Analytic number theory for 0-cycles

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.

19 pages, v3: paper reorganized, introduction rewritten, and a main theorem added

References in corpus (2)

• Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
• Polynomial splitting measures and cohomology of the pure braid group

Cited by in corpus (2)

• Counting points on ${\text{Hilb}}^m{\mathbb{P}}^2$ over function fields
• Generalizations of Alladi's formula for arithmetical semigroups