Function Fields · Analytic Number Theory and Arithmetic Statistics over Function Fields

An archetypal problem in multiplicative number theory is to determine the factorization statistics in a given set, such as the set of values of an integral polynomial, or of the function raising an integer to a non-integral power and taking the integral part, the set of integers n for which there exists an integer 0 < a < n/10 such that n divides a^3-2, or the set of discriminants of cubic extensions of a number field. One is particularly interested in estimating the number of primes in such sets. We study analogs of such problems over function fields (in one variable over a finite field). Almost every problem over number fields admits a sensible (although not necessarily obvious) analog over function fields, and solutions to such problems carry over as well. On the other hand, the Riemann Hypothesis has been resolved in a most definitive form by Deligne in the function field setting. While solutions of problems over function fields do not translate to solutions of analogous problems over number fields, new insights are gained, and connections to geometry, topology, and homological stability emerge. Function field analytic number theory problems often reduce to obtaining cancellation in sums of trace functions of l-adic etale sheaves over the points of a variety over a finite field. The Grothendieck--Lefschetz trace formula, in conjunction with Delignes theorem, gives us cancellation once strong upper bounds on the dimensions of the cohomology groups of our sheaves are available. The main proposed innovation is a bound on the dimensions of cohomology groups of sheaves built using the six operations from more basic sheaves, approached using Massey's bound involving the characteristic cycle. Our methods involve also judicious choices of l for which the reduction of our sheaves mod l simplifies them. These will be combined with the circle method to serve as an off-the-shelf approach to function field problems.