Lectures on Moduli Spaces of Elliptic Curves

TL;DR

This paper provides an accessible introduction to the moduli space of elliptic curves, covering various approaches and foundational concepts like orbifolds and stacks, aimed at students and researchers new to the field.

Contribution

It offers a comprehensive, multi-perspective overview of moduli spaces of elliptic curves, emphasizing explicit constructions and foundational concepts, suitable for educational purposes.

Findings

Clarifies the role of orbifolds and stacks in moduli theory

Demonstrates all four approaches to moduli spaces in the elliptic curve case

Provides explicit examples illustrating theoretical concepts

Abstract

These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are considered. Topics covered reflect my own biases. Very little, if anything, in these notes is original, except perhaps the selection of topics and the point of view.