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Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms
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Let
p
p
and
q
q
be anisotropic quasilinear quadratic forms over a field
F
F
of characteristic
2
2
, and let
i
i
be the isotropy index of
q
q
after scalar extension to the function field of the affine quadric with equation
p
=
0
p=0
. In this article, we establish a strong constraint on
i
i
in terms of the dimension of
q
q
and two stable birational invariants of
p
p
, one of which is the well-known “Izhboldin dimension”, and the other of which is a new invariant that we denote
Δ
(
p
)
\Delta (p)
. Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.
American Mathematical Society (AMS)
Title: Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms
Description:
Let
p
p
and
q
q
be anisotropic quasilinear quadratic forms over a field
F
F
of characteristic
2
2
, and let
i
i
be the isotropy index of
q
q
after scalar extension to the function field of the affine quadric with equation
p
=
0
p=0
.
In this article, we establish a strong constraint on
i
i
in terms of the dimension of
q
q
and two stable birational invariants of
p
p
, one of which is the well-known “Izhboldin dimension”, and the other of which is a new invariant that we denote
Δ
(
p
)
\Delta (p)
.
Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively.
This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.
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