Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7

Yoshikawa, Sho

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 729-741.

Let $K$ be a totally real field which is a finite abelian extension over $\mathbb{Q}$ and is unramified at 3,5, and 7. We prove that any elliptic curve $E$ over $K$ is modular, by reducing modularity of $E$ to known modularity lifting theorems.

Soit $K$ un corps totalement réel qui est une extension abélienne finie de $\mathbb{Q}$ non ramifiée en $3,5$ et $7$. Nous prouvons que toute courbe elliptique $E$ sur $K$ est modulaire, en réduisant la question de modularité de $E$ aux théorèmes de relèvement modulaire connus.

Classification: 11F80, 11G05, 11F41
Keywords: elliptic curves, Hilbert modular forms, Galois representations

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