Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_q$. Hasse’s theorem says that $\#E(\mathbb{F}_q) = q + 1 - t_E$ where $|t_E| \le 2\sqrt{q}$. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of $t_E$ in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as $q$ goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula.
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In this talk we discuss finer counting questions for elliptic curves over $\mathbb{F}_q$. For example, what is the probability that the number of rational points is divisible by $5$? What is the probability that the group of rational points is cyclic? If we choose a curve at random, and then pick a random point on that curve, what is the probability that the order of the point is odd? We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. We will also discuss some open problems. This is joint with work Ian Petrow (ETH Zurich).