## Special Colloquium

Ari Shnidman

Boston College

Rational points on elliptic curves and beyond

**Abstract:**Given a system of polynomial equations, we may ask whether there exist solutions whose coordinates are all rational numbers, and if so, how many. In the language of arithmetic geometry, we wish to describe the set of rational points on an algebraic variety. I will survey results concerning the behavior of rational points in families of algebraic varieties. I'll start with plane conics, and then move on to families of elliptic (cubic) curves, for which there are many beautiful results and many unresolved questions. I'll finish with some recent work on higher degree curves and their Jacobians.

Tuesday November 28, 2017 at 3:00 PM in SEO 636