{"title":"On the holes in $I^n$ for symmetric bilinear forms in characteristic 2","authors":"Stephen Scully","doi":"arxiv-2409.02061","DOIUrl":null,"url":null,"abstract":"Let $F$ be a field. Following the resolution of Milnor's conjecture relating\nthe graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in\nthe theory of symmetric bilinear forms is to understand, for any positive\ninteger $n$, the low-dimensional part of $I^n(F)$, the $n$th power of the\nfundamental ideal in the Witt ring of $F$. In a 2004 paper, Karpenko used\nmethods from the theory of algebraic cycles to show that if $\\mathfrak{b}$ is a\nnon-zero anisotropic symmetric bilinear form of dimension $< 2^{n+1}$\nrepresenting an element of $I^n(F)$, then $\\mathfrak{b}$ has dimension $2^{n+1}\n- 2^i$ for some $1 \\leq i \\leq n$. When $i = n$, a classical result of Arason\nand Pfister says that $\\mathfrak{b}$ is similar to an $n$-fold Pfister form. At\nthe next level, it has been conjectured that if $n \\geq 2$ and $i= n-1$, then\n$\\mathfrak{b}$ is isometric to the tensor product of an $(n-2)$-fold Pfister\nform and a $6$-dimensional form of trivial discriminant. This has only been\nshown to be true, however, when $n = 2$, or when $n = 3$ and $\\mathrm{char}(F)\n\\neq 2$ (another result of Pfister). In the present article, we prove the\nconjecture for all values of $n$ in the case where $\\mathrm{char}(F) =2$. In\naddition, we give a short and elementary proof of Karpenko's theorem in the\ncharacteristic-2 case, rendering it free from the use of subtle\nalgebraic-geometric tools. Finally, we consider the question of whether\nadditional dimension gaps can appear among the anisotropic forms of dimension\n$\\geq 2^{n+1}$ representing an element of $I^n(F)$. When $\\mathrm{char}(F) \\neq\n2$, a result of Vishik asserts that there are no such gaps, but the situation\nseems to be less clear when $\\mathrm{char}(F) = 2$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"13 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02061","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $\mathrm{char}(F) = 2$.