Brief description:  Exponential sums are sums of complex numbers of absolute value one. They appear in many branches of Mathematics. We will be focusing on Exponential sums in Number Theory and, in particular, on rational exponential sums which are sums of roots of unity. We will discuss how they are used and the basic questions about exponential sums, for instance, estimates for their absolute values (both complex and p-adic).
 

Topics to be covered will include:

• Fourier analysis on finite abelian groups.
• Gauss sums and Jacobi sums. Hasse-Davenport formula. Evaluation of Gauss sums. Stickelberger formula.
• Exponential sums, L-series and the Riemann hypothesis for curves.
• Incomplete exponential sums.
• p-adic properties of sums of roots of unity.

Prerequisite:  Graduate Algebra
 

Textbook:  None.    
 

Course notes:  Each student will be required to take notes for a week and TeX them. The whole set of notes is now available here

Links

• Lecture notes by P. Roquette.
• Lecture notes by I. Shparlinski.