Three-dimensional Galois Representations with Conjectural Connections to Arithmetic Cohomology

In [4], Ash and Sinnott conjecture that any Galois representation having niveau 1 which satisfies a certain parity condition is attached in a specific way to a Hecke eigenclass in cohomology, and they make a prediction about exactly where the relevant cohomology class should lie. They give examples of reducible three-dimensional representations which appear to be attached to cohomology eigenclasses, in the sense that the characteristic polynomials of Frobenius elements for small primes correspond exactly to the Hecke eigenvalues of certain eigenclasses. The question of proving this connection for all primes seems to be difficult; however, in [6] Ash and Tiep develop techniques for proving that certain irreducible three-dimensional symmetric square representations are in fact attached to cohomology classes. Until recently there were no known examples of three-dimensional irreducible non-symmetric square characteristic p Galois representations which seem to be attached to cohomology eigenclasses. In this paper we extend the original conjecture of Ash and Sinnott to include irreducible niveau 2 representations and give an example of a niveau 2 Galois representation which is neither reducible nor obtained as the symmetric square of a two dimensional representation, but for which the conjectured connection with arithmetic cohomology appears to hold, at least for prime ` ≤ 47. We also briefly discuss the computational techniques needed to demonstrate the apparent connection. We note that the forthcoming paper [3] of the author, Avner Ash and David Pollack extends the conjecture given here to include reducible (but semisimple) representations, deals with higher dimensional and higher niveau representations, and includes many more computational verifications of the conjecture.