On a theorem of Ax and Katz

TL;DR

This paper demonstrates that the Adolphson-Sperber p-divisibility bound for rational points on algebraic varieties over finite fields can be achieved at special fibers for infinitely many primes, extending the original Ax-Katz theorem.

Contribution

It proves that the Adolphson-Sperber bound can be realized at special fibers for generic varieties over number fields, with conditions for achieving the bound at all but finitely many primes.

Findings

The bound is achieved for a set of primes of positive density.

The bound is achieved for all but finitely many primes under certain conditions.

The results extend the Ax-Katz theorem to more general algebraic varieties.

Abstract

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. It was strengthened by Adolphson-Sperber in terms of Newton polytope of the support set G of V. In this paper we prove that for every generic algebraic variety over a number field supported on G the Adolphson-Sperber bound can be achieved on special fibre at p for a set of prime p of positive density in SpecZ. Moreover we show that if G has certain combinatorial conditional number nonzero then the above bound is achieved at special fiber at p for all but finitely many primes p.