On a conjecture of Watkins

Dummigan, Neil

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 345-355.

Watkins a conjecturé que si $R$ est le rang du groupe des points rationnels d’une courbe elliptique $E$ définie sur le corps des rationels, alors $2^R$ divise le degré du revêtement modulaire. Nous démontrons, pour une classe de courbes $E$ choisie pour que ce soit le plus facile possible, que cette divisibilité découlerait de l’énoncé qu’un anneau de déformation $2$-adique est isomorphe à un anneau de Hecke, et est un anneau d’intersection complète. Mais nous démontrons aussi que la méthode de Taylor, Wiles et autres pour démontrer de tels énoncés ne s’applique pas à notre situation. Il semble alors qu’on ait besoin d’une nouvelle méthode pour que cette approche de la conjecture de Watkins puisse marcher.

Watkins has conjectured that if $R$ is the rank of the group of rational points of an elliptic curve $E$ over the rationals, then $2^R$ divides the modular parametrisation degree. We show, for a certain class of $E$, chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain $2$-adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove such statements, is necessarily inapplicable to our situation. It seems then that some new method is required if this approach to Watkins’ conjecture is to work.

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