Modularity of certain potentially Barsotti-Tate Galois representations

Abstract

We show that certain potentially semistable lifts of modular mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.