2024/09/03 by Stephen Scully, Scully, Stephen
Mathematics · #11E04 #11E39 #11E81 #13N05 #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.02061
openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 In(F), the nth 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 \mathfrakb is a non-zero anisotropic symmetric bilinear form of dimension < 2n+1 representing an element of In(F), then \mathfrakb has dimension 2n+1 - 2i for some 1 ≤ i ≤ n. When i = n, a classical result of Arason and Pfister says that \mathfrakb is similar to an n-fold Pfister form. At the next level, it has been conjectured that if n ≥ 2 and i= n-1, then \mathfrakb 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 char(F) ≠ 2 (another result of Pfister). In the present article, we prove the conjecture for all values of n in the case where 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 ≥ 2n+1 representing an element of In(F). When char(F) ≠ 2, a result of Vishik asserts that there are no such gaps, but the situation seems to be less clear when char(F) = 2.