PCMI 2009, Undergraduate Summer School Program

• Dirichlet L-functions, Generalizations, and Applications: 
  Keith Conrad, University of Connecticut. See the website for his course.
  
  
• Elliptic Curves, Modular Forms, and L-functions:  
  Álvaro Lozano-Robledo, University of Connecticut
  
  Description:
  

  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.

• About the instructor.
  
• Basics: Prerequisites, Textbooks and references.
  
• Schedule: Here you will find a tentative schedule for the course. The pace may vary.
  
• Assignments: Assignments will be posted here.