Modularity of certain potentially Barsotti-Tate Galois representations

Abstract

We show that certain potentially semistable lifts of modular mod l l representations are themselves modular. As a result we show that any elliptic curve over the rational numbers with conductor not divisible by 27 is modular.

Ternary Diophantine Equations via Galois Representations and Modular Forms

“…Proof Applying work of Breuil, Conrad, Diamond and Taylor [4] (or even, since, by part (b) of Lemma 2.1, the curve E does not have conductor divisible by 27, earlier work of Conrad, Diamond and Taylor [12]) we may conclude that E is modular. In particular this means that the the representation ρ E n is modular.…”

Untitled

“…Recall that if K ‫ޑ/‬ p is a finite extension the inertial type of a potentially semistable Galois representation ρ : G K → GL n ‫ޑ(‬ p ) is the restriction to I K of the corresponding Weil-Deligne representation. In this paper we normalise this definition as in the appendix to [Conrad et al 1999], so that, for example, the inertial type of a finite order character is just the restriction to inertia of that character. We refer the reader to Definition 2.1.2 and the discussion immediately following it for our definition of "Hodge type 0."…”

Transcendental Nature of Special Values of L-Functions

“…Let E be an elliptic curve defined over Q with conductor N. Since every elliptic curve over Q is modular (see [3,4,6,16,20 Then S has at most one algebraic element.…”