# Modular Curves (TCC 2014)

## Course aims

The aim of this course is to give an introduction to the theory of modular curves -- the quotients of the upper half-plane by finite-index subgroups of $SL_2(\mathbf{Z})$ -- and some of the insights into the theory of modular forms that can be obtained by applying geometric techniques to these spaces.

## Prerequisites

Initially, the course will assume knowledge of the basic concepts of modular forms, at the level of the Warwick MA4H9 Modular Curves course, and some familiarity with algebraic curves. More demanding background material from algebraic geometry (e.g. schemes) will be needed for some later sections. The language of algebraic stacks will not be used.

## Provisional Syllabus

• Modular curves as Riemann surfaces: coordinate charts, elliptic points and cusps.
• Genus formulae via Riemann-Hurwitz.
• Modular forms as differentials, dimension formulae via Riemann-Roch.
• Modular curves as algebraic curves, via the curves-fields correspondence.
• Introduction to representable functors and moduli spaces.
• Modular curves over $\mathbf{Q}$ as moduli spaces for elliptic curves.
• Correspondences between curves; Hecke operators.
• The Tate elliptic curve over $\mathbf{Q}((q))$ and the algebraic approach to cusps.
• Modular curves and modular forms over $\mathbf{Z}[1/N]$.
• The Deligne--Rapoport model of $X_0(p)$, and its reduction modulo $p$.
• The Eichler--Shimura relation.

## References

• Diamond & Shurman, A First Course in Modular Forms
• Rohrlich, Modular curves, Hecke correspondences and L-functions (chapter III of Cornell, Silverman & Stevens, Modular Forms and Fermat's Last Theorem)
• Deligne & Rapoport, Les schemas de modules de courbes elliptiques (in Deligne & Kuyk, Modular Functions of One Variable II, Springer Lecture Notes #349)
• Katz & Mazur, Arithmetic Moduli of Elliptic Curves

(The last two references are very advanced, and we will only cover a very small subset of the material in them!)