Math 693a: Topics in Number Theory

Stephen D. Miller

TuTh 1-2:15 pm
206 LOM (12 Hillhouse)

Starts Tuesday September 12th.

Text: A Classical Introduction to Modern Number Theory, Ireland and Rosen, Springer Graduate Texts in Mathematics, volume 84. CLICK HERE FOR A PRICE-LIST. Make sure you order the most-recent edition! (circa 1990+). I am told that the Yale bookstore/COOP will have the text in a few weeks, but it will probably not be so necessary until then.

Summary: I want to give an overview of number theory, in particular some the recent accomplishments and conjectures.  Since these involve applications of many different areas of mathematics, I will sacrifice rigor for breadth.  In the beginning of the course I will give a fairly-solid background of classical number theory, presented with an analytic viewpoint.  After studying diophantine equations, we will then turn to elliptic curves and discuss modular forms.  At the end of the course, I will explain the Modularity Theorem and sketch its applications.

This is the first semester of a two-semester course.  The second semester will be on topics to be determined later; it will be more technical than this course and will probably concentrate mostly on the analytic theory of modular forms.  Potential topics include:


  Part of Ireland and Rosen 

The Riemann Zeta Function, Prime Number Theorem

Ch. 1 & 2

The structure Z/nZ and its multiplicative group; an analytic proof that every prime of the form 4n+1 is the sum of two squares

Ch. 3 & 4, parts of 8

Kronecker-Jacobi-Legendre symbols; Quadratic Reciprocity

Ch. 5 & 6

Fermat curve mod p

Ch. 8 & 10

Introduction to Elliptic Curves

Ch. 10 & 18

Zeta Functions for Finite Fields; Weil's work

Ch. 11

Algebraic Number Theory, esp. Quadratic Fields

Ch. 12 & 13

Diophantine Equations

Ch. 17

Dirichlet L-functions; Proof that there are infinitely many primes in most arithmetic progressions

Ch. 16

Elliptic Curves and the Modularity Conjecture (Wiles-Taylor-Conrad-Diamond-Breuil Theorem).  Summary of major recent discoveries.

Ch. 18-20

 Stephen Miller, Aug. 31, 2000.