# An application of Selberg's sieve to elliptic curves

Speaker:

Jyothsnaa Sivaraman, University of Toronto

Date and Time:

Thursday, September 26, 2019 - 2:10pm to 3:30pm

Location:

Fields Institute, Stewart Library

Abstract:

Let $E$ be an elliptic curve over the rationals. Koblitz conjectured that the number of primes $p \le x $ for which $|E(\mathbb{F}_p)|$ is prime, is asymptotic to $C_E \frac{x}{\log^2(x)}$, for some constant $C_E$. In this talk, we will try to discuss a result of S. Ali Miri and V. Kumar Murty which addresses the above conjecture. The result states that if we only consider elliptic curves with complex multiplication and assume the Riemann hypothesis for all Dedekind zeta functions, then it can be shown that $|\{p \le x :\nu (|E(\mathbb{F}_p)|) \le 16\}| \ll \frac{x}{\log^2 x}$. Here $\nu(n)$ is used to denote the number of distinct prime divisors of $n$.