Matt Baker

We determine the exact set of prime numbers $N$ such that the modular curve $X_0(N)$ is trigonal, i.e. admits a degree 3 map to ${\bf P}^1$ defined over an algebraic closure of ${\bf Q}$. The method generalizes the one used by Ogg to study hyperelliptic modular curves. We also say something about degree $d$ maps for arbitrary $d$. We give two applications, one to a question about the cardinality of $X_0(N)(L)$ as $L$ ranges over all cubic extensions of a given number field $K$, and the other to a conjecture of Coleman, Kaskel, and Ribet concerning torsion points on $X_0(N)$. Specifically, we explain how one can now verify their conjecture for a large set of prime numbers $N$.