Topology and arithmetic of resultants , II : the resultant = 1 hypersurface Benson Farb and Jesse Wolfson ∗ With an appendix by Christophe Cazanave

Topology and arithmetic of resultants , II : the resultant = 1 hypersurface Benson Farb and Jesse Wolfson

We consider the moduli space Rn of pairs of monic, degree n polynomials whose resultant equals 1. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their étale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of Fq-points. When q and n are coprime, we show that the étale cohomology of Rn/Fq is pur...

Topology and arithmetic of resultants, I

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in P. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen–Cohen–Mann–Milgr...

Étale homological stability and arithmetic statistics

We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over C in the example of configuration spaces of n points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory of étale cohomology; in partic...

Coincidences between homological densities, predicted by arithmetic

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil’s number field/ function field analogy to predict coincidences for limiting homological densities of various sequences Z1m n pXq of spaces of 0-cycles on manifolds X. The main theorem in this paper is that these topo...

Concise Algebraic Topology localization , completion , and model categories