| When f is a newform, its associated automorphic representation pi f = ⊗vpi f,v has finite components pip which determine the local behavior of the Galois representations attached to f. In particular, we can talk about the "type" of pi p as the restriction of the corresponding Weil-Deligne representation to the inertia subgroup. Let a weight k ≥ 2 be given. We prove that under certain mild hypotheses, a newform f can be found of weight k for which each pi p is of a prescribed type. Our strategy is to reduce the problem to finding a certain representation of SL2(Z/ NZ) inside a space of cusp forms Sk(Gamma( N)), and then to use an "equivariant" version of the Riemann-Roch formula to calculate the latter space inside the Grothendieck group of SL 2(Z/NZ).; We present a corollary to our main theorem whereby for a broad class of number fields K, there exists a modular abelian variety A for which AK has good reduction at all places. We also calculate the minimal extension field of Q p for which the modular Jacobian J1( pn) becomes semi-stable. In the final chapter, we formulate and provide evidence for a conjecture for the shape of the stable reduction of the modular curve X(p 2). |
