I propose to determine semi-stable models of the modular curves. One now knows that every elliptic curve defined over the rationals is a quotient of a modular curve. It is said to be semi-stable if the singularities are only on its reductions and are ordinary double points. One knows, thanks to Mum-ford, that every curve has a semi-stable model, after a base extension, but for most (eg., those of prime cubed level) one doesn't know one. I plan to remedy this defect.
I propose to determine semi-stable models of the modular curves. Modular curves play a prominent role in contemporary algebraic number theory. For example, they were crucial in Wiles' proof of Fermat's last theorem. One now knows that every elliptic curve defined over the rational numbers (a.k.a. the rationals) (i.e., defined by equations with only rational coefficients) is a quotient of a modular curve and this has been used to provide most of the theoretical evidence for the Birch-Swinnerton-Dyer conjecture on the structure of the set of rational points on such a curve. Modular curves first arose as smooth curves defined over the complex numbers. Shimura proved they could be defined over he rationals. A model for a curve defined the rationals is essentially a set of equations with integer coefficients. It is said to be semi-stable if its singularities incorporated in the equations are of the simplest sort. One knows, thanks to David Mumford, that every curve has a semi-stable model but for most modular curves one doesn't know one. I plan to remedy this defect.
