Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms

TL;DR

This paper extends fundamental theorems on isotropy of quasilinear quadratic forms over function fields in characteristic 2, introducing new invariants and unifying previous results.

Contribution

It establishes new constraints on isotropy indices using stable birational invariants, including a novel invariant, extending and unifying prior key results.

Findings

Unified and extended quasilinear analogues of key theorems by Karpenko and Merkurjev.

Introduced a new invariant elta(p) for quasilinear forms.

Proved strong constraints on isotropy indices in terms of dimension and invariants.

Abstract

Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$ of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar extension to the function field of the affine quadric with equation $p=0$. In this article, we establish a strong constraint on $i$ in terms of the dimension of $q$ and two stable birational invariants of $p$, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote $\Delta(p)$. Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.