Analytic number theory for 0-cycles

TL;DR

This paper extends the analogy between integers and polynomials to effective 0-cycles on general varieties over finite fields, analyzing their prime factorization and demonstrating Poisson distribution of prime factors.

Contribution

It generalizes the prime factorization analysis from polynomials to 0-cycles on arbitrary varieties, revealing their prime factors follow a Poisson distribution.

Findings

Prime factors of 0-cycles are typically Poisson distributed.

Extends the integer-polynomial analogy to more general algebraic varieties.

Abstract

There is a well-known analogy between integers and polynomials over $F_q$, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorization of effective 0-cycles on an arbitrary connected variety $V$ over $F_q$, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles are typically Poisson distributed.