I run a reading course on arithmetic geometry at the Institute for Research in Fundamental Sciences (IPM). The aim of this reading course, geared towards graduate students, is to learn important topics -required for research- in arithmetic geometry, starting from basic algebraic number theory. Given the wide range of materials that one needs to know in this area, choosing what topics to read and in which order seemed to be a difficult task. Therefore, I decided to make a more concrete goal: learn, as much as possible, the proof of Fermat’s Last Theorem. This theorem with its epic history and whose proof was finally settled by marvelous work of Andrew Wiles in 1994 (and built on 358 years of efforts by great minds in mathematics) occupies a special place in most number theorist’s heart. Deep ideas and sophisticated techniques from algebraic geometry and number theory are employed in the proof and so, understanding it requires a good knowledge of these disciplines. I hope that, by moving towards this goal, many subjects will be naturally covered and we will all enjoy this journey.
The structure of this reading course is as follows: at the end of each class we specify some part of the reading material. During the coming week, we will read this part very carefully, write down any questions we have, come up with cool examples and ideas, solve exercises, etc. The next class, someone (who has been chosen before) will overview this part. As the presenter goes on, we ask our questions and discuss this part. Please note that the presenter does not have any duty other than skimming over the material (without giving details), and that everybody will participate in the discussions (if they wish so). It is therefore very important that participants prepare and read the material carefully, if they really want to benefit from the class. However, everybody is welcome to assist the course, even if they just want to enjoy the scenery superficially.
Dynamic information of the class, especially the part that we have to read each week, are posted on the News page.
The course takes place Wednesdays at IPM, Lecture Hall 1, from 10 to 12.
Reading material: The books that we are going to follow are:
- K. Kato, N. Kurokawa, T. Saito. Number Theory 1: Fermat’s Dream, AMS, 2000.
- K. Kato, N. Kurokawa, T. Saito. Number Theory 2: Introduction to Class Field Theory, AMS, 2011.
- N. Kurokawa, M. Kurihara, T. Saito. Number Theory 3: Iwasawa Theory and Modular Forms, AMS, 2012.
- T. Saito. Fermat’s Last Theorem: Basic Tools, AMS, 2013.
- T. Saito. Fermat’s Last Theorem: The Proof, AMS, 2014.
The first three books cover more elementary materials, whereas the last two books are dedicated to more advanced topics. Let me just mention some of the topics that will be covered in these books (not in any particular order):
Dedekind domains, p-adic numbers, local fields, elliptic curves over fields, Mordell’s theorem, zeta and L-functions, class number formula, prime number theorem, adeles and ideles, local and global class field theory, Brauer group, division algebras over local and global fields, cyclotomic fields, Iwasawa theory, modular forms, Eisenstein series, Hecke operators, automorphic forms, Poisson summation formula, Selberg trace formula, p-adic L-functions, Rankin-Selberg method, elliptic curves of schemes, Tate modules, finite flat groups schemes, Galois representations, deformation rings, modular curves, Igusa curves, Jacobians, Néron models, Galois cohomology, Selmer groups, the 3-5 trick, R=T.
In these books some topics are not covered in depth and some are completely missing. Therefore, as we go forward, we will digress from these books and study other books. Please see the What to Read page for a non-exhaustive related bibliography.
Prerequisites: I want this course to be accessible to most students, and therefore students are only required to have a good knowledge of commutative algebra (e.g. at the level of Atiyah-MacDonald’s book). More advanced topics require basic knowledge of algebraic geometry (scheme theory) and it is not our plan to cover it in this reading course. However, I will help students organize a parallel work group on algebraic geometry.