Hermitian u-invariants over function fields of p-adic curves

TL;DR

This paper computes the exact u-invariants of hermitian forms over central simple algebras with involution over function fields of p-adic curves, advancing understanding of quadratic form invariants in this setting.

Contribution

It provides explicit calculations of the u-invariant for hermitian forms over specific algebras, refining previous bounds and contributing to algebraic and arithmetic theory.

Findings

Exact u-invariant values for hermitian forms over (A, σ)

Refinement of known upper bounds for u-invariants

Enhanced understanding of hermitian forms over function fields of p-adic curves

Abstract

Let $p$ be an odd prime. Let $F$ be the function field of a $p$-adic curve. Let $A$ be a central simple algebra of period 2 over $F$ with an involution $\sigma$. There are known upper bounds for the $u$-invariant of hermitian forms over $(A, \sigma)$. In this article we compute the exact values of the $u$-invariant of hermitian forms over $(A, \sigma)$.