Â鶹¹ÙÍø

Calabi-Yau Manifolds Seminar

Speaker: Kamryn Spinelli (Â鶹¹ÙÍø's University)

Title: Counting points with invariant theory: elliptic curves

Abstract: Consider a family of elliptic curves E parametrized by a base X. When E is in Legendre form, X is 1-dimensional and the point count \# E(\mathbb{F}_p) is given by a simple formula involving Legendre symbols of the parameter of X. Extending this formula to families of elliptic curves over a higher-dimensional base X is more complicated, yet highly relevant to the study of hypergeometric motives. In this talk, we will collect some thoughts on a new method for lifting the Legendre family point count to the full family of elliptic curves branched over four points in \mathbb{P}^1, using invariant theory over finite fields. This is joint work in progress with Noriko Yui.