ESSENTIAL DIMENSION OF TWISTED C_{4}

MARKUS ROST

Introduction

In this note we show that over a field of characteristic different from 2 the essential dimension of any Galois module which is cyclic of order 4 is equal to 2, except for μ_{4}.

The method is consists of a (non-)ramification argument, which is also used in the text [1] for the computation of of the essential dimension of PGL_{4}.

For the essential dimension of Z/4 with trivial Galois action see also [2].

1. Statement of result

Let k be a field of characteristic different from 2 and let G be a Galois module over k, which as an abstract group is cyclic of order 4. Let d ∈ k^{∗} be an element such that G is isomorphic to μ_{4} twisted by the quadratic character given by the square class of d. Thus G ≃ μ_{4} if d is a square and G ≃ Z/4 if −d is a square.

One has H^{1}(k, μ_{4}) = k^{∗}/(k^{∗})^{4} and it easy to see that ed(μ_{4}) = 1.

Proposition. If d is not a square, then ed(G) = 2.

2. Galois cohomology

We assume in the following that d is not a square.

Then G can be identified with the elements of order 4 in the norm 1 sub group of the multiplicative group of K = k(\sqrt{d}).

Moreover there is an exact sequence

1 → G → (K ⊗_{k} k̄)^{∗} ⟶^{π} k̄^{∗} × \frac{(K ⊗_{k} k̄)^{∗}}{k̄^{∗}} → 1

where

π(λ) = (N_{K/k}(λ), [λ^{2}])

This is easy to check.

By Hilbert’s Theorem 90 one gets an exact sequence

K^{∗} ⟶^{π} k^{∗} × \frac{K^{∗}}{k^{∗}} ⟶^{δ_{k}} H^{1}(k, G) → 0

Note that H^{1}(k, G) classifies the extensions L/k of degree 4 together with a monomorphism (of algebraic groups!) G → Aut(L/k). In fact, let a ∈ k^{∗} and μ ∈ K^{∗} and let μ̄ be the conjugate of μ under the Galois group of K/k. Consider the Kummer extension

E = K[u], u^{4} = a^{2} \frac{μ}{μ̄}

Date: March 2, 2002.

It has a K-semilinear automorphism

σ: E → E, σ(u) = \frac{a}{u}

Then

L = E^{σ}

is an extension of k of degree 4 with a natural G-action. This extension corresponds to δ_{k}(a, [μ]).

3. Invariants

We consider the cohomological invariants

η_{1} : H^{1}(k, G) → H^{1}(k, μ_{2})
η_{2} : H^{1}(k, G) → H^{2}(k, μ_{2})

defined as follows: The invariant η_{1} is just the map induced from the projection G → μ_{2}. The invariant η_{2} is given by the embedding

G → PGL(2)

ζ → [(\begin{matrix}1 & −\sqrt{−d} \\ \sqrt{−d}^{−1} & 1 \end{matrix})]

where ζ ∈ G(k̄) is a generator, followed by the standard invariant H^{1}(k, PGL(2)) → H^{2}(k, μ_{2}).

One has for a ∈ k^{∗} and μ ∈ K^{∗}:

η_{1}(δ_{k}(a, [μ])) = (N_{K/k}(μ))
η_{2}(δ_{k}(a, [μ])) = (a) ∪ (d)

One may take these formula also for a definition of η_{1} and η_{2}.

4. Versal elements

Let F = k(x, y). Then

α = δ_{F} (x, [1 + y\sqrt{d}]) ∈ H^{1}(F, G)

is a versal element for H^{1}(?, G).

Let v be the valuation on F which is trivial on k(y) and with prime element x. Then κ(v) = k(y). One has

η_{1}(α) is unramified at v.

η_{2}(α) is ramified at v.

5. Proof of the Proposition

Now let α′ ∈ H^{1}(F′, G) be some versal element for H^{1}(?, G). We have to show that F′ has transcendence degree at least 2.

Let

φ: F′ ⇝ F

be a k-place with

φ_{∗}(α′) = α

(see [1] for places).

Further let v′ = φ^{∗}(v). Since η_{2}(α) = φ_{∗}(η_{2}(α′)) is ramified at v (here we use that d is not a square), we must have that η_{2}(α′) is ramified at v′. Hence the ramification index of v′|v is odd. In particular, v′ is not trivial. Since η_{1}(α) = φ_{∗}(η_{1}(α′)) is unramified at v, it follows then that η_{1}(α′) is unramified at v′.

Next note that the specialization

η_{1}(α)(v) ∈ H^{1}(k(y), μ_{2})

is a versal element for square classes which are norms from K/k.

Let

φ̄: κ(v′) → κ(v) = k(y)

be the induced homomorphism on the residue class fields. Then

φ̄_{∗}(η_{1}(α′)(v′)) = η_{1}(α)(v)

Hence

η_{1}(α′)(v′) ∈ H^{1}(κ(v′), μ_{2})

is also a versal element for square classes which are norms from K/k. It follows that κ(v′) has transcendence degree at least 1. Since v′ is non-trivial it follows that F′ has transcendence degree at least 2. This finishes the proof.

References

[1] M. Rost, Computation of some essential dimensions, notes, 2000, http://www.math.ohiostate.edu/˜ rost/ed.html.

[2] J.-P. Serre, Topics in Galois theory, Jones and Bartlett Publishers, Boston, MA, 1992, Lecture notes prepared by Henri Damon [Henri Darmon], With a foreword by Darmon and the author.

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://math.ohio-state.edu/~rost