A mod five approach to modularity of icosahedral Galois representations

Kevin Buzzard and William A. Stein

Vol. 203 (2002), No. 2, 265–282

DOI: 10.2140/pjm.2002.203.265

Abstract

We give eight new examples of icosahedral Galois representations that satisfy Artin’s conjecture on holomorphicity of their L-function. We give in detail one example of an icosahedral representation of conductor 1376 = 2⁵ ⋅ 43 that satisfies Artin’s conjecture. We briefly explain the computations behind seven additional examples of conductors 2416 = 2⁴ ⋅ 151, 3184 = 2⁴ ⋅ 199, 3556 = 2² ⋅ 7 ⋅ 127, 3756 = 2² ⋅ 3 ⋅ 313, 4108 = 2² ⋅ 13 ⋅ 79, 4288 = 2⁶ ⋅ 67, and 5373 = 3³ ⋅ 199. We also generalize a result of Sturm on computing congruences between eigenforms.

Milestones

Received: 28 June 2000

Revised: 18 October 2000

Published: 1 April 2002