A bound for the index of a quadratic form after scalar extension to the\n function field of a quadric

2016/07/25 by Stephen Scully, Scully, Stephen · 1 citation
Mathematics · #11E04 #14E05 #15A03 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1607.07529

openalex publication_date 2016/07/25 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let q be an anisotropic quadratic form defined over a general field F. In\nthis article, we formulate a new upper bound for the isotropy index of q\nafter scalar extension to the function field of an arbitrary quadric. On the\none hand, this bound offers a refinement of a celebrated bound established in\nearlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct\ngeneralization of Karpenko's theorem on the possible values of the first higher\nisotropy index. We prove its validity in two important cases: (i) the case\nwhere \char(F) \≠ 2, and (ii) the case where \char(F) = 2\nand q is quasilinear (i.e., diagonalizable). The two cases are treated\nseparately using completely different approaches, the first being\nalgebraic-geometric, and the second being purely algebraic.\n

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