# 18.782 Arithmetic Geometry

Welcome to the course webpage for 18.782, Introduction to Arithmetic Geometry, taught in Fall 2017. This is a course intended for undergraduate mathematics majors who have had one year of undergraduate abstract algebra (18.701 and 18.702 at MIT). This page maintains a schedule for the class, for the problem sets, notes and references.

**Instructor: **Dhruv Ranganathan (2-241). **Meeting time: **TuTh 930-11 (2-146). **Office hours: **By appointment and open door. I will be available after each class and on Wednesdays.

**Overview: **Arithmetic geometry is an old and incredibly active area of mathematical research. The subject represents the intersection of two other amazingly active fields -- algebraic geometry and number theory. In this course, students will acquire familiarity with the basic questions and techniques in arithmetic geometry, including the p-adic numbers, local-to-global principles, The course description, pre-requisites, requirements, and policies may be found in **PDF form**.

**Textbook: **There is no official textbook for the class, but notes and detailed references will be provided. Excellent references include *p-adic Numbers* *by* *Fernando Gouvea, Algebraic curves by Bill Fulton, Elliptic curves by Milne, and Undergraduate algebraic geometry by Miles Reid.*

**Problem sets: **Homework will be assigned roughly every Tuesday (with exceptions) and collected the following Tuesday. These will not be onerous, as your main task is to grapple with the material.

**Final project:** Your final project will be a short expository paper (5 pages) on a topic close to the class, and a short presentation on the same.

**Micro-presentations: **Ever now and then, the end of a class will consist of a student giving a short lecture on a topic assigned the previous week. Everyone should present at some point during the semester. The topic will be either an interesting aside, an illuminating example, or a difficult problem from a previous homework set. If during lecture you think of an interesting topic for a micro-presentation, talk to me soon after or send me an email.

**Grading:** Your grade will be based on the problem sets and micro-presentations (70%), the final paper (15%), the final presentation (10%), and participation (5%). There will be no exams.

**Lecture schedule:**

- September 7: Introduction to the course, diophantine geometry, rational points on curves, norms on fields.
- September 12: Completions of normed fields, inverse systems, construction of the p-adics.
- September 15: p-adic integers and p-adic expansions, algebraic properties of
**Z**_{p}. - September 19: Completeness of
**Q**_{p}, solutions to polynomial equations, and Hensel's lemma. - September 21: Squares in
**Q**_{p}, field extensions, and algebraic closures. - September 26: Affine varieties, introduction and examples.
- September 28: Affine varieties continued, irreducibility, prime ideals, Nullstellensatz.
- October 3: A review of the details of squares in
**Q**_{p}, smoothness of hypersurfaces. - October 5: Guest lecture by Isabel Vogt. Quadratic forms, basic properties, equivalence, Hasse-Minkowski.
- October 10: No classes at MIT, so Dhruv went to Germany.
- October 12: Reviewing where we are, and where we want to go. Proof Hasse-Minkowski.
- October 17: A proof of the Chevallay-Warning theorem.
- October 26: Properties of varieties, smoothness, irreducibility, affine patches.
- October 31: Function fields, and valuations on the function field of a curve.
- November 2: Genus, divisor theory and the Picard group.
- November 7: Degeneration methods, the degree-genus formula for plane curves.
- November 9: Giulia Sacca lectured about Riemann-Roch and Weierstrass equations.
- November 14: Review of Riemann-Roch and some consequences.
- November 21: Groups varieties, and abelian varieties and finite generation for groups of rational points.
- November 28: Proof of the weak Mordell-Weil theorem for curves with rational 2-torison.
- November 30: Isabel lectured about height functions as they apply to Mordell-Weil.
- December 5: Conclusion of the proof of the Mordell-Weil theorem for elliptic curves.
- December 7: (Presentation Days 1) Rachel lectured about Riemann-Roch for graphs and Abhijit lectured about Newton Polygons.
- December 12: (Presentation Days 2) Sammy and Ashley will prove the matrix tree theorem and Guanjie will discuss p-adic zeta functions.

**Problem sets: **will be posted here with due date.

**Problem Set 1**(due September 22): Valuations and valuation rings in a general context, some topology in the p-adic metric, and another example of inverse systems.**Problem Set 2**(due October 5): Squares in the 2-adic numbers, examples and non-examples of affine varieties, and roots of unity a.k.a fun with Hensel's lemma.**Problem Set 3**(due October 26): Chevallay-Warning revisited, and a proof of the local-global principle for quadratic forms.**Problem Set 4**(Due November 23): Integral dependence (algebra) and blowups (geometry). These are both secretly about resolution of singularities. We'll discuss this more in class.