# Lei Fu

## Course Description

In his works on the quadratic reciprocity law, Gauss introduced the so-called Gauss sums. In order to compute these sums, Gauss was lead to evaluate the number of solutions of congruences of the following types:

\(ax^3 - by^3 \equiv 1 \quad mod \quad p, \) \(\quad ax^4 - by^4 \equiv 1 \quad mod \quad p, \) \(\quad y^2 \equiv ax^4 - b \quad mod \quad p. \)

Weil studied this problem for the more general equation

\[ a_0x_0^{k_0} + \cdots + a_rx_r^{k_r} \equiv b \quad mod \quad p. \quad\quad (1) \]

The number of solutions is encoded in the ζ-function for the algebraic variety defined by this equation. Weil conjectured that the ζ-function of a smooth projective variety could be expressed in terms of the actions of the Frobenius correspondence on the cohomology groups of the algebraic variety, and the ζ-function should satisfy the Riemann hypothesis. These properties of the ζ- function give rise to the optimal estimate for number of rational points of the algebraic variety. The Weil conjecture was later proved by Grothendieck and Deligne using the ‘-adic cohomology theory. In this course, we will express the ζ-function of (1) in terms of Gauss sums, and verify the Weil conjecture directly.