Title: Imaginary quadratic fields over which X_0(15) has rank 0.

Abstract: In January this year, Caraiani and Newton proved that if F is an imaginary quadratic field such that X_0(15) has rank 0 over F, then every elliptic curve over F is modular. In this talk we will consider the case F=Q(sqrt(-p)). By doing 2-Selmergroup computations over various quadratic fields like Q(sqrt(p)) and Q(sqrt(17p)), we will obtain a rankbound that is equivalent to the one obtained from a 4-descent of the quadratic twist by -p of X_0(15) over Q. The proof of this equivalence uses ideas from visualisation of Sha[2]. The 2-Selmergroup calculations make heavy use of Rédei symbols, a number theoretic symbol encapsulating splitting behaviour in D_4-extensions of Q which satisfies a powerful reciprocity law based on the product formula for Hilbert symbols over quadratic fields.