Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

• Published in: Documenta mathematica Journal der Deutschen Mathematiker-Vereinigung. Vol. 14; pp. 259 - 296
• Main Author: Bergström, Jonas
• Format: Journal Article
• Language: English
• Published: 2009
• Subjects: MATEMATIK; MATHEMATICS; curves over finite fields; Cohomology of moduli spaces of curves
• ISSN: 1431-0635; 1431-0643; 1431-0643
• DOI: 10.4171/dm/273

We consider the moduli space \mathcal H_{g,n} of n -pointed smooth hyperelliptic curves of genus g . In order to get cohomological information we wish to make \mathbb S_n -equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfil. Thus, if we can make \mathbb S_n -equivariant counts of \mathcal H_{g,n} for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for \mathcal H_{g,n} for all g and for n \leq 7 . These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the \mathbb S_n -equivariant Galois (resp. Hodge) structure of their \ell -adic (resp. Betti) cohomology.