5-dimensional minimal quadratic and bilinear forms over function fields of conics

TL;DR

This work resolves the 5-dimensional K-minimal classification problem for quadratic and bilinear forms over function fields of conics in characteristic $2$. By leveraging Pfister-neighbor structure, even Clifford algebras, and Kato’s $e^2$ invariant, the authors give explicit, checkable criteria for minimality over $F(C)$ in both types $(2,1)$ and $(1,3)$, covering singular and nonsingular conics. The results extend Faivre’s earlier char-$2$ findings and integrate quasi-Pfister neighbor theory, norm-degree arguments, and isotropy refinements to deliver a complete picture for dimension five, including dedicated treatments of bilinear forms and SPN phenomena. The paper also develops a parallel framework for bilinear SPN forms and provides concrete examples illustrating the criteria and methods, highlighting the role of even Clifford algebras and cohomological invariants in characteristic $2$.

Abstract

Over a field of characteristic 2, we give a complete classification of quadratic and bilinear forms of dimension 5 that are minimal over the function field of an arbitrary conic. This completes the unique known case due to Faivre concerning the classification of minimal quadratic forms of dimension 5 and type (2,1) over function fields of nonsingular conics.

Theorems & Definitions (37)

• Theorem 1
• Proposition 1
• Theorem 2
• Theorem 3
• Proposition 2
• Corollary 1
• Theorem 4
• Theorem 5
• Proposition 3
• proof