Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)

TL;DR

This paper studies the topology, arithmetic, and point counts of the moduli space of polynomial pairs with resultant 1, revealing conditions for purity and Tate type cohomology, and connecting to monopole moduli spaces.

Contribution

It computes the étale cohomology, Frobenius eigenvalues, and point counts of the moduli space of polynomial pairs with resultant 1, linking algebraic geometry, arithmetic, and monopole moduli spaces.

Findings

Étale cohomology is pure when q and n are coprime.

Cohomology is of Tate type iff q ≡ 1 mod n.

Explicit point counts for the moduli spaces and their finite field counterparts.

Abstract

We consider the moduli space $\mathcal{R}_n$ 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 \'{e}tale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the \'etale cohomology of $\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$ monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space $X_n$ of strongly centered monopoles. An appendix by Cazanave gives an alternative and elementary computation of the point counts.