On icosahedral Artin representations

Self-dual Artin Representations

Functional equations in number theory are relations between an L-function and some sort of dual L-function, and in general, the L-function and its dual need not coincide. For example, if χ is a primitive Dirichlet character then the functional equation relates L(s, χ) to L(1− s, χ), and L(s, χ) = L(s, χ) if and only if χ = 1. Or if f is a primitive cusp form of weight two for Γ1(N) and f∨ is th...

Icosahedral Galois Representations

Even Icosahedral Galois Representations of Prime Conductor

In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all prime conductors less than 10,000 of even icosahedral Galois representations.

Mod 2 and Mod 5 Icosahedral Representations

We shall call a simple abelian variety A/Q modular if it is isogenous (over Q) to a factor of the Jacobian of a modular curve. In this paper we shall call a representation ρ̄ : GQ→GL2(F̄l) modular if there is a modular abelian variety A/Q, a number field F/Q of degree equal to dimA, an embedding OF ↪→ End(A/Q) and a homomorphism θ : OF→F̄l such that ρ̄ is equivalent to the action of GQ on the ker θ...

A Mod Five Approach to Modularity of Icosahedral Galois Representations

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 = 2 · 43 that satisfies Artin’s conjecture. We briefly explain the computations behind seven additional examples of conductors 2416 = 2 · 151, 3184 = 2 · 199, 3556 = 2 · 7 · 127, 3...