Given a curve X with an action of the finite group G by automorphisms, the \ell-adic cohomology of X decomposes as a G-representation. We study this decomposition in two different ways. Firstly: as a way of translating between representation-theoretic and arithmetic data, thereby establishing formulae for conductor exponents of curves (which need not have extra automorphisms!). We also study the components of this decomposition as \ell-adic representations in their own right, using the earlier conductor results to demonstrate L-function computations and numerically verify a BSD-type conjecture.
