In joint work with Gergely Harcos we strengthened a result of Taylor on associating Galois representations to cuspforms over imaginary quadratic fields. We further present a result (joint with Kris Klosin) on the modularity of ordinary, residually reducible Galois representations of imaginary quadratic fields.
An overview will be given of the contents of http://arxiv.org/abs/0706.0272.
The Tamagawa Number Conjecture predicts that arithmetic motives admit special zeta elements that account for the special values of their L functions on the one hand, and which form Euler/ Kolyvagin systems on the other hand. The general philosophy of this approach suggests that the zeta elements of a motive should have coefficients in the universal deformation ring of the Galois representation associated with it. Using works of Kato, Ochiai, Howard, Buyukboduk and myself, I will describe three situations where this is almost known and, if time permits, I will suggest another more striking one.
This is a joint work with Florian Breuer (University of Stellenbosch). We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring $\BF_q[T]$.
In joint work with W. Stein and K. Koo we study the set of primes p such that the coefficient a_p(f) generates the coefficient field of a given newform f. Under the assumption that f is of weight at least 2 and that f nor has CM nor non-trivial inner twists, we prove that this set has density 1. We also provide an analogue in the presence of non-trivial inner twists. In the talk we present an overview of the proof as well as more precise heuristic and computational indications about the asymptotic behaviour of the set of primes p such that a_p(f) does not generate the coefficient field.
Last modification: 18 January 2008.