Modular forms constructed from moduli of elliptic curves, with applications to explicit models of modular curves

TL;DR

This paper provides an expository overview of modular forms derived from elliptic curve moduli, exploring their relation to Eisenstein series and methods for explicitly modeling modular curves.

Contribution

It offers a clear exposition of Katz's definition of modular forms and introduces techniques for constructing explicit models of modular curves.

Findings

Standard results on modular forms and Eisenstein series

Explicit methods for modeling modular curves

Insights into the Katz definition of modular forms

Abstract

These are the lecture notes from my portion of a mini-course for the summer school "Building Bridges 3" that was held in Sarajevo during July 2016. My lectures covered the Katz definition of modular forms, a family of forms defined from this perspective and their relation to Eisenstein series, and methods of finding explicit models of modular curves. The treatment is purely expository, and the results are mostly standard, although a few points of view may not be as widely known as they deserve to be.