seems to be a divisor of 24.
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François Brunault. On the ramification of modular parametrizations at the cusps. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2016, 28 (3), pp.773-790. ⟨10.5802/jtnb.963⟩. ⟨ensl-00707488v2⟩

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