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Pacific Journal of Mathematics 203 (2002), 265-282.

A mod five approach to modularity of icosahedral Galois representations

Kevin Buzzard and William A. Stein

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 ${\bf 1376}=2^5\cdot 43$ that satisfies Artin's conjecture. We briefly explain the computations behind seven additional examples of conductors ${\bf 2416}=2^4\cdot 151$, ${\bf 3184}=2^4\cdot 199$, ${\bf 3556}=2^2\cdot 7\cdot 127$, ${\bf 3756}=2^2\cdot 3\cdot 313$, ${\bf 4108}=2^2\cdot 13\cdot 79$, ${\bf 4288}=2^6\cdot 67$, and ${\bf 5373}=3^3\cdot 199$. We also generalize a result of Sturm on computing congruences between eigenforms.