On the holes in $I^n$ for symmetric bilinear forms in characteristic 2

Abstract

Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer $n$, the low-dimensional part of $I^n(F)$, the $n$th power of the fundamental ideal in the Witt ring of $F$. In a 2004 paper, Karpenko used methods from the theory of algebraic cycles to show that if $\mathfrak{b}$ is a non-zero anisotropic symmetric bilinear form of dimension $< 2^{n+1}$ representing an element of $I^n(F)$, then $\mathfrak{b}$ has dimension $2^{n+1} - 2^i$ for some $1 \leq i \leq n$. When $i = n$, a classical result of Arason and Pfister says that $\mathfrak{b}$ is similar to an $n$-fold Pfister form. At the next level, it has been conjectured that if $n \geq 2$ and $i= n-1$, then $\mathfrak{b}$ is isometric to the tensor product of an $(n-2)$-fold Pfister form and a $6$-dimensional form of trivial discriminant. This has only been shown to be true, however, when $n = 2$, or when $n = 3$ and $\mathrm{char}(F) \neq 2$ (another result of Pfister). In the present article, we prove the conjecture for all values of $n$ in the case where $\mathrm{char}(F) =2$. In addition, we give a short and elementary proof of Karpenko's theorem in the characteristic-2 case, rendering it free from the use of subtle algebraic-geometric tools. Finally, we consider the question of whether additional dimension gaps can appear among the anisotropic forms of dimension $\geq 2^{n+1}$ representing an element of $I^n(F)$. When $\mathrm{char}(F) \neq 2$, a result of Vishik asserts that there are no such gaps, but the situation seems to be less clear when $\mathrm{char}(F) = 2$.