Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms

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.