We discuss the Weil conjectures, especially the Riemann hypothesis, for varieties over finite fields.
Particular detail is devoted to the proof of the Riemann hypothesis for cubic threefolds in projective 4-space, as given by Bombieri and Swinnerton-Dyer.
The main idea is to relate the number of points on a cubic threefold to the trace of Frobenius on an associated abelian variety, and we develop the necessary machinery of abelian varieties.
Description:
We discuss the Weil conjectures, especially the Riemann hypothesis, for varieties over finite fields.
Particular detail is devoted to the proof of the Riemann hypothesis for cubic threefolds in projective 4-space, as given by Bombieri and Swinnerton-Dyer.
The main idea is to relate the number of points on a cubic threefold to the trace of Frobenius on an associated abelian variety, and we develop the necessary machinery of abelian varieties.