Modularity of certain potentially Barsotti-Tate Galois representations

1999/01/01 by Brian Conrad, Fred Diamond, Richard Taylor · 170 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Homotopy and Cohomology in Algebraic Topology #Modular design #Modularity (biology) #Mathematics #Annotation #Semantics (computer science) #Elliptic curve #Galois module #Pure mathematics #Computer science #Programming language #Artificial intelligence

paper · pdf · doi:10.1090/s0894-0347-99-00287-8

published in Journal of the American Mathematical Society 12(2), 521-567 (American Mathematical Society)

openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Sur les représentations modulaires de degré 2 de Gal(Q¯/Q)
• Good Reduction of Abelian Varieties
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Ring-Theoretic Properties of Certain Hecke Algebras
• On Deformation Rings and Hecke Rings
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• p-adic Hodge-theoretic properties of étale cohomology with mod pcoefficients, and the cohomology of Shimura varieties
• Serre’s modularity conjecture (II)
• Proof of de Smit’s conjecture: a freeness criterion
• Modularity theorem [wikipedia]