Arithmetic statistics of elliptic curves over a fixed finite field

03/22/2016 - 12:15pm
03/22/2016 - 1:10pm
Nathan Kaplan (UC Irvine)

In this talk we will discuss the distribution of rational point counts for elliptic curves over a fixed finite field of size q. Hasse’s theorem says that such a curve has q+1-t points where |t| is at most 2*q^{1/2}. A theorem of Birch, the ‘vertical’ version of the Sato-Tate theorem, explains how t varies as q goes to infinity. We discuss joint work with Ian Petrow in which we generalize Birch’s theorem and give applications to several statistical questions. What is the probability that the number of points on an elliptic curve is divisible by 5? Surprisingly, the answer is not 1/5. What is the average number of points on a curve containing a rational 5-torsion point? What is the expected exponent of the group of rational points of a randomly chosen elliptic curve?

Millikan 2099, Pomona College