Elliptic curves and L-functions

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My research interests are mainly elliptic curves and L-functions. In the last years, I studied several distribution problems associated with elliptic curves over the rationals, in collaboration with several co-authors. This is a very active research area, and I intend to extend my previous work in various different directions in the next years. For example, a conjecture of N. Koblitz gives an exact asymptotic for the number of primes such that the group of points modulo p of a given elliptic curve over Q has prime order. This has applications to public key cryptography. In a recent work with A. Balog and A. Cojocaru, we proved that Koblitz's conjecture is true on average, when one takes the average over all elliptic curves in a box. This work raises a lot of new interesting questions, which are addressed in this proposal.