Diagonal forms of higher degree over function fields of p-adic curves

S Pumplün

doi:10.1142/s1793042120500098

Abstract

We investigate diagonal forms of degree [Formula: see text] over the function field [Formula: see text] of a smooth projective [Formula: see text]-adic curve: if a form is isotropic over the completion of [Formula: see text] with respect to each discrete valuation of [Formula: see text], then it is isotropic over certain fields [Formula: see text], [Formula: see text] and [Formula: see text]. These fields appear naturally when applying the methodology of patching; [Formula: see text] is the inverse limit of the finite inverse system of fields [Formula: see text]. Our observations complement some known bounds on the higher [Formula: see text]-invariant of diagonal forms of degree [Formula: see text]. We only consider diagonal forms of degree [Formula: see text] over fields of characteristic not dividing [Formula: see text].

Highlights

The fact that Springer’s Theorem holds for diagonal forms of higher degree over fields of characteristic not dividing d! [9] guarantees that on occasion diagonal forms of higher degree defined over function fields behave to quadratic forms
Let v be a rank one discrete valuation of F, and Fv the completion of F with respect to v. It was shown by Colliot-Thelene, Parimala and Suresh [2, Theorem 3.1] that a quadratic form which is isotropic over Fv for each v is already isotropic over F, using the methodology of patching developed by Habater and Hartmann [4], i.e. viewing F as the inverse limit of a finite inverse system of certain fields {FU, FP, Fp}
As a consequence of Springer’s Theorem for diagonal forms, any diagonal form of degree d and dimension > d3 + 1 over a function field in one variable F = K(t), where K is a p-adic field with residue field k, char(k) d!, is isotropic over Fv for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K

Summary

Introduction

The fact that Springer’s Theorem holds for diagonal forms of higher degree over fields of characteristic not dividing d! [9] guarantees that on occasion diagonal forms of higher degree defined over function fields behave to quadratic forms. As a consequence of Springer’s Theorem for diagonal forms, any diagonal form of degree d and dimension > d3 + 1 over a function field in one variable F = K(t), where K is a p-adic field with residue field k, char(k) d!, is isotropic over Fv for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K It is isotropic over FU for each reduced, irreducible component U ⊂ Y of the complement of S in the special fibre Y = X ×Ak of X/A, and isotropic over FP for each P ∈ S (Corollary 3.4), and isotropic over Fp for each p = (U, P ). S is the inverse image under a finite A-morphism f : X → P1A of the point at infinity of the special fibre P1k

Forms of higher degree

Higher degree u-invariants

Cr0 fields

Diagonal forms over Henselian valued fields