Mod 4 Galois representations and elliptic curves

Abstract

Galois representations ρ ¯ : G Q → G L 2 ( Z / n ) \bar {\rho }: G_{\mathbb Q} \rightarrow GL_{2}(\mathbb Z/n) with cyclotomic determinant all arise from the n n -torsion of elliptic curves for n = 2 , 3 , 5 n=2,3,5 . For n = 4 n=4 , we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.