Grant: Discovery Grants Program - Individual

I propose to continue my investigation of both the analytic and arithmetic invariants of the L-functions (and their twists) attached to elliptic curves over the rational field. In [FKK], it is proved that the central value vanishes for an infinite number of cubic (order 3) twists, for most (conjecturally all) elliptic curves defined over the rational field. In certain cases, lower bounds for the frequency of vanishing were proved which approach (remarkably closely) to the values predicted by ``random matrix theory'' obtained in [DFK1]. In recent work, I have undertaken a study of the critical value of the derivative of twisted elliptic L-functions in the case that the twisted L-function vanishes to rank 1. Computational evidence was found [FK] supporting the existence of a remarkable ``algebraic part'' of these values which show hoped for Galois equivariance after ``factoring the elliptic regulator.'' This is predicted by the ``Equivariant Tamagawa Number Conjecture''. A related problem is that of establishing (when feasible) the non-vanishing of the central value of twisted L-functions. Although this is probably the more common state, it seems very difficult to establish in either its analytic or its algebraic form.