MATH 7390 - Topics in Algebra: Arithmetic of Curves Fall 2013

Instructor: David Zywina

This course will survey several areas of arithmetic geometry with an emphasis on curves. Arithmetic geometry, loosely defined, is the study of varieties over fields of number theoretic interest (like the rationals or finite fields). We will be interested in understanding how the rational points of an algebraic curve are governed by its geometry.

We will start with a review of the theory of curves and their Jacobians, before describing a web of theorems and conjectures underlying much of modern number theory. Topics include: the Mordell-Weil theorem, the Weil conjectures, L-functions and the Birch and Swinnerton-Dyer conjecture, the Sato-Tate conjecture, Mordell’s conjecture (Faltings’ theorem). Additional topics will depend on the makeup and interests of the class.

Some basic algebraic geometry will be assumed, though not that much since we will largely focus on curves. Previous exposure to algebraic number theory will be useful but not assumed.