Grant: Discovery Grants Program - Individual

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. Another facet of my research is the study of the vanishing of special values of L-functions of elliptic curves in families. The vanishing of L-functions is a major problem of number theory. For L-functions of elliptic curves, the order of vanishing at the central critical point is related to the rank of the curve by the conjecture of Birch and Swinnerton-Dyer. It is believed that very few L-functions would vanish in a given reasonable family, but there are almost no such results proven up to now. Over the previous years, in collaboration with J. Fearnley and H. Kisilevsky, we used random matrix theory to predict the number of twists of a given order of an elliptic curve vanishing at the central critical point. In another work in progress, with collaborators D. Milicevic and G. Ricotta, we are finding a lower bound for the number of non-vanishing cubic twists of an elliptic curve over the quadratic extension of Q obtained by adding a third root of unity.