Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves

TL;DR

This paper explores the relationship between strong approximation properties of spin groups and the Hasse principle for integral quadratic forms over affine curves, extending classical representation theory to new algebraic settings.

Contribution

It extends representation theory for integral quadratic forms to coordinate rings of affine curves and links strong approximation to the Hasse principle using genus theory.

Findings

Identifies cases where spin groups do not satisfy strong approximation.

Establishes a connection between strong approximation and the Hasse principle.

Provides applications and examples in the context of affine curves.

Abstract

We extend some parts of the representation theory for integral quadratic forms over the ring of integers of a number field to the case over the coordinate ring $k[C]$ of an affine curve $C$ over a general base field $k$. By using the genus theory, we link the strong approximation property of certain spin groups to the Hasse principle for representations of integral quadratic forms over $k[C]$ and derive several applications. In particular, we give an example where a spin group does not satisfy strong approximation.