On rational points of certain affine hypersurfaces

Alexander S. Sivatski

Vol. 292 (2018), No. 1, 239–256

DOI: 10.2140/pjm.2018.292.239

Abstract

Let $F$ be a field with $charF\ne 2$, let $a_1,\dots ,a_n\in F^{\ast }$, and let $f\in F\left[y\right]$ be a monic polynomial of degree $2m$. Let further $S$ be an affine hypersurface over $F$ determined by the equation $f\left(y\right)={\sum }_{i=1}^n{a}_i{x}_i^2$. In the first part of the paper we prove a certain version of Springer’s theorem. Namely, we show that if the form $\psi \simeq \langle 1,-a_1,\dots ,-a_n\rangle$ is anisotropic and $S$ has an $L$-rational point for some odd-degree extension $L∕F$, then $S$ has an $L$-rational point for some odd-degree extension $L∕F$ with $\left[L:F\right]\le m$, and the last inequality is strict in general.

In the second part we consider the case where the polynomial $f$ is quartic. We show that the surface $S$ has a rational point if and only if the quadratic form $\psi \perp \langle -x,g\left(x\right)\rangle$ is isotropic over $F\left(x\right)$, where $g\left(x\right)\in F\left[x\right]$ is a certain polynomial of degree at most $3$, whose coefficients are expressed in a polynomial way via the coefficients of $f$.

In the third part we describe all Pfister forms that belong to the Witt kernel $W\left(F\left(C\right)∕F\right)$, where $C$ is the plane nonsingular curve determined by the equation $y^2=a_4{x}^4+a_2{x}^2+a_{1}x+a_0$. In the case where the $u$-invariant of $F$ is at most $10$, we describe generators of the ideal $W\left(F\left(C\right)∕F\right)$.

Keywords

quadratic form, Springer's theorem, Brauer group, Pfister form, field extension

Mathematical Subject Classification 2010

Primary: 11E04

Secondary: 11E81

Milestones

Received: 7 January 2017

Revised: 13 February 2017

Accepted: 22 March 2017

Published: 22 September 2017