Mod p representations on elliptic curves

Frank Calegari

Vol. 225 (2006), No. 1, 1–11

DOI: 10.2140/pjm.2006.225.1

Abstract

Modular Galois representations ρ : Gal(ℚ∕ℚ) → GL₂(𝔽_p) with cyclotomic determinant arise from elliptic curves for small p. We show that ρ does not necessarily arise from an elliptic curve whose conductor is as small as possible outside p. For p = 3 this disproves a conjecture of Lario and Rio.

Keywords

elliptic curves, modular forms, Galois representations

Mathematical Subject Classification 2000

Primary: 11G05

Milestones

Received: 31 August 2004

Accepted: 17 February 2005

Published: 1 May 2006