Rational points on modular curves

The study of arithmetical properties of elliptic curves has been a tremendously active and fruitful area of investigation in recent decades, and plays a pervasive role in the recent solution to Fermat's Last Theorem. One way to study certain arithmetical properties of elliptic curves is through an object called a modular curve, whose arithmetic and geometry reflects that of the elliptic curves it classifies. Modular curves also give rise to the notion of a modular form, another object of extensive study which classically has a definition in terms of complex function theory and yet has an intimate connection with the arithmetic of elliptic curves. The recent solution to Fermat's Last Theorem draws upon a mysterious interplay between the Fermat equation, elliptic curves, modular curves, and modular forms. This talk will address certain questions about the arithmetic of modular curves, and the role such questions have in Fermat's Last Theorem, variations of Fermat's Last Theorem, and related problems in number theory.

*Dr. Chen is a candidate for a position in our department. All faculty are urged to attend this lecture.