In this talk, we delve into the arithmetic moduli problem for elliptic curves, aiming to understand modular curves as algebraic objects. Drawing from Katz and Mazur’s work, along with insights from Deligne-Rapoport, we explore the theories. In the beginning, I will introduce the basic definitions and questions. In the second part, I will present the famous construction M_{1,1}, which is a solution to the naive moduli problem, as an example. There will be a second talk in the algebraic stacks seminar.

Talk by Fengyuan Lin.