Introduction to arithmetic geometry

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:

Each topic represents 1-2 weeks of lectures

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.

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.

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

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

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

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.

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.

Don't show me this again

This is one of over 2,200 courses on OCW. Find materials for this course in the pages linked along the left.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)

Learn more at Get Started with MIT OpenCourseWare