ON THE ESSENTIAL DIMENSION OF CYCLIC GROUPS

WANSHUN WONG

Abstract. We find an upper bound for the essential dimension of finite cyclic groups ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}} ℤ over a field F of characteristic different from p_{i} containing all the primitive p_{i}-th roots of unity, where p_{i} are distinct prime numbers.

1. Introduction

The essential dimension of an algebraic structure is a numerical invariant that measures the complexity of the structure. Informally, the essential dimension of an algebraic structure over a field F is the smallest number of algebraically independent parameters required to define the structure over a field extension of F (see [1] and [8]).

Let 𝔉 : F ields/F → Sets be a functor (an algebraic structure) from the category F ields/F of field extensions of F and field homomorphisms over F to the category of sets. Let K ∈ F ields/F and a ∈ 𝔉(K). The essential dimension of a, denoted ed(a), is the least transcendence degree tr.deg_{F} (K_{0}) over all subfields K_{0} of K over F such that a is in the image of the map 𝔉(K_{0}) → 𝔉(K). The essential dimension of the functor 𝔉 is

ed(𝔉) = sup{ed(a)}

where the supremum is taken over all K ∈ F ields/F and all a ∈ 𝔉(K).

If G is a finite group, we view G as a constant group scheme over a field F. The essential dimension of G is defined as

ed(G) = ed(H^{1}(−, G)).

Thus the essential dimension of G measures the complexity of the category Gtorsors. If G is a finite cyclic p-group, and F is a field of characteristic different from p containing primitive p-th roots of unity, then the essential dimension of G is computed in [4] and [5]. In this paper we prove in Thm. 3.1 that if G = ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ is a finite cyclic group, p_{i} are distinct prime numbers, and F is a field of characteristic different from p_{i} containing all the primitive p_{i}-th roots of unity, then

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) ≤ [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1

where ξ_{r} denotes r-th primitive root of unity for any positive integer r.

Let A be a central division F -algebra of degree q_{1}^{a_{1}} ⋯ q_{k}^{a_{k}}, where q_{i} are distinct prime numbers, a_{i} are non-negative integers. It is a conjecture (see [3]) that

cdim(SB(A)) = q_{1}^{a_{1}} + ⋯ + q_{k}^{a_{k}} − k

2000 Mathematics Subject Classification. 20G15; 14L30.

Key words and phrases. Essential dimension, finite cyclic groups, algebraic tori.

where cdim(SB(A)) is the canonical dimension of the Severi-Brauer variety of A (see Section 4 for definition). If the conjecture is valid, then we show in Thm. 4.4 that

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) = [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1.

Acknowledgements: The author would like to thank his advisor Alexander Merkurjev for his help along the way.

2. Affine Group Schemes

References for affine group schemes are [6] and [10].

Let p be a prime number, F be a field such that char(F) ≠ p and ξ_{p} ∈ F. Let Γ be the absolute Galois group of F, i.e. Γ = Gal(F_{sep}/F) where F_{sep} is a separable closure of F.

For any non-negative integer r, let G_{r} = ℤ/p^{r}ℤ be a constant group scheme over F, F_{r} = F (ξ_{p^{r}}) be a field extension of F and Γ_{r} = Gal(F_{r}/F) be the corresponding Galois group. Then let T_{r} = R_{F_{r}/F} (𝔾_{m}) be the corestriction of the multiplicative group 𝔾_{m} from F_{r} to F.

For any γ ∈ Γ_{r}, γ(ξ_{p^{r}}) = ξ_{p^{r}}^{χ_{r}(γ)} for some χ_{r}(γ) ∈ (ℤ/p^{r}ℤ)^{×}. Then χ_{r} is a Γ-homomorphism χ_{r} : Γ_{r} → (ℤ/p^{r}ℤ)^{×}. It extends linearly to a surjective Γhomomorphism f_{r} : (T_{r,sep})^{∗} = ℤ[Γ_{r}] → ℤ/p^{r}ℤ = (G_{r,sep})^{∗},

f_{r}(∑ a_{γ}γ) = ∑ a_{γ}χ_{r}(γ) (mod p^{r}).

Fix a positive integer n. Let s = min{n, sup{m ∈ ℕ| ξ_{p^{m}} ∈ F }}. Define a surjective Γ-homomorphism g : ℤ[Γ_{n−s}] ⊕ ℤ[Γ_{n}] → ℤ/p^{n}ℤ by

g(x, y) = p^{s} · f_{n−s}(x) + f_{n}(y)

for every (x, y) ∈ ℤ[Γ_{n−s}] ⊕ ℤ[Γ_{n}]. Let V be the factor group scheme

V = (T_{n−s} × T_{n})/G_{n}

with (V_{sep})^{∗} = ker(g), so we have an exact sequence of group schemes

1 ⟶ G_{n} ⟶ T_{n−s} × T_{n} ⟶ V ⟶ 1.

For every K ∈ F ields/F, passing to cohomology and applying Hilbert’s Theorem 90 give

V (K) ⟶ H^{1}(K, G_{n}) ⟶ H^{1}(K, T_{n−s} × T_{n}) = 1.

Consider the composition

(1)  \xymatrix{V (K) \ar@{->>}[r] & H^{1}(K, G_{n}) \ar[r] & H^{1}(K, G_{n−s}),}

where the latter homomorphism is induced by the exact sequence

1 ⟶ G_{s} ⟶ G_{n} ⟶ G_{n−s} ⟶ 1.

Let 𝔉(K) be the image of V (K) in H^{1}(K, G_{n−s}), which is the same as the image of H^{1}(K, G_{n}). Then we get a subfunctor 𝔉 of H^{1}(−, G_{n−s}). The main result of this section is the following

Proposition 2.1. There exists a closed subscheme Y ⊆ V of dimension [F_{n} : F]−1 such that for every infinite K ∈ F ields/F, the image of Y (K) in H^{1}(K, G_{n−s}) is equal to 𝔉(K).

Proof. If n = s, the result follows immediately from H^{1}(K, G_{n−s}) = 1 and [F_{n} : F] = 1.

If n > s, first we want to show that (1) is part of a commutative diagram. Consider the following commutative diagram of Γ-modules

(2)  \xymatrix{ℤ/p^{s}ℤ & ℤ/p^{n}ℤ \ar[l]_{π} & ℤ/p^{n−s}ℤ \ar[l]_{p^{s}} \\ ℤ[Γ_{n}] \ar[u]_{π∘f_{n}} & ℤ[Γ_{n−s}] ⊕ ℤ[Γ_{n}] \ar[u]_{g} \ar[l]_{ϖ} & ℤ[Γ_{n−s}] \ar[l]_{i} \ar[u]_{f_{n−s}} \\ ker(π ∘ f_{n}) \ar[u] & ker(g) \ar[u] \ar[l]_{ϖ} & ker(f_{n−s}) \ar[l]_{i} \ar[u]}

where π is the canonical projection, i is the canonical inclusion i(x) = (x, 0), ϖ is the canonical projection ϖ(x, y) = y for every x ∈ ℤ[Γ_{n−s}], y ∈ ℤ[Γ_{n}]. Note the all the rows and columns in (2) are short exact sequences. Let U = T_{n}/G_{s} with (U_{sep})^{∗} = ker(π ∘ f_{n}), and S = T_{n−s}/G_{n−s} with (S_{sep})^{∗} = ker(f_{n−s}). The commutative diagram of group schemes dual to (2) is

\xymatrix{G_{s} \ar[r] \ar[d] & G_{n} \ar[r] \ar[d] & G_{n−s} \ar[d] \\ T_{n} \ar[r] \ar[d] & T_{n−s} × T_{n} \ar[r] \ar[d] & T_{n−s} \ar[d] \\ U \ar[r] & V \ar[r] & S}

which gives the following commutative diagram

(3)  \xymatrix{T_{n−s}(K) × T_{n}(K) \ar[r] \ar[d] & T_{n−s}(K) \ar[d] \\ V (K) \ar[r] \ar@{->>}[d] & S(K) \ar@{->>}[d] \\ H^{1}(K, G_{n}) \ar[r] & H^{1}(K, G_{n−s})}

for every K ∈ F ields/F.

In order to construct Y, we consider

(4)  \xymatrix{ℤ[Γ_{n−s}] ⊕ ℤ & ℤ[Γ_{n−s}] \ar[l]_{j} \\ ker(g) \ar[u]_{φ} & ker(f_{n−s}) \ar@{^{(}->}[u] \ar@{^{(}->}[l]_{i}}

where j is the inclusion j(x) = (x, 0), φ is defined by φ(x, y) = (x, ϵ(y)/p^{s}) for every (x, y) ∈ ker(g) ⊆ ℤ[Γ_{n−s}] ⊕ ℤ[Γ_{n}], where ϵ is the augmentation map of a group ring.

Lemma 2.2. φ : ker(g) → ℤ[Γ_{n−s}] ⊕ ℤ is well-defined and surjective.

Proof. Let (x, y) be any element in ker(g).

(1) For p ≠ 2, and for p = 2 and s ≥ 2, Γ_{n} is a cyclic group. Let σ be the generator of Γ_{n} such that σ(ξ_{p^{n}}) = ξ_{p^{n}}^{p^{s}+1}. Then we can write y = ∑ a_{m}σ^{m}, and f_{n}(y) = ∑ a_{m}(p^{s} + 1)^{m} (mod p^{n}). Since (x, y) ∈ ker(g), p^{s} · f_{n−s}(x) + ∑ a_{m}(p^{s} + 1)^{m} = 0 (mod p^{n}). Therefore p^{s} divides ∑ a_{m}.

(2) For p = 2 and s = 1, write y = ∑ a_{γ}γ. Since (x, y) ∈ ker(g), 2f_{n−1}(x) + ∑ a_{γ}χ_{n}(γ) = 0 (mod 2). Note that χ_{n}(γ) ∈ (ℤ/2^{n}ℤ)^{×} for every γ, in particular χ_{n}(γ) is odd. Hence ∑ a_{γ} is even.

Therefore φ is well-defined. Claim: If ϵ/p^{s} : ker(f_{n}) → ℤ is surjective, then φ is surjective.

To prove the claim let (x, m) be any element in ℤ[Γ_{n−s}] ⊕ ℤ. Recall that f_{n} : ℤ[Γ_{n}] → ℤ/p^{n}ℤ is surjective. There exists y ∈ ℤ[Γ_{n}] such that f_{n}(y) = −p^{s} · f_{n−1}(x), which implies (x, y) ∈ ker(g). Let m′ = ϵ(y)/p^{s}. Note that for every y′ ∈ ker(f_{n}), (0, y′) ∈ ker(g). If ϵ/p^{s} : ker(f_{n}) → ℤ is surjective, let y′ ∈ ker(f_{n}) such that ϵ(y′)/p^{s} = 1. Then (x, y+(m−m′)y′) ∈ ker(g) and φ(x, y+(m−m′)y′) = (x, m), proving the claim. It remains to show that ϵ/p^{s} : ker(f_{n}) → ℤ is surjective.

(1) For p ≠ 2, and for p = 2 and s ≥ 2, simply note that σ − p^{s} − 1 ∈ ker(f_{n}).

(2) For p = 2 and s = 1, consider Im(χ_{n}) ⊆ (ℤ/2^{n}ℤ)^{×}. Note that (ℤ/2^{n}ℤ)^{×} is a direct product of two cyclic subgroups generated by 5 and −1 respectively. We claim that −5^{r} ∈ Im(χ_{n}) for some integer r. Suppose not, then all elements of Im(χ_{n}) are powers of 5, which implies Γ_{n} fixes ξ_{4} = ξ_{2^{n}}^{2^{n−2}} and contradicts the fact that ξ_{4} ∉ F. Let γ ∈ Γ_{n} such that χ_{n}(γ) = −5^{r}. Since γ + 5^{r} and 2^{n} ∈ ker(f_{n}), (1 + 5^{r})/2 and 2^{n−1} ∈ Im(ϵ/2). As 5^{r} = 1 (mod 4), (1 + 5^{r})/2 is odd. Hence (1 + 5^{r})/2 and 2^{n−1} are coprime and Im(ϵ/2) = ℤ. □

It is clear that the diagram (4) is commutative, so we have the dual commutative digram of group schemes

(5)  \xymatrix{T_{n−s} × 𝔾_{m} \ar@{^{(}->}[d] \ar[r] & T_{n−s} \ar[d] \\ V \ar@{->>}[d]^{π} \ar[r] & S \\ V′}

where V′ = V/(T_{n−s} × 𝔾_{m}).

Let E = F (V′) be the function field of V′. From the exact sequence of cohomology

V (E) ⟶ V′(E) ⟶ H^{1}(E, T_{n−s} × 𝔾_{m}) = 1,

the generic point of V′, Spec (E) → V′ factors through V. Therefore there exists a rational map α : V′ → V such that the composition with the projection π is the identity map on the largest open set U which α is defined. Let Y = \overline{Im(α)}. Clearly dim Y = dim V′ = dim V − dim(T_{n−s} × 𝔾_{m}) = [F_{n} : F] − 1. It remains to check that the images of Y (K) and V (K) in H^{1}(K, G_{n−s}) are equal.

Lemma 2.3. For every v ∈ V (K), there exists u ∈ Y (K) such that the images of v and u in H^{1}(K, G_{n−s}) are equal.

Proof. Since T_{n−s} × T_{n} is a quasi-split torus, it is an open subset of an affine space. Therefore T_{n−s}(K)×T_{n}(K), viewed as the set of K-rational points, is a dense subset of T_{n−s} × T_{n} because K is infinite. Then the image of T_{n−s}(K) × T_{n}(K) in V (K) is dense, and V′(K) is dense in V′. As U is open in V′, U(K) = U ∩ V′(K) ≠ ∅. We can find some v′ in the image of T_{n−s}(K) × T_{n}(K) such that π(v · v′) ∈ U(K). By (3) v and v · v′ have the same image in H^{1}(K, G_{n}). Therefore by replacing v by v · v′ we may assume π(v) ∈ U(K).

Let u = α ∘ π(v) ∈ Y (K). As π(u) = π(v), by (5) u and v differ by an element in T_{n−s}(K) × 𝔾_{m}(K). Then by commutativity of (5) the images of u and v in S(K) differ by the image of an element in T_{n−s}(K). Hence by (3) u and v have the same image in H^{1}(K, G_{n−s}). □

This completes the proof of the proposition. □

Corollary 2.4. ed(𝔉) ≤ ed(ℤ/p^{n}ℤ) − 1.

Proof. First we show that ed(𝔉) ≤ ed(Y). Let K ∈ F ields/F and a ∈ 𝔉(K). If K is a finite field, then tr.deg_{F} (K) = 0 and ed(a) = 0 ≤ ed(Y). If K is infinite, by Prop. 2.1 Y (K) ⟶ 𝔉(K) is a surjection. Then ed(a) ≤ ed(Y) by the proof of [1] Lemma 1.9.

By [1] Prop. 1.17 and [5] Cor. 5.2, we have

ed(𝔉) ≤ ed(Y) = dim(Y) = [F_{n} : F] − 1 = ed(ℤ/p^{n}ℤ) − 1.  □

3. Main Theorem

Theorem 3.1. Let p_{1}, …, p_{r} be distinct prime numbers, n_{1}, …, n_{r} be positive integers. Let F be a field such that char(F) ≠ p_{i} and ξ_{p_{i}} ∈ F for every i. Then

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) ≤ ed(ℤ/p_{1}^{n_{1}}ℤ) + ⋯ + ed(ℤ/p_{r}^{n_{r}}ℤ) − r + 1
= [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1.

Proof. Let s_{i} = min{n_{i}, sup{m ∈ ℕ| ξ_{p_{i}^{m}} ∈ F }} for every i. For each prime number p_{i}, let 𝔉_{i} be the corresponding 𝔉 defined above. Let C_{N} = ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ be a constant group scheme over F, C_{S} = ℤ/p_{1}^{s_{1}} ⋯ p_{r}^{s_{r}}ℤ and C_{N−S} = ℤ/p_{1}^{n_{1}−s_{1}} ⋯ p_{r}^{n_{r}−s_{r}}ℤ. The exact sequence of group schemes

1 ⟶ C_{S} ⟶ C_{N} ⟶ C_{N−S} ⟶ 1

induces an exact sequence

(6)  ⋯ ⟶ H^{1}(K, C_{S}) ⟶ H^{1}(K, C_{N}) ⟶ H^{1}(K, C_{N−S}) ⟶ ⋯

for every K ∈ F ields/F. Note that

H^{1}(K, C_{N}) = H^{1}(K, ℤ/p_{1}^{n_{1}}ℤ) × ⋯ × H^{1}(K, ℤ/p_{r}^{n_{r}}ℤ),

and similarly

H^{1}(K, C_{N−S}) = H^{1}(K, ℤ/p_{1}^{n_{1}−s_{1}}ℤ) × ⋯ × H^{1}(K, ℤ/p_{r}^{n_{r}−s_{r}}ℤ).

Then the exact sequence (6) implies that we have a fibration of functors ([1] Def. 1.12)

\xymatrix{H^{1}(−, C_{S}) \ar[r] & H^{1}(−, C_{N}) \ar@{->>}[r] & 𝔉_{1} × ⋯ × 𝔉_{r}.}

By Cor. 2.4, [1] Lemma 1.11 and Prop. 1.13, and [5] Cor. 5.2,

ed(C_{N}) ≤ ed(C_{S}) + ed(𝔉_{1} × ⋯ × 𝔉_{r})
≤ 1 + ed(𝔉_{1}) + ⋯ + ed(𝔉_{r})
≤ 1 + (ed(ℤ/p_{1}^{n_{1}}ℤ) − 1) + ⋯ + (ed(ℤ/p_{r}^{n_{r}}ℤ) − 1)
= ed(ℤ/p_{1}^{n_{1}}ℤ) + ⋯ + ed(ℤ/p_{r}^{n_{r}}ℤ) − r + 1
= [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1,

where ed(C_{S}) = 1 as ξ_{p_{1}^{s_{1}} ···p_{r}^{sr}} ∈ F. □

Example 3.2. If s_{i} = n_{i} for 2 ≤ i ≤ r, then ξ_{p_{r}^{nr}} ∈ F for 2 ≤ i ≤ r. Thm. 3.1 implies

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) ≤ [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1
= [F (ξ_{p_{1}^{n_{1}}} ) : F]
= ed(ℤ/p_{1}^{n_{1}}ℤ).

On the other hand,

H^{1}(−, ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) = H^{1}(−, ℤ/p_{1}^{n_{1}}ℤ) × ⋯ × H^{1}(−, ℤ/p_{r}^{n_{r}}ℤ).

By [1] Remark 1.16 max{ed(ℤ/p_{i}^{n_{i}})} ≤ ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) where the maximum is taken over 1 ≤ i ≤ r. Therefore

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) = ed(ℤ/p_{1}^{n_{1}}ℤ).

Remark 3.3. Let m = p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}, and G = ℤ/mℤ. If V is a faithful linear representation of G over F then ed(G) ≤ dim(V) ([1] Prop. 4.15). We want to compare the least dimension of a faithful representation of G over F with the upper bound of ed(G) given by Thm. 3.1.

Let n_{i} > s_{i} for 1 ≤ i ≤ a, and n_{i} = s_{i} for a < i ≤ r for some integer 1 ≤ a ≤ r. By Maschke’s Theorem ([7] Ch. XVIII Thm. 1.2), F [G] is semisimple. Since F [G] is a commutative ring, F [G] ≅ F [t]/⟨t^{m} − 1⟩ is isomorphic to a product of fields E_{1} × … × E_{k}. For every divisor d of m, there exists a surjection \xymatrix{F [t]/⟨t^{m} − 1⟩ \ar@{->>}[r] & F (ξ_{d})} , t ⟼ ξ_{d}. Therefore F (ξ_{d}) ≅ E_{i} for some i. On the

other hand, for every E_{j} clearly there exists a surjection \xymatrix{F [t]/⟨t^{m} − 1⟩ \ar@{->>}[r] & E_{j}} . Let ξ be the image of t under this surjection. Then E_{j} = F [ξ] = F (ξ), and ξ^{m} = 1. Hence E_{j} ≅ F (ξ_{d}) for some divisor d of m. Therefore F [G] is isomorphic to a product of F (ξ_{d}), d|m. Note that there can be more than one copy of a particular F (ξ_{d}) in the product.

For every divisor d of m, the kernel of the natural representation G → GL(F (ξ_{d})), 1 ⟼ ξ_{d}, is the subgroup ⟨d⟩. Then the kernel of the natural representation G ⟶ GL(∐ F (ξ_{d_{j}})) is ⋂⟨d_{j}⟩ = ⟨lcm{d_{j}}⟩, where d_{j} divides m for every j. By choosing d_{j} to be p_{1}^{n_{1}}, …, p_{a−1}^{n_{a−1}}, p_{a}^{n_{a}} ⋯ p_{r}^{n_{r}}, we can see that the natural representation of G in the F -space V = F (ξ_{p_{1}^{n_{1}}} ) ⊕ ⋯ ⊕ F (ξ_{p_{a}^{na}}) is a faithful representation of the least dimension, as F (ξ_{p_{a}^{na}}) = F (ξ_{p_{a}^{na} ···p_{r}^{nr}}). We have

dim V = [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{a}^{na}}) : F]
≥ [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1

where equality holds if and only if a = 1 (see Example 3.2). In particular, if a ≥ 2, then ed(G) < dim(V). This is different from the case for p-groups, where the essential dimension of a p-group G′ is equal to the least dimension of a faithful representation of G′ over F (see [5] Thm. 4.1).

4. A conjecture for ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ)

4.1. Canonical dimension. Let F be a field and ℭ be a class of field extensions of F. A field E ∈ ℭ is called generic if for any K ∈ ℭ there exists an F -place of E with values in K. The canonical dimension of the class ℭ is

cdim(ℭ) = min{tr.deg_{F} E}

where the minimum is taken over all generic fields E ∈ ℭ.

Example 4.1. If X is a separated scheme of finite type over F, let ℭ_{X} be the class of field extensions K of F such that X(K) ≠ ∅. The canonical dimension of X is defined as cdim(X) = cdim(ℭ_{X}). It can be shown that cdim(X) ≤ dim(X) (see [3] and [8]).

Example 4.2. If θ is an element of Br(F) the Brauer group of F, let ℭ_{θ} be the class of splitting fields of θ. The canonical dimension of θ is defined as cdim(θ) = cdim(ℭ_{θ}).

Similarly, if D is a finite subgroup of Br(F), let ℭ_{D} be the class of common splitting fields of all elements in D. The canonical dimension of D is defined as cdim(D) = cdim(ℭ_{D}).

Let θ ∈ Br(F) be represented by a central simple F -algebra A. Let SB(A) be the Severi-Brauer variety of A. K ∈ F ields/F splits A if and only if SB(A)(K) ≠ ∅ ([6] Prop. 1.17). Therefore cdim(θ) = cdim(SB(A)).

Conjecture 4.3. Let A be a central division F -algebra of degree q_{1}^{a_{1}} ⋯ q_{k}^{a_{k}}, where q_{i} are distinct prime numbers, a_{i} are non-negative integers. Write A as a tensor product A_{i} ⊗ ⋯ ⊗ A_{k} where A_{i} is a central division F -algebra of degree q_{i}^{a_{i}}. Let X = SB(A) be the Severi-Brauer variety of A, and let Y = SB(A_{1})×⋯×SB(A_{k}). K ∈ F ields/F splits A if and only if it splits A_{i} for every i, therefore X(K) ≠ ∅ if and only if Y (K) ≠ ∅. Hence

(7)  cdim(SB(A)) = cdim(Y) ≤ dim(Y) = q_{1}^{a_{1}} + ⋯ + q_{k}^{a_{k}} − k.

It is conjectured in [3] that the inequality in (7) is actually an equality.

4.2. Algebras and Representations. Let G be a finite group, C be a central subgroup of G and set H = G/C. Then we have an exact sequence

(8)  1 ⟶ C ⟶ G ⟶ H ⟶ 1.

Let E → Spec(L) be a generic H-torsor, L ∈ F ields/F (see [1] section 6, [5] section 4). Let C^{∗} denote the character group Hom(C, 𝔾_{m}) of C. Define a homomorphism β^{E} : C^{∗} → Br(L) by taking χ : C → 𝔾_{m} to the image of the class of E under the composition

H^{1}(L, H) ⟶^{∂} H^{2}(L, C) ⟶^{χ_{∗}} H^{2}(L, 𝔾_{m}) = Br(L),

where ∂ is the connecting homomorphism for the exact sequence (8).

Let χ : C → 𝔾_{m} be a character, Rep^{(χ)}(G) be the category of all finite dimensional representations ρ of G such that ρ(c) is multiplication of χ(c) for any c ∈ C. It is proved in [5] Thm. 4.4 that

ind(β^{E}(χ)) = gcd dim(V)

over all representations V ∈ Rep^{(χ)}(G).

4.3. Gerbes. Let X be a gerbe over F (see [8] section 3, [9] p. 144). Define a functor \overline{X} : F ields/F → Sets by mapping K to the set of isomorphism classes of objects in the category X (K). The essential dimension of X is defined as ed(X) = ed(\overline{X}). Let ℭ_{X} be the class of field extensions K of F such that \overline{X} (K) ≠ ∅. Then the canonical dimension of X is defined as cdim(X) = cdim(ℭ_{X}).

4.4. A conjecture for ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ).

Theorem 4.4. Let p_{1}, …, p_{r} be distinct prime numbers, n_{1}, …, n_{r} be positive integers. Let F be a field such that char(F) ≠ p_{i} and ξ_{p_{i}} ∈ F for every i. If Conjecture 4.3 is valid, then

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) = ed(ℤ/p_{1}^{n_{1}}ℤ) + ⋯ + ed(ℤ/p_{r}^{n_{r}}ℤ) − r + 1
= [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1.

Proof. By Thm. 3.1 we only need to prove

ed(ℤ/p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}ℤ) ≥ [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1

when Conjecture 4.3 is valid.

Let m = p_{1}^{n_{1}} ⋯ p_{r}^{n_{r}}, G = ℤ/mℤ, C = ℤ/p_{1} ⋯ p_{r}ℤ be a subgroup of G, and set H = G/C. Let E → Spec(L) be a generic H-torsor, L ∈ F ields/F. Recall that we have a homomorphism β^{E} : C^{∗} → Br(L).

Consider the gerbe E/G banded by C. Since ξ_{p_{i}} ∈ F for every i and L ∈ F ields/F, C ≅ μ_{p_{1}···p_{r}} and H^{2}(L, C) ≅ Br_{p_{1}···p_{r}}(L). Then the element in H^{2}(L, C) corresponding to E/G can be represented by a central division L-algebra A with [A] ∈ Br_{p_{1}···p_{r}}(L). Note that Im(β^{E}) is generated by the class of A. It follows that

cdim_{L}(E/G) = cdim_{L}(Im(β^{E})).

By [5] Thm. 4.2 and [2] Thm. 7.1, we have

(9) ed(G) ≥ ed_{L}(G) ≥ ed_{L}(E/G) = cdim_{L}(E/G) + 1 = cdim_{L}(Im(β^{E})) + 1.

Let χ : C → 𝔾_{m} be the character such that β^{E}(χ) = [A], and

a = ind(β^{E}(χ)) = gcd dim(V)

over all V ∈ Rep^{(χ)}(G). For every V ∈ Rep^{(χ)}(G), by the calculation in Remark 3.3 V = ∐ F (ξ_{d_{j}}), d_{j} divides m for every j. For every c ∈ C, c acts on V by

multiplication of χ(c). Therefore ξ_{d_{j}}^{p_{1}^{n_{1}−1}···p_{r}^{nr −1}} is a primitive p_{1} ⋯ p_{r}-root of unity. Combining with the fact that d_{j} divides m, we have d_{j} = m for every j, which implies

a = [F (ξ_{m}) : F] = ∏[F (ξ_{p_{i}^{n_{i}}}) : F],

where the second equality follows from the fact that ξ_{p_{i}} ∈ F, [F (ξ_{p_{i}^{n_{i}}}) : F] is a power of p_{i} for every i.

If Conjecture 4.3 is valid,

cdim_{L}(Im(β^{E})) = cdim_{L}(β^{E}(χ)) = cdim(SB(A))
= [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r.

By combining the above inequality with (9), we have

ed(G) ≥ cdim_{L}(Im(β^{E})) + 1 ≥ [F (ξ_{p_{1}^{n_{1}}} ) : F] + ⋯ + [F (ξ_{p_{r}^{nr}}) : F] − r + 1. □

Example 4.5. Let ξ_{2}, ξ_{3} ∈ F but ξ_{4}, ξ_{9} ∉ F. Consider ed(ℤ/36ℤ) = ed(ℤ/2^{2}3^{2}ℤ). In this case ind(β^{E}(χ)) = 6, and Conjecture 4.3 is proven when A is of degree 6 by [3] Thm. 1.3. Therefore ed(ℤ/36ℤ) = 4 (see [2] Remark 14.2).

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Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: wanshunwong@math.ucla.edu