Singularities and vanishing cycles in number theory over function fields
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RESEARCH
Singularities and vanishing cycles in number theory over function fields Will Sawin * Correspondence:
[email protected] Columbia University, New York, NY, USA
Abstract This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to.
1 Introduction This article concerns number theory over function fields—the study of classical problems in number theory, transposed to the function field setting. These proposed problems are then amenable to attack by topological, and more precisely cohomological, methods. The goal of this article is to give a broad, conceptual introduction to a specific set of topological techniques that have proven useful to recent works in this area. We begin with a brief review of the general study of function field number theory, though we recommend that anyone unfamiliar with it read another article on the subject (e.g., [5,10]) before tackling this one 1.1 Number theory over function fields
We start by expressing a problem in number theory as the problem of finding an approximate formula (or proving an already conjectured one) for the number of elements in a fixed set. Many problems in analytic number theory and arithmetic statistics have this form or can be reduced to it. For instance, a solution to the twin primes conjecture on the infinitude of twin primes would follow from a counting formula for the number of twin primes at most N , for an arbitrary integer N . If this problem seems intractable, we can consider a finite field analogue, which may be easier. We take the definition of the set we wish to count and modify it by replacing each occurrence of the integers in the definition with the ring Fq [T ] of polynomials over a finite field, but otherwise change it as little as possible. For instance, we could replace twin primes (numbers n such that n and n + 2 are prime) with polynomials f such that f and f + 2 are both irreducible polynomials. (More generally, if our problem involves the ring of integers of a number field, we can replace it with the ring of functions on an affine algebraic curve over Fq .)
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We can almost always recognize this new set as the set of points of an algebraic variety defined over Fq , i.e., the solutions in Fq of some finite list of polynomial equations, possibly after removing the solutions to another list of polynomial equations. The Grothendieck– Lefschetz fixed-point formula then gives an approximate formula for the number of points in this new set if we can calculate the (high-degree, compactly supported, étale) cohomology of this algebraic variety. If we do not know how to calculate this cohomology, we could transfer the problem to yet another setting. We can consider the analogous, pur
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