Rational Points on Curves, Compositions of Polynomials, and Nevanlinna’s Theorem

Faltings’ theorem (the Mordell Conjecture) affirms that no curve with genus larger than one can have infinitely many rational points. In this talk, we describe how to classify all curves defined by F(x) = G(y) with genus at most 1. We will draw parallels to the subject of compositions of polynomials, and a surprising theorem of Nevanlinna on unique range sets.