The classification of free algebras of orthogonal modular forms

We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free. From the necessary condition, we derive a classification result. Let $M$ be an even lattice of signature $(2,n)$ splitting two hyperbolic planes. Suppose $\Gamma$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\Gamma$ such that the space of modular forms for $\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.

Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes

We study the homology of Riemannian manifolds of finite volume that are covered by an r-fold product $$({\mathbb {H}}^2)^r = {\mathbb {H}}^2 \times \cdots \times {\mathbb {H}}^2$$ ( H 2 ) r = H 2 × ⋯ × H 2 of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-Phan, we show that any such manifold M possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic r-dimensional submanifolds whose fundamental classes are linearly independent in the homology group $$H_r(M;{\mathbb {Z}})$$ H r ( M ; Z ) .

Zeros of pairs of quadratic forms

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in eight variables. The argument develops work of Colliot-Thélène, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off three hyperbolic planes.

Quadratic Forms

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.