ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11G25, 14G15, 14F20, 14M10 Ghorpade, Sudhir R., Indian Institute of Technology, Bombay, India and Lachaud, Gilles, CNRS Institut de Mathématiques de Luminy, Marseille, France Number of Points of Singular Varieties over Finite Fields Let X be a projective algebraic subvariety of ℙ^{N} of dimension n defined over the finite field 𝔽_{q} of q elements. In case X is a nonsingular complete intersection, we have a sharp estimate for |X(𝔽_{q})| due to Deligne (1981). This was extended to the singular case by Hooley and Katz (1991). We obtain a refinement of the latter by proving a precise inequality for ||X(𝔽_{q})| − π_{n}|, where π_{n} denotes |ℙ^{n}(𝔽_{q})| = q^{n} + q^{n−1} + ⋯ + 1. The proof is primarily based on a suitable generalization to singular varieties of the Weak Lefshetz Theorem together with some Bertini-type arguments and the Grothendieck-Lefshetz Trace Formula. In the process, we also obtain a sharper version of the classical Lang-Weil inequality.