Katz-Mazur seminar
The goal of this seminar is to learn about the integral regular models for congruence modular curves using Drinfeld level structures. We will mostly follow Katz-Mazur’s book Arithmetic moduli of elliptic curves. We meet weekly on Monday’s at 5pm in 347 Altgeld. We also have a discord server. Email me to join.
Some additional resources are:
Motivation and overview of Katz-Mazur’s regular models: Notes by Wei Ho
Modular curves as moduli spaces over C : sections 1.3-1.5 of Diamond-Shurman — A first course on modular forms
Background on finite group schemes: Schoof - Introduction to finite group schemes and Tate - Finite flat group schemes
Schedule
Week 1 (9/23, Langwen) - Classical moduli theory of elliptic curves over \(\mathbb{C}\), definition of the level structures associated to the subgroups \(\Gamma_1(n), \Gamma_0(n)\) and \(\Gamma(n)\) over \(\mathbb{C}\) and over \(\mathbb{Z}[1/n].\)
Week 2 (9/30, Ken) - \(A\)-structures, \(A\)-generators, and Drinfeld level structures, examples of \(\mathbb{G}_m, \mu_n, \mathbb{G}_a, \alpha_p\) in the étale and nonétale cases. (Chapter 1 of KM).
Week 3 (10/7) - Skipped.
Week 4 (10/14, Ken) - A "full set of sections", intrinsic definitions of \(A\)-structures and \(A\)-generators, more examples.
Week 5 (10/21, Fengyuan) - The four basic moduli problems for elliptic curves \(\Gamma(N),\Gamma_1(N),\text{bal-}\Gamma_1(n)\) and \(\Gamma_0(n)\) over \(\mathbb{Z}\) (Chapter 3), the formalism of moduli problems. (Chapter 4)
Week 6 (10/28, Fengyuan) - Rigidity and representability (Chapter 4). Axiomatic regularity (Chapter 5)
Week 7 (11/4, Ken) - Applying axiomatic regularity: Prove that the first three basic moduli problems are regular. (Chapter 5)
Week 8 (11/11, Langwen) - Cyclic subgroups, regularity of \(\Gamma_0(n)\).
Week 9 (11/18, Fengyuan) - Quotients by finite groups, relations between the various moduli problems.
To come next semester:
Chapter 8 - coarse moduli schemes, the Tate curve, cusps, compactification (Langwen)
Chapter 10 - the calculus of cusps and components, level structures on the Tate curve (Langwen)
Chapter 12 - new moduli problems in characteristic \(p\)
Chapter 13 - reductions mod \(p\) of the basic moduli problems