Description: A basic way to study the integer solutions to a diophantine equation like the the Fermat equation is to first study the solutions modulo a carefully chosen prime p. A refinement of this idea is to study compatible systems of solution modulo all powers of p. This idea leads to the notion of the p-adic numbers and more generally local methods in number theory.
An analogous idea appears in algebraic geometry when one studies a plane curve by analyzing the power series solutions to its defining equation. In this course, we will learn about these ideas, focusing either on arithmetic or geometry depending on student interest. See the syllabus for a list of specific potential topics.