Seminar at SMU Delhi

Let E be an elliptic curve defined over Q and let N be a positive integer. Now, M_E(N) counts the number of primes p such that the group E_p(F_p) is of order N. In an earlier joint work with Balasubramanian, we showed that M_E(N) follows Poisson distribution when an average is taken over a family of elliptic curve with parameters A and B where A, B > N^{l/2} (log N)^{1+\gamma} and $AB>N^{3l/2}(\log N)^{2+\gamma} for a fixed integer l and any \gamma>0. In this talk, we explain that the same result holds even if we take A and B in the range \exp(N^{\epsilon^2 /20l})> A, B>N^\epsilon and AB>N^{3l/2}(log N)^{6+\gamma} for any \epsilon>0.