Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


18.702 Algebra II


Arithmetic geometry lies at the intersection of algebraic geometry and number theory. Its primary motivation is the study of classical Diophantine problems from the modern perspective of algebraic geometry. Topics include:

  1. Rational points on conics
  2. p-adic numbers
  3. Quadratic forms
  4. Affine and projective varieties
  5. Curves and function fields
  6. Divisors on curves
  7. The Riemann-Roch theorem
  8. Elliptic curves and abelian varieties
  9. Rational points on elliptic curves

Each topic represents 1-2 weeks of lectures

Textbook and Notes

There is no required text; lecture notes will be provided.

We may make reference to material in the following books and online resources

Fulton, William. Algebraic Curves: An Introduction to Algebraic Geometry.
This book is available for free on Fulton's website.

Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishers, 2006. ISBN: 9781419652578.
This book is also available for free on Milne's website, along with addendum/erratum.

Buy at Amazon Serre, Jean-Pierre. A Course in Arithmetic. Springer-Verlag, 1996. ISBN: 9783540900405.

Buy at Amazon Shafarevich, I. R. (translated by Miles Reid). Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon Stichtenoth, H. Algebraic Function Fields and Codes. Berlin: Springer, 2008. ISBN 9783540768777. [Preview with Google Books]

Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (errata) [Preview with Google Books]


Some of the theorems presented in lecture will be demonstrated using the Sage computer algebra system, which is based on Python™. Sage is an open-source system that provides both a command-line interface and a browser-based GUI (the Sage notebook). Tutorials and many examples can be found online. You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the SageMathCloud™.

Enrolled students will be given an account on a Sage notebook server set up for this course. But you are free to use other packages such as Magma, Maple™ or Mathematica™ if you wish.

Problem Sets

There will be weekly problem sets. Problem sets are to be prepared in typeset form (typically via LaTeX) and submitted electronically as PDF files. Collaboration is permitted/encouraged, but you should first attempt to solve the problems on your own, and in any case, you must write up your own solutions. Any collaborators should be identified, as well as any resources you consulted that are not listed above.


Your grade will be determined by your average problem set score, after dropping your lowest score. There are no exams and no final.