Number Theory

First lecture: Wednesday, September 22, 2010

Contents

We will take the book "Rational Points on Elliptic Curves" by Silverman and Tate as a guideline with the goal of reaching the last chapter on Complex mutlitplication. In particular we will give an explicit construction of an abelian extention of an imaginary quadratic field using division points of an elliptic curve. Along the way we will study elliptic and modular functions.

Prerequisites

Complex Analysis (Funktionentheorie).

Exercises

First exercise session: Wednesday, September 29, 2010

Requirements for the certificate (Testat): Reasonable attempt to solve 70% of all tasks.

Organisation: Exercises will be available on this page (approximately one exercise every second week).
You may hand in your written solutions at the beginning of the exercise classes (resp. in class) on the due date given above.
In general, you will have your marked exercises returned at the beginning of the subsequent exercise class.
If there are any questions and/or corrections please send an email to jjermann@math.ethz.ch.

Literature

• Joseph H. Silverman and John Tate, "Rational Points on Elliptic Curves"
• K. Chandrasekharan, "Elliptic Functions"
• Serge Lang, "Elliptic Functions"
• Neal I. Koblitz, "Introduction to Elliptic Curves and Modular Forms" (this book also has a lot on the congruent number problem)
• Lawrence C. Washington, "Elliptic Curves: Number Theory and Cryptography"
• Joseph H. Silverman, "The Arithmetic of Elliptic Curves" (more advanced)
• Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves" (more advanced)