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 Zp. 

• September 19: Completeness of Qp, solutions to polynomial equations, and Hensel's lemma.

• September 21: Squares in Qp, 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 Qp, 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.