On rational quadratic cocycles

TL;DR

This paper constructs cohomology classes for arithmetic groups related to quadratic spaces and shows their generating series are Siegel modular forms in specific cases, revealing deep connections between geometry and modular forms.

Contribution

It introduces a new method to associate cohomology classes to quadratic spaces and demonstrates their generating series form Siegel modular forms in extremal cases.

Findings

Constructed classes in cohomology of arithmetic groups for quadratic spaces.

Generated series are Siegel modular forms of genus k.

Results connect geometric cycles with automorphic forms.

Abstract

Let $(V,q)$ be a non-degenerate $n$-dimensional quadratic space over the rationals of real signature $(r,s)$. For every integer $1\leq k \leq \min\{r,n-2\}$ we construct classes in the cohomology of arithmetic subgroups of $\mathrm{O}(V)$ with values in the group of codimension $k$ cycles on the quadric of isotropic lines in $V$. Generating series of images of these classes in an equivariant version of the $k$-th Chow group are shown to be Siegel modular forms of genus $k$ in the extremal cases $k=1$ and $k=r$. Soit $(V,q)$ un espace quadratique non d\'eg\'en\'er\'e de dimension $n$ sur les rationnels, de signature r\'eelle $(r,s)$. Pour tout entier $1 \leq k \leq \min\{r,n-2\}$, nous construisons des classes dans la cohomologie des sous-groupes arithm\'etiques de $\mathrm{O}(V)$ \`a valeurs dans le groupe des cycles de codimension $k$ sur la quadrique des droites isotropes dans $V$. Les s\'eries g\'en\'eratrices des images de ces classes dans une version \'equivariante du $k$-i\`eme groupe de Chow sont des formes modulaires de Siegel de genre $k$ dans les cas extr\'emaux $k=1$ et $k=r$.