We investigate the ramification of modular parametrizations of elliptic curves over 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 . We also report on numerical computations indicating that in general, the ramification index at a cusp seems to be a divisor of 24.
Nous étudions la ramification aux pointes des paramétrisations modulaires des courbes elliptiques sur . Nous montrons que si le forme modulaire associée à la courbe elliptique est de niveau minimal parmi ses tordues par les caractères de Dirichlet, alors la paramétrisation modulaire est non ramifiée aux pointes. La preuve utilise la formule de Bushnell pour la constante locale de Godement-Jacquet d’une représentation automorphe supercuspidale de . Nous présentons également des calculs numériques indiquant qu’en général, l’indice de ramification en une pointe semble être un diviseur de 24.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.963
Keywords: Elliptic curve, Modular parametrization, Ramification index, Automorphic representation, Local constant
@article{JTNB_2016__28_3_773_0,
author = {Fran\c{c}ois Brunault},
title = {On the ramification of modular parametrizations at the cusps},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {773--790},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {28},
number = {3},
year = {2016},
doi = {10.5802/jtnb.963},
zbl = {1417.11102},
mrnumber = {3610697},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.963/}
}
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%0 Journal Article %A François Brunault %T On the ramification of modular parametrizations at the cusps %J Journal de théorie des nombres de Bordeaux %D 2016 %P 773-790 %V 28 %N 3 %I Société Arithmétique de Bordeaux %U https://doi.org/10.5802/jtnb.963 %R 10.5802/jtnb.963 %G en %F JTNB_2016__28_3_773_0
François Brunault. On the ramification of modular parametrizations at the cusps. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 3, pp. 773-790. doi : 10.5802/jtnb.963. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.963/
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