Elliptic curves and L-functions

我的研究兴趣主要是椭圆曲线和l函数。在过去的几年中，我与几位合著者合作，研究了几个与有理数上的椭圆曲线相关的分布问题。这是一个非常活跃的研究领域，我打算在未来几年将我之前的工作扩展到各个不同的方向。例如，N. Koblitz的一个猜想给出了素数的精确渐近，使得给定的椭圆曲线在Q上的点群模p具有素数阶。这对公钥加密也有应用。在最近与a . Balog和a . Cojocaru合作的一项工作中，我们证明了Koblitz猜想在平均情况下是正确的，当我们取一个盒子中所有椭圆曲线的平均值时。这项工作提出了许多新的有趣的问题，这些问题在本提案中得到了解决。

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.