This course will be an introduction to elliptic curves and modular forms, with an emphasis on examples. We will begin with some motivating problems, such as the congruent number problem, and the definitions, and then explain how a link between elliptic curves and modular forms is suggested through L-functions. Students will learn how to manipulate elliptic curves, modular forms and L-functions to extract interesting arithmetic information such as rational points, the rank, or congruences (using the free software SAGE). We will discuss some of the big theorems and conjectures - such as the Mordell-Weil theorem and Birch and Swinnerton-Dyer conjecture -, and their consequences. For example, we will sketch how the modularity of elliptic curves is used to prove Fermat's Last Theorem.

The prerequisites for this course are elementary number theory, linear algebra and group theory.

A List of Topics you should be familiar with before PCMI:

• Elementary Number Theory: modular arithmetic from the basic properties of congruences, Fermat's little theorem, Euler's theorem, up to quadratic reciprocity, familiarity with the fact that Z/pZ, for a prime p, is a field.
  
• Linear Algebra: vector spaces (concept of dimension), linear maps and linear isomorphisms, multiplication and other operations with matrices, determinants, the definitions of GL(2,R) and SL(2,R), where R is the field of real numbers.
  
• Group Theory and/or Abstract Algebra: groups, subgroups, group homomorphisms and isomorphisms, familiarity with the structure theorem for finite abelian groups, concepts of field and ring.
  

Additional topics that will be used (but we will introduce in the lecture or through exercises): the projective plane (projective coordinates, affine charts, homogeneous equations); a little bit of complex analysis (complex numbers, the complex plane, the complex exponential function); basics of algebraic geometry; convergence of infinite series, Taylor series.

REFERENCES: I will be following my own set of notes during the course, but there are a number of excellent references on the subject:

• Rational Points on Elliptic Curves, by J. Silverman and J. Tate: This is an introduction to the theory of elliptic curves, for undergraduates (and it will be our main reference);
  
• Introduction to Elliptic Curves and Modular Forms, by N. Koblitz: This is a book for graduate students, but it contains a very good introduction to modular forms;
• Elliptic Curves: Number Theory and Cryptography, Second Edition, by L. Washington: Another introductory text, for undergraduates;
  
• The Arithmetic of Elliptic Curves, by J. Silverman: The best graduate text on elliptic curves;
  
• Elliptic Curves, by J. Milne (see Milne's website for an -old- electronic copy);
• Modular Forms, a computational approach, by W. Stein: A very nice introduction to modular forms, and computations using SAGE;
• A Course in Arithmetic, by J-P. Serre: Chapter VII is a very nice introduction to modular forms.