On the essential dimension of cyclic p-groups

Mathieu Florence^{1}

December 2006

Abstract. Let p be a prime number and r ≥ 1 an integer. We compute the essential dimension of ℤ/p^{r}ℤ over fields of characteristic not p, containing the p-th roots of unity (theorem 3.1). In particular, we have ed_{ℚ}(ℤ/8ℤ) = 4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished).

Keywords and Phrases: Galois cohomology, essential dimension, Severi-Brauer varieties.

A la mémoire de mon père.

Acknowledgements

We thank Jean Fasel and Giordano Favi for fruitful discussions about essential dimension.

Contents

Acknowledgements 1 1. Introduction 1 2. Some auxiliary results 2 3. The main theorem 5 References 6

1. Introduction

The notion of essential dimension was introduced by Buhler and Reichstein for finite groups in [BR]. It was later generalized by Reichstein to arbitrary linear algebraic groups ([Re]). Throughout this paper, we shall assume that the reader is somewhat familiar with this concept. A convenient and comprehensive reference on this subject is [BF].

Definition 1.1. Let k be a field, and G a (smooth) linear algebraic k-group. The essential dimension of G over k, denoted by ed_{k}(G), is the smallest nonnegative integer n with the following property. For each field extension K/k, and each G_{K}-torsor T ⟶ Spec(K), there exists a subfield K′ of K, containing k, and a G_{K′}-torsor T′ ⟶ Spec(K′), such that

i) The G_{K}-torsors T and T′_{K} are isomorphic, ii) The transcendence degree of K′/k is equal to n.

1The author gratefully acknowledges support from the Swiss National Science Fundation, grant no. 200020-109174/1 (project leader: E. Bayer-Fluckiger)

Thus, ed_{k}(G) is the smallest number of algebraically independent parameters required to define G-torsors. It turned out that this number, even in apparently simple cases, is extremely difficult to compute. Focusing of finite abelian groups, let us mention some known results. Over fields containing all roots of unity, the essential dimension of a finite abelian group equals its rank, at least if the characteristic does not divide the order of the group under consideration ([BR], theorem 6.1). Over general fields, the answer was known only for cyclic groups of small order. To the author’s knowledge, the results obtained so far over the field of rational numbers may be summarized as follows. The number ed_{ℚ}(ℤ/nℤ) equals 1 for n = 2, 3 (easy exercise); it is 2 for n = 4 (Lenstra, Serre, see also [BF], theorem 7.6 for an alternate proof); it is 2 for n = 5 ([JLY] , see also [BF], corollary 7.9); it is 2 for n = 7 ([Le]). For n odd, Jensen, Ledet and Yui proved that ed_{ℚ}(ℤ/2nℤ) = 1 + ed_{ℚ}(ℤ/nℤ) ([JLY]). This settles the cases n = 6, 10 and 14. Let us also mention the following result of Rost ([Ros]): let k be a field of characteristic not 2, and G/k a linear algebraic group, geometrically isomorphic to μ_{4}. Then ed_{k}(G) = 1 if G is isomorphic to μ_{4}, and ed_{k}(G) = 2 otherwise. For arbitrary n ≥ 4, it seems that the best known lower bound for ed_{ℚ}(ℤ/nℤ) is 2, and that the best upper bound is given by a result of Ledet ([Le], see also [FF], theorem 4.1): for a prime number p and a positive integer r, we have ed_{ℚ}(ℤ/p^{r}ℤ) ≤ ϕ(p − 1)p^{r−1}, where ϕ denotes Euler’s function.

2. Some auxiliary results

In this section, we introduce the material required to prove the main theorem. Before going any further, we briefly recall the notion of friendly open subset. Let G/k be an algebraic group, where k is any field, and X/k an irreducible Gvariety, on which G acts generically freely, in the sense that there is a G-invariant dense open subvariety V ⊂ X such that the scheme-theoretic stabilizer of any point of V is trivial. By a theorem of Gabriel, there exists a G-invariant dense open subset U ⊂ X such that the categorical quotient U ⟶ U/G exists, and is a G-torsor for the fppf topology (see [BF], theorem 4.7).

Definition 2.1. Such an open subset U ⊂ X is called a friendly open subset (for the action of G on X).

If V is a finite dimensional vector space over some field k, we denote by 𝔸(V) the k-variety representing the functor A ↦ V ⊗_{k} A, where A runs through all kalgebras. The following proposition seems to be well-known; as we lack a suitable reference, we include a proof.

Proposition 2.2. Let k be a field, G/k a linear algebraic group, and V a finitedimensional representation of G over k, which is generically free. Let U ⊂ 𝔸(V) be a friendly open subset. Then ed_{k}(G)+dim(G) = min(dim(ϕ(U))), where ϕ runs through all G-equivariant rational maps \xymatrix{U \ar@{-->}[r] & U} . In particular, if every such ϕ is dominant, we have ed_{k}(G) = dim(V) − dim(G).

Proof. We first prove that ed_{k}(G) + dim(G) ≥ min(dim(ϕ(U))). By [BF], proposition 4.11, we know that the G-torsor U ⟶ U/G is versal. Let X ⟶ Y be a versal G-torsor, with dim(Y) = ed_{k}(G). By the very definition of a versal torsor, there exists two commutative squares

\xymatrix{U \ar[d] \ar@{-->}[r]^{f} & X \ar[d] \\ U/G \ar@{-->}[r]^{g} & Y}

and

\xymatrix{X \ar[d] \ar@{-->}[r]^{f′} & U \ar[d] \\ Y \ar@{-->}[r]^{g′} & U/G,}

where the horizontal arrows are rational maps, and where f and f′ are Gequivariant. Note that g (and hence f) is necessarily dominant. Indeed, let Y′ be the closure of the image of g, and X′ := X ×_{Y} Y′. Then the G-torsor X′ ⟶ Y′ is versal (as a compression of the versal torsor U ⟶ U/G), which implies that dim(Y′) ≥ ed_{k}(G) = dim(Y). In other words, Y = Y′. Thus, the composite ϕ := f′ ∘ f is well-defined, and dim(ϕ(U)) ≤ dim(X) = ed_{k}(G) + dim(G). The desired inequality follows. For the other inequality, let ϕ : \xymatrix{U \ar@{-->}[r] & U} be a Gequivariant rational map. We have a commutative square

\xymatrix{U \ar[d] \ar@{-->}[r]^{ϕ} & U \ar[d] \\ U/G \ar@{-->}[r]^{ϕ̄} & U/G.}

Let S be the closure of the image of ϕ̄, and T = U ×_{U/G}S. Then T ⟶ S is a versal G-torsor. Therefore, dim(ϕ(U)) = dim(T) = dim(S)+dim(G) ≥ ed_{k}(G)+dim(G). □

The next lemma will be applied in order to prove the main theorem. It allows us to restrict our attention to a particular type of G-equivariant rational maps; namely, homogeneous ones.

Definition 2.3. Let V be a finite dimensional vector space over some field k, and d an integer. A non-zero rational map ϕ : 𝔸(V) ⟶ 𝔸(V) is said to be d-homogeneous if the following diagram commutes:

\xymatrix{𝔾_{m} × 𝔸(V) \ar@{-->}[d]^{Id×ϕ} \ar[r]^{(λ,v)↦λv} & 𝔸(V) \ar@{-->}[d]^{ϕ} \\ 𝔾_{m} × 𝔸(V) \ar[r]^{(λ,v)↦λ^{d}v} & 𝔸(V).}

We shall also say that ϕ is homogeneous if it is d-homogeneous for some (unique) d ∈ ℤ.

Lemma 2.4. Let k be a field, G/k a linear algebraic group, and V a finitedimensional irreducible representation of G over k. Assume there exists a G-

equivariant rational map \xymatrix{𝔸(V) \ar@{-->}[r]^{ϕ} & 𝔸(V)} which is not dominant and non-zero.

Then there exists \xymatrix{𝔸(V) \ar@{-->}[r]^{ψ} & 𝔸(V)} with the same properties, and which is moreover homogeneous.

Proof. Identify 𝔸(V) with 𝔸^{N} using a basis of V/k. The map ϕ is now given by

(x_{1}, ..., x_{N}) ∈ 𝔸^{N} ↦ (\frac{P_{1}}{Q} , ..., \frac{P_{N}}{Q}),

where Q and the P_{i}’s are non-zero polynomials in the x_{j}’s. From the Gequivariance of ϕ and the irreducibility of V, it follows that the P_{i}’s have the same degree d. Let d′ be the degree of Q. Let H_{i} be the homogeneous component of degree d of P_{i} and S the homogeneous component of degree d′ of Q. The hypothesis made on ϕ implies that there exists a non zero polynomial R in N variables such that R(\frac{P_{1}}{Q} , ..., \frac{P_{N}}{Q})=0. Write R = R_{n} + R_{n−1} + ... + R_{m}, where R_{i} is homogeneous of degree i and R_{n},R_{m} are both non-zero. We have ∑_{i=m}^{n} R_{i}(P_{1}, ..., P_{N})Q^{n−i} = 0. If d′ > d, an easy degree consideration yields that R_{m}(H_{1}, ..., H_{N}) = 0. Similarly, if d′ < d, we have R_{n}(H_{1}, ..., H_{N}) = 0. Finally, if d = d′, one gets ∑_{i=m}^{n} R_{i}(H_{1}, ..., H_{N})S^{n−i} = 0. In all cases, we see that the map

ψ : (x_{1}, ..., x_{N}) ∈ 𝔸^{N} ↦ (\frac{H_{1}}{S} , ..., \frac{H_{N}}{S})

is not dominant. As it is obviously G-equivariant and homogeneous, we are done. □

The following theorem, due to Karpenko, is a key argument in the proof of our main theorem. It can be viewed as a consequence of Rost’s degree formula. For the convenience of the reader, we outline a proof.

Theorem 2.5 (see [Ka], theorem 2.1). Let k be a field, p a prime number, and A/k a central division algebra of index p^{n} for some n ≥ 1. Then any rational map

\xymatrix{SB(A) \ar@{-->}[r]^{f} & SB(A)} is dominant.

Proof. Let \xymatrix{SB(A) \ar@{-->}[r]^{f} & SB(A)} be a rational map. We want to apply Rost’s rational degree formula ([Me], theorem 3.3). To this end, we have to compute the numbers I(SB(A)) and η_{p}(SB(A)) (cf. loc. cit. for the definition of these numbers). The first one is easily seen to be p^{n}, as A is a division algebra. By loc. cit., Remark 6.5, the number η_{p}(SB(A)) ∈ ℤ/p^{n}ℤ equals p^{n−1}. Hence, Rost’s degree formula yields p^{n−1} = deg(f)p^{n−1} mod p^{n}. In particular, deg(f) is non-zero; that is to say, f is dominant. □

From now on, p will be a fixed prime number, K will be a field of characteristic not p, and K̄ a separable closure of K. For any n ≥ 1, let G_{n} = μ_{p^{n}}(K̄) (viewed as a finite abstract group) and K_{n} = K(μ_{p^{n}}(K̄)). Choose primitive p^{n}-th roots of unity ζ_{n} ∈ K_{n} such that ζ_{n+1}^{p} = ζ_{n}. We will sometimes identify G_{n} with ℤ/p^{n}ℤ by sending ζ_{n} to 1. Let

s = max{n ∈ ℕ, such that K = K_{n}}

(we assume that s is finite; if not, the question we are dealing with has a trivial answer). We make the following assumptions on s:

i) We have s ≥ 1, ii) If p = 2 and s = 1, then K_{2} ≠ K_{3}.

It is easy to check that i) and ii) imply that the polynomial X^{p^{n}} − ζ_{s} is irreducible over K for any n ≥ 0 (in other words, K_{s+n}/K is a Galois field extension, of degree p^{n}). Let r ≥ 1 be any integer. Our goal is to compute the number ed_{K}(ℤ/p^{r}ℤ). It is 1 if r ≤ s, so we assume r ≥ s.

Consider ℙ(K_{r}), the projective space of K_{r} viewed as a K-vector space. It is naturally endowed with an action of G_{r}. For this action, G_{s} ⊂ G_{r} acts trivially. Thus, we have an induced action of G_{r−s} = G_{r}/G_{s}, which is easily seen to be faithful. The following easy lemma will be useful in the sequel.

Lemma 2.6. Let L/K be any field extension, and M/L a cyclic field extension, of group G_{r−s}. Then the twist of ℙ(K_{r})_{L} by the G_{r−s}-torsor associated to M/L is the Severi-Brauer variety corresponding to the cyclic algebra (M/L, σ, ζ_{s}), where σ = ζ_{r−s} ∈ G_{r−s}. In other words, this algebra is generated by M and an indeterminate X, with relations X^{p^{r−s}} = ζ_{s} and XmX^{−1} = σ(m), for m ∈ M.

Proof. Let L_{r} = L ⊗_{K} K_{r}. We have to describe the algebra A obtained by twisting End_{L}(L_{r}) by the G_{r−s}-torsor T associated to M/L. First of all, this algebra clearly contains L_{r} as a maximal étale subalgebra (indeed, G_{r−s} acts trivially on L_{r} ⊂ End_{L}(L_{r})). Now consider the maximal (split) étale subalgebra E = L^{p^{r−s}} of End_{L}(L_{r}) consisting of linear maps admitting 1, ζ_{r}, ..., ζ_{r}^{p^{r−s}−1} as eigenvectors. One easily sees that ζ_{r−s} ∈ G_{r−s} = G_{r}/G_{s} acts on E by cyclic permutation of the coordinates. Hence, the twist of E by T is a maximal subfield of A isomorphic to M. It follows that A is presented as stated in the lemma, with X = ζ_{r} ∈ L_{r}. □

In the proof of theorem 3.1, we shall apply theorem 2.5 to some particular division algebra, arising as a generalized version of the ’generic’ division algebra of Brauer-Rowen ([Row]). Mutatis mutandis, the proof of the next theorem is the same as that of loc. cit., theorem 7.3.8. We tried to harmonize our notations with those of Rowen, with one minor change: the letters q, n, t, k used in loc. cit. correspond here to p^{q}, p^{n}, p^{t} and p^{k}, respectively.

Theorem 2.7. Let n, t and q ≥ s be three integers, with n ≤ t ≤ q + n. Let E_{q,t} = K_{q}(x_{1}, ..., x_{p^{t}}) be purely transcendental over K_{q}. Let σ be the K_{q}-automorphism of order p^{t} of E_{q,t}, permuting the x_{i}’s cyclically. Let K_{q,n,t} be the subfield of E_{q,t} fixed by σ^{p^{n}}. Then the cyclic algebra R_{q,n,t} := (K_{q,n,t}, σ, ζ_{q}) is a division algebra.

Proof. We adopt the notations of loc. cit., page 246. More precisely, let F_{0} = K_{q}, H = F_{0}[x_{1}, ..., x_{p^{t}}], H_{1} = ∑_{i=1}^{p^{t}} F_{0}x_{i}. Write R_{q,n,t} = ∑_{i=0}^{p^{n}−1} K_{q,n,t}z^{i} where zaz^{−1} = σ(a) for each a ∈ K_{q,n,t} and z^{p^{n}} = ζ_{q}. Choose a decomposition F_{0}[X]/ < X^{p^{t}} − 1 >= L_{1} × ... × L_{u} (direct product of fields). Let V_{i} = L_{i}H_{1}; these are simple σ-modules. Given j = (j_{1}, ..., j_{u}) ∈ ℕ^{u}, put H_{j} = V_{1}^{j_{1}}...V_{u}^{j_{u}}. First of all, note that lemma 7.3.4 of loc. cit. remains valid in our setting (for 7.3.4 ii), note that, in fact, all p^{t−k}-th roots of unity occur as eigenvalues of σ^{p^{k}} acting on H). It is also clear that lemma 7.3.6 i) still holds, since the polynomial X^{p^{m}} − ζ_{q} is irreducible in k_{q}[X], for any m ≥ 0. With these lemmas at our disposal, the rest of the proof is exactly the same as that of loc. cit., theorem 7.3.8. □

3. The main theorem

We are now ready to prove the theorem announced in the abstract. We keep the notations of the previous section.

Theorem 3.1. We have ed_{K}(ℤ/p^{r}ℤ) = p^{r−s}.

Proof. By proposition 2.2, it is enough to show that every G_{r}-equivariant non-

zero rational map \xymatrix{𝔸(K_{r}) \ar@{-->}[r]^{ϕ} & 𝔸(K_{r})} is dominant. Assume the contrary: let

\xymatrix{𝔸(K_{r}) \ar@{-->}[r]^{ϕ} & 𝔸(K_{r})} be a non-zero, non-dominant G_{r}-equivariant rational map. Then by lemma 2.4, we can assume that ϕ is homogeneous: there exists an integer d such that ϕ(λv) = λ^{d}ϕ(v) for all v ∈ 𝔸(K_{r}) and all λ ∈ K. Thus, we have an

induced G_{r−s}-equivariant rational map \xymatrix{ℙ(K_{r}) \ar@{-->}[r]^{ϕ̄} & ℙ(K_{r})} (remember that here G_{r−s} is viewed as the quotient G_{r}/G_{s}). From the relation ϕ(ζ_{s}v) = ζ_{s}ϕ(v), it follows that d is non-zero. Hence, ϕ̄ cannot be dominant. Let M = K(x_{g}, g ∈ G_{r−s}) be purely transcendental over K. The group G_{r−s} acts on M the obvious way; let L = M^{G_{r−s}}. Now extend scalars to L. We get a non-dominant G_{r−s}-equivariant rational map ϕ̄_{L} : \xymatrix{ℙ(L_{r}) \ar@{-->}[r] & ℙ(L_{r})} , where L_{r} = K_{r} ⊗_{K} L. We can twist both sides by the G_{r−s}-torsor corresponding to the cyclic extension M/L. Thanks to lemma 2.6, we obtain a non-dominant rational selfmap of SB(A), where A/L is the cyclic algebra (M/L, σ, ζ_{s}), with σ = ζ_{r−s} ∈ G_{r−s}. By theorem 2.7, A is a division algebra. This contradicts theorem 2.5. □

Corollary 3.2. For any n ≥ 1, we have ed_{ℚ}(ℤ/2^{n}ℤ) = 2^{n−1} and ed_{ℚ}(ℤ/3^{n}ℤ) = 3^{n−1}.

Proof. The first equality directly follows from the theorem. For the second one, note that, by a result of Ledet ([Le]), we have ed_{ℚ}(ℤ/p^{n}ℤ) ≤ ϕ(p−1)p^{n−1}. Hence, ed_{ℚ}(ℤ/3^{n}ℤ) ≤ 3^{n−1}. But, as essential dimension decreases after a field extension, we also have ed_{ℚ}(ℤ/3^{n}ℤ) ≥ ed_{ℚ(μ_{3})}(ℤ/3^{n}ℤ) = 3^{n−1}.

□

Remark 3.3. At first sight, it seems that our strategy cannot be applied to compute the value of ed_{ℚ}(ℤ/11ℤ).

References

[BF] G. Berhuy, G. Favi.— Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279-330.

[BR] J. Buhler, Z. Reichstein. — On the essential dimension of a finite group, Compos. Math. 106(1997), 159-179.

[FF] G. Favi, M. Florence. — Tori and Essential Dimension, preprint, available at http://www.math.uni-bielefeld.de/LAG/man/208.pdf

[JLY] C. U. Jensen, A. Ledet, N. Yui. — Generic Polynomials: Constructive Aspects of the Inverse Galois Problem, MSRI Pub. Series 45 (2002), Cambridge University Press.

[Ka] N. Karpenko. — On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15, no. 1 (2002), 1-22.

[Le] A. Ledet. — On the essential dimension of some semi-direct products, Canad. Math. Bull. 45(2002), 422-427.

[Me] A. Merkurjev. — Degree Formula, available at http://www.mathematik.uni-bielefeld.de/ rost/degree-formula.html

[Re] Z. Reichstein. — On the Notion of Essential Dimension for Algebraic Groups, Transform. Groups 5, no. 3 (2000), 265-304.

[Ros] M. Rost. — Essential dimension of twisted C_{4}, available at http://www.mathematik.uni-bielefeld.de/ rost/ed.html#C4

[Row] L. Rowen. — Ring Theory, vol. 2, Pure and Applied Mathematics 128 (Boston, Academic Press, 1988).

Mathieu Florence, Ecole Polytechnique Fédérale de Lausanne, FSB-MA, CH-1015 Lausanne, Switzerland.

E-mail address: mathieu.florence@gmail.com