Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves

Abstract

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the disc with $n$ punctures. The group homology $H_{\ast }(B_n;V_n)$ of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over ${\mathbb{F}}_q$ is exactly $q$.

18 pages. v3: minor changes

References in corpus (3)

• Homological stability for automorphism groups
• Topology and arithmetic of resultants, I: spaces of rational maps
• Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)

Cited by in corpus (2)

• Homological stability for automorphism groups
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups