## MAT6630, Elliptic curves and Modular Forms, Fall Semester 2011

### (Courbes elliptiques et formes modulaire)

Prof: Andrew Granville
Bureau: 6153 André Aisenstadt, Tel: 343-6583; Courriel: andrew@dms.umontreal.ca

Course Book: The Arithmetic of Elliptic Curves by Joe Silverman is the standard reference in the subject and should be owned by every number theorist. It takes an approach that is well in-line with modern arithmetic geometry, which is not exactly how this course will proceed. You will find this available in the U de M bookstore

Alternatives: Rational Points on Elliptic Curves by Joe Silverman and John Tate is more elementary but covers many of the key introductory issues from as elementary a viewpoint as possible. We will use the approaches from this book in much of the first half of the semester.

Introduction to Elliptic Curves and Modular Forms by Neal Koblitz was an early book on precisely the subject of this course. It is a beautiful coherent account of many ideas leading up to some extraordinary developments with modular forms.

I am providing some notes entitled Rational and Integral points on curves.

Classtimes: Monday 10h30-12h30; Wednesday 10h30-12h30, in Andre-Aisenstadt 5183.
Class dates: From Sept 7 to Dec 7, 2011

En Francais, et page web complet

## Course Contents/Syllabus

We will begin by reviewing what is known about linear and quadratic Diophantine equations, highlighting the remaining big mysteries about quadratic equations. In particular we will look at solutions in the rationals, in the integers, and in finite fields, and the relations between these. Only then will we begin to study cubic and higher degree equations. First finding that there is a convenient simplification of the underlying equations, and then finding that there are analogies to many of the beautiful structures that are well-known for quadratic equations. We will aim to cover the following topics

• Linear equations in linear algebra and number theory
• Quadratic equations over the rationals. The geometry of Pythagoras' Theorem
• Quadratic equations over the integers. Descent and approximations
• Quadratic equations in finite fields, and the definition of Dirichlet L-functions
• The general cubic equation, the Weierstrass model and reduction mod p
• Poincare's group law -- geometry, algebra and geometry again
• Points of finite order --- the Nagell-Lutz theorem
• Mordell's Theorem and Weil's proof: Heights and descent, and not a proof of Fermat's Last Theorem
• Cubic curves over finite fields
• Integer points on cubic curves and applications. An introduction to Diophantine appoximation and Siegel's Theorem
• Lattices and Weierstrass parametrization
• Endomorphism structure and Complex multiplication
• Constructing L-functions for elliptic curves
• Questions of torsion and rank. The Birch-Swinnerton-Dyer conjecture.
• The construction of modular functions and forms
• The j-function and singular moduli
• The modularity of y^2=x^3-x
• A rough statement of the Taniyama conjecture.
• Congruent numbers and the coefficients of modular forms