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$q$-bic forms

Raymond Cheng

TL;DR

The paper develops an intrinsic geometric theory of $q$-bic forms, introducing two intrinsic filtrations arising from Frobenius-twisted orthogonals and extracting numerical invariants that classify forms by type $(a;b_m)_{m\ge1}$. It proves a Classification Theorem: after a suitable Frobenius twist, a $q$-bic form decomposes into a nonsingular part of dimension $a$ plus a direct sum of blocks $\mathbf{N}_m^{\oplus b_m}$, yielding a normal form over algebraically closed fields with Gram matrix $\mathbf{1}^{\oplus a}\oplus\bigoplus_m\mathbf{N}_m^{\oplus b_m}$. The work analyzes automorphism group schemes, giving dimension formulas in terms of the type and revealing nonreduced structures tied to the canonical filtrations; it also develops a moduli space stratified by type and studies specialization relations between strata. Together, these results provide deformation- and moduli-theoretic tools for understanding $q$-bic hypersurfaces and related positive characteristic phenomena. The framework connects to Hermitian forms, unitary groups, and related geometric structures, enabling tractable computations of isomorphism classes and automorphism groups in a broad arithmetic setting.

Abstract

A $q$-bic form is a pairing $V \times V \to \mathbf{k}$ that is linear in the second variable and $q$-power Frobenius linear in the first; here, $V$ is a vector space over a field $\mathbf{k}$ containing the finite field on $q^2$ elements. This article develops a geometric theory of $q$-bic forms in the spirit of that of bilinear forms. I find two filtrations intrinsically attached to a $q$-bic form, with which I define a series of numerical invariants. These are used to classify, study automorphism group schemes of, and describe specialization relations in the parameter space of $q$-bic forms.

$q$-bic forms

TL;DR

The paper develops an intrinsic geometric theory of -bic forms, introducing two intrinsic filtrations arising from Frobenius-twisted orthogonals and extracting numerical invariants that classify forms by type . It proves a Classification Theorem: after a suitable Frobenius twist, a -bic form decomposes into a nonsingular part of dimension plus a direct sum of blocks , yielding a normal form over algebraically closed fields with Gram matrix . The work analyzes automorphism group schemes, giving dimension formulas in terms of the type and revealing nonreduced structures tied to the canonical filtrations; it also develops a moduli space stratified by type and studies specialization relations between strata. Together, these results provide deformation- and moduli-theoretic tools for understanding -bic hypersurfaces and related positive characteristic phenomena. The framework connects to Hermitian forms, unitary groups, and related geometric structures, enabling tractable computations of isomorphism classes and automorphism groups in a broad arithmetic setting.

Abstract

A -bic form is a pairing that is linear in the second variable and -power Frobenius linear in the first; here, is a vector space over a field containing the finite field on elements. This article develops a geometric theory of -bic forms in the spirit of that of bilinear forms. I find two filtrations intrinsically attached to a -bic form, with which I define a series of numerical invariants. These are used to classify, study automorphism group schemes of, and describe specialization relations in the parameter space of -bic forms.
Paper Structure (30 sections, 36 theorems, 156 equations, 2 figures)

This paper contains 30 sections, 36 theorems, 156 equations, 2 figures.

Table of Contents

  1. Basic notions
  2. Definitions
  3. Gram matrices
  4. Standard forms
  5. Orthogonals
  6. Frobenius twists and orthogonals
  7. Total orthogonals and complements
  8. Hermitian forms
  9. Hermitian elements
  10. Scheme of Hermitan vectors
  11. Examples
  12. Canonical filtrations
  13. $\perp$-filtration
  14. $\perp^{[ \mathord{ \vcenter{\hbox{$\m@th\bullet$}}}]}$-filtration
  15. Example
  16. ...and 15 more sections

Key Result

Theorem A

Let $(V,\beta)$ be a $q$-bic form of type $(a;b_m)_{m \geq 1}$ over an algebraically closed field $\mathbf{k}$. Then there exists a basis $V = \langle e_1,\ldots,e_n \rangle$ such that

Figures (2)

  • Figure 1: Immediate specialization relations amongst $5$-dimensional $q$-bic forms and dimensions of the corresponding strata, up to the first few with nontrivial radical.
  • Figure 2: Immediate specialization relations amongst for $6$-dimensional $q$-bic forms, up to the first few with nontrivial radical.

Theorems & Definitions (63)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.5
  • ...and 53 more