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Cohomological invariants of hermitian forms that detect hyperbolicity

Yong Hu, Alexandre Lourdeaux

Abstract

By using unramified cohomology groups, we construct a full sequence of cohomological invariants for hermitian forms of any type (orthogonal, symplectic or unitary) that can be used to detect hyperbolicity. The base central simple algebras can have arbitrary degree and the base field can have arbitrary characteristic. In the orthogonal case, we work with hermitian pairs, and we apply our construction to show that over fields of separable dimension 3, hermitian pairs over quaternion algebras with trivial classical invariants are hyperbolic. This last result extends a result of Berhuy to arbitrary characteristic.

Cohomological invariants of hermitian forms that detect hyperbolicity

Abstract

By using unramified cohomology groups, we construct a full sequence of cohomological invariants for hermitian forms of any type (orthogonal, symplectic or unitary) that can be used to detect hyperbolicity. The base central simple algebras can have arbitrary degree and the base field can have arbitrary characteristic. In the orthogonal case, we work with hermitian pairs, and we apply our construction to show that over fields of separable dimension 3, hermitian pairs over quaternion algebras with trivial classical invariants are hyperbolic. This last result extends a result of Berhuy to arbitrary characteristic.

Paper Structure

This paper contains 15 sections, 16 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Involutions, hermitian forms and hyperbolicity
  3. Involutorial algebras and hermitian forms
  4. Quadratic and hermitian pairs
  5. Generic index reduction and hyperbolicity theorems
  6. Unramified cohomology and invariants of quadratic forms
  7. Unramified cohomology for generic index reduction fields
  8. Cohomological invariants of quadratic forms
  9. Cohomological invariants in symplectic and unitary cases
  10. Symplectic case
  11. Unitary case
  12. Cohomological invariants of hermitian pairs
  13. Construction of invariants
  14. Link with classical invariants
  15. A hyperbolicity criterion in the quaternion case

Key Result

Proposition 2.7

Every CSAQP is a direct sum of an anisotropic one and an hyperbolic one. Also, if $\mathfrak{q}$ is an orthogonal sum of two central simple algebras with quadratic pair $\mathfrak{q}_1$ and $\mathfrak{q}_2$, and if $\mathfrak{q}$ and $\mathfrak{q}_1$ are hyperbolic, then $\mathfrak{q}_2$ is hyperbol

Theorems & Definitions (35)

  • Proposition 2.7: BerhuyFringsTignol
  • Definition 2.8
  • Remark 2.9
  • Definition 2.14: BlackQueguinerMathieu14
  • Theorem 2.15: Karpenko10DocMath, Karpenko12SciChina
  • Theorem 2.16
  • proof
  • Remark 2.17
  • Theorem 3.4
  • proof
  • ...and 25 more