Welcome to the NYU Arithmetic Geometry Seminar webpage! This semester we will be studying elliptic curves and modular forms.
Leaders: Jacob Monzel, Jake Zweifler, and Zeid Ghalyoun
Email: [email protected], [email protected], and [email protected]
Class: Wednesday 6:30-7:30pm (WWH 202)
References: Arithmetic of Elliptic Curves by Silverman (AECI), Introduction to Modular Forms by Diamond and Shurman (DS)
(We will also use chapter 1 from Silverman II as a guide in the second half of the seminar)
| Week | Topic | Reading | Remarks |
|---|---|---|---|
| 1 | Review of Varieties and Algebraic Curves | Skim chapters 1 and 2 in AECI | Summary of chapters 1 and 2 in AECI. The goal for next week will be to start reading about elliptic curves in chapter 3. |
| 2 | Weierstrass Equations, Group Law | Sections 3.1-3.2 in AECI (start to look at 3.3) | Discussion on chapter 3. |
| 3 | Elliptic curves, Isogenies | Sections 3.3-3.5 in AECI | Continued discussion on chapter 3. |
| 4 | Elliptic curves, Isogenies, Dual Isogeny | Sections 3.3-3.6 in AECI | Continued discussion on chapter 3. |
| 5 | Tate Modules | Sections 3.7 in AECI | Continued discussion on chapter 3. |
| 6 | Elliptic Curves over Finite Fields | Chapter 5 in AECI | Introduction to elliptic curves over finite fields. |
| 7 | Elliptic Curves over C | Chapter 6 in AECI | Analysis of elliptic curves over complex numbers. |
| 8 | Weak Mordell-Weil, Kummer Pairing, Descent, Mordell-Weil over Q | Sections 8.1-8.4 in AECI | Introduction to descent and Mordell-Weil theorem. |
| 9 | Heights, Torsion Points | Sections 8.5-8.7 in AECI | Discussion on heights and torsion points. |
| 10 | Minimal Discriminant, Canonical Height, Rank, ABC Conjecture | Sections 8.8-8.11 in AECI | Exploring the ABC conjecture and related topics. |
| 11 | Modular Forms and Modular Curves | Chapter 1 in DS | Introduction to modular forms. |
| 12 | Modular Curves as Riemann Surfaces | Chapter 2 in DS | Discussion on modular curves. |
| 13 | Dimension Formulas | Chapter 3 in DS | Introduction to dimension formulas. |
| 14 | Eisenstein Series | Chapter 4 in DS | Analysis of Eisenstein series. |
| 15 | Hecke Operators | Chapter 5 in DS | Discussion on Hecke operators. |