p-adic variation of arithmetic and modular forms

Given a polynomial equation with rational coefficients, what are its rational solutions? This is an ancient and vexing question in number theory that has seen ongoing progress over the last 200 years thanks to the application of ideas from geometry and topology, developing into the field of arithmetic algebraic geometry. First I will summarize a few aspects of the confluence of geometry, topology, and number theory that comprises arithmetic algebraic geometry. Then I will describe some results that apply holomorphic modular forms and p-adic variation to answer questions in arithmetic algebraic geometry.