Exponential sums over finite fields

October - December 2010.  Dr. Tim Browning.

Syllabus

1. Introduction
2. Finite fields
[Chevalley-Warning theorem]
3. Characters of finite abelian groups
4. Gauss and Jacobi sums
[Gauss sums; Jacobi sums; Salié sums]
5. Varieties over finite fields
[counting points on hypersurfaces; Hasse-Davenport relation; zeta function of a hypersurface]
6. Riemann hypothesis for sums in one variables
[short character sums; Stepanov's method]

Course Notes

• Exponential sums over finite fields

Course Assessment

The course will be assessed by problems, with one given at the end of each section. Solutions should be returned to me at the end of the term at Tim Browning, School of Mathematics, University of Bristol, Bristol, BS8 1TW

Reading

• W. Schmidt, Equations over finite fields, SLNM 536, 1974.
• K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., Springer 1990.
• H. Iwaniec and E. Kowalski, Analytic number theory, AMS 2004.