MATH 596: Topics in Algebra and Number Theory
This is the webpage for MATH 596, taught at McGill in the winter of 2017.
Room: Burnside 1214
Time: MWF, 9.30 - 10.30am
We start with Fermat's classical result about when a prime number is the sum of two squares, and develop the theory of quadratic forms motivated by possible generalisations of Fermat's criterion. We treat both definite and indefinite forms, and develop the classical theory of reduction, genera, and Gauss composition, taking all the while a 'visual' approach through Conway's topograph and Bhargava cubes. Finally, we formulate an inexplicit criterion for when a prime number is of the form x^2 + ny^2.
We ended the previous chapter with a resolution of the question of when a prime p is of the form x^2 + ny^2, and now turn to the question of how to check our criterion in practice. After a short review of class field theory, we develop the theory of complex multiplication and show how it it may be used to make the class field theory of imaginary quadratic fields explicit. By investigating the j-function and the Weber functions, we give a proof of Gauss' class number one problem following Heegner.
We discuss Serre's reformulation of the class number one problem in geometric terms. This motivates us to look for general techniques for finding all rational points on modular curves. We review some classical general methods such as descent via isogeny and the p-adic method of Chabauty-Coleman, and work out lots of explicit examples. Finally, we will look at the arguments of Mazur-Tate that show elliptic curves over Q cannot have rational 13-torsion, and give some indications how certain arguments forshadow parts of the proof of Mazur's Torsion theorem.
Preliminary notes. These will be updated as we go along.
Primes of the form x^2 + ny^2 (Cox)
The sensual (quadratic) form (Conway)
The topology of numbers (Hatcher)
Rational quadratic forms (Cassels)
Introduction to the arithmetic theory of automorphic forms (Shimura).
Advanced topics in the arithmetic of elliptic curves (Silverman).
Rational points on modular curves
Selmer groups, Tate-Shafarevich groups, and the weak Mordell--Weil theorem (Poonen).
The method of Chabauty and Coleman (MacCallum, Poonen)
Points of order 13 on elliptic curves (Mazur, Tate)