Local-global principles for hermitian spaces over semi-global fields

TL;DR

This paper proves a local-global principle for rational points on projective homogeneous spaces under certain algebraic groups over semi-global fields, extending classical results to more complex hermitian and involution settings.

Contribution

It establishes a local-global principle for projective homogeneous spaces associated with hermitian forms over semi-global fields, considering involutions of any kind and specific algebraic conditions.

Findings

Projective homogeneous spaces satisfy local-global principles over semi-global fields.

The results apply to algebras with involutions of both first and second kind.

The work extends classical local-global principles to hermitian spaces with involutions.

Abstract

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A \in {}_2Br(F)$ be a central simple algebra with an involution $\sigma$ of any kind and $F_0 =F^{\sigma}$. Let $h$ be an hermitian space over $(A, \sigma)$ and $G = SU(A, \sigma, h)$ if $\sigma$ is of first kind and $G = U(A, \sigma, h)$ if $\sigma$ is of second kind. Suppose that $\text{char}(k) \neq 2$ and ind$(A)\leq 4$. Then we prove that projective homogeneous spaces under $G$ over $F_0$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$.