On the modularity of supersingular elliptic curves over certain totally real number fields

Abstract

We study generalisations to totally real fields of the methods originating with Wiles and Taylor and Wiles [A. Wiles, Modular elliptic curves and Fermat's Last ] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over Q( √ 2 ).

Elliptic curves over real quadratic fields are modular

“…Work of Jarvis and Manoharmayum [40] establishes modularity of semistable elliptic curves over Q( √ 2) and Q( √ 17), and there are various other works [1], [26], which establish modularity under local assumptions on the curve E and the field K. In this paper, we prove modularity of all elliptic curves over all real quadratic fields.…”

Moduli of finite flat group schemes, and modularity

“…also [JM,2]), since these properties can be forced by imposing local conditions on the extension F 0 =F at a finite set of primes of F . To see that F 0 can be chosen to satisfy the last condition in (vi) note that in constructing the intermediate field F iC1 from F i we can always choose a finite set of primes T of F i , which are disjoint from any given finite set of primes, and such that v 2 T splits in G F i .…”

Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms

“…The following key lemma is a known result and a consequence of the work of Langlands-Tunnell. For references where it is used see [21], [26], [14] and [17].…”