A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field

Jean-Pierre Serre

Let k be a field. Let Cr(k) be the Cremona group of rank 2 over k, i.e. the group of k-automorphisms of k(X, Y), where X and Y are two indeterminates.

We shall be interested in the finite subgroups of Cr(k) of order prime to the characteristic of k. The case k = C has a long history, going back to the 19-th century (see the references in [Bl 06] and [DI 07]), and culminating in an essentially complete (but rather complicated) classification, see [DI 07]. For an arbitrary field, it seems reasonable to simplify the problem à la Minkowski, as was done in [Se 07] for semisimple groups ; this means giving a sharp multiplicative bound for the orders of the finite subgroups we are considering.

In §6.9 of [Se 07], I had asked a few questions in that direction, for instance the following :

If k = Q, is it true that Cr(k) does not contain any element of prime order ⩾ 11 ?

More generally, what are the prime numbers ℓ, distinct from char(k), such that Cr(k) contains an element of order ℓ?

This question has now been solved by Dolgachev and Iskovskikh ([DI 08]), the answer being that there is equivalence between :

Cr(k) contains an element of order ℓ and

[k(z_{ℓ}) : k] = 1, 2, 3, 4 or 6, where z_{ℓ} is a primitive ℓ-th root of unity.

As we shall see, a similar method can handle arbitrary ℓ-groups and one obtains an explicit value for the Minkowski bound of Cr(k), in terms of the size of the Galois group of the cyclotomic extensions of k (cf. Th.2.1 below). For instance :

Theorem - Assume k is finitely generated over its prime subfield. Then the finite subgroups of Cr(k) of order prime to char(k) have bounded order. Let M(k) be the least common multiple of their orders .

a) If k = Q, we have M(k) = 120960 = 2^{7}.3^{3}.5.7.

b) If k is finite with q elements, we have :

M(k) = \{ \begin{matrix}3.(q^{4} − 1)(q^{6} − 1) & if q ≡ 4 or 7 (mod 9) \\ (q^{4} − 1)(q^{6} − 1) & otherwise. \end{matrix}

For more general statements, see §2. These statements involve the cyclotomic invariants of k introduced in [Se 07, §6] ; their definition is recalled in §1. The proofs are given in §3 (existence of large subgroups) and in §4 (upper bounds). For the upper bounds, we use a method introduced by Manin ([Ma 66]) and perfected by Iskovskikh ([Is 79], [Is 96]) and Dolgachev-Iskovskikh ([DI 08]) ; it allows us to realize any finite subgroup of Cr(k) as a subgroup of Aut(S), where S is either a del Pezzo surface or a conic bundle over a conic. A few conjugacy results are given in §5. The last § contains a series of open questions on the Cremona groups of rank > 2.

§1 The cyclotomic invariants t and m

In what follows, k is a field, k_{s} is a separable closure of k and \overline{k} is the algebraic closure of k_{s}.

Let ℓ a prime number distinct from char(k) ; the ℓ-adic valuation of Q is denoted by v_{ℓ}, or sometimes simply by v. If A is a finite set, with cardinal |A|, we write v_{ℓ}(A) instead of v_{ℓ}(|A|).

There are two invariants t = t(k, l) and m = m(k, l) which are associated with the pair (k, l), cf. [Se 07, §4]. Recall their definitions :

1.1 Definition of t

Let z ∈ k_{s} be a primitive ℓ-th root of unity if l > 2 and a primitive 4-th root of unity if ℓ = 2. We put

t = [k(z) : k].

If ℓ > 2, t divides ℓ − 1. If ℓ = 2 or 3, then t = 1 or 2.

1.2 Definition of m

For ℓ > 2, m is the upper bound (possibly infinite) of the n’s such that k(z) contains the ℓ^{n}-th roots of unity. We have m ⩾ 1.

For ℓ = 2, m is the upper bound (possibly infinite) of the n’s such that k contains z(n) + z(n)^{−1} , where z(n) is a primitive 2^{n}-root of unity. We have m ⩾ 2. [The definition of m given in [Se 07, §4.2] looks different, but it is equivalent to the one here.] Remark. Knowing t and m amounts to knowing the image of the ℓ-th cyclotomic character Gal(k_{s}/k) → Z_{l}^{∗}, cf. [Se 07, §4].

1.3 Example : k = Q

Here, t takes its largest possible value, namely t = ℓ − 1 for ℓ > 2 and t = 2 for ℓ = 2. And m takes its smallest possible value, namely m = 1 for ℓ > 2 and m = 2 for ℓ = 2.

1.4 Example : k finite with q elements

If ℓ > 2, one has :

t = order of q in the multiplicative group F_{ℓ}^{∗} m = v_{ℓ}(q^{t} − 1) = v_{ℓ}(q^{l−1} − 1).

If ℓ = 2, one has :

t = order of q in (Z/4Z)^{∗} m = v_{2}(q^{2} − 1) − 1.

§2 Statement of the main theorem

Let K = k(X, Y), where X, Y are indeterminates, and let Cr(k) be the Cremona group of rank 2 over k, i.e. the group Aut_{k} K . Let ℓ be a prime number, distinct from char(k), and let t and m be the cyclotomic invariants defined above.

2.1 Notation

Define a number M(k, ℓ) ∈ {0, 1, 2, ..., ∞} as follows :

For ℓ = 2, M(k, ℓ) = 2m + 3.

For ℓ = 3, M(k, ℓ) = \{ \begin{matrix}4 & if t = m = 1 \\ 2m + 1 & otherwise. \end{matrix}

For ℓ > 3, M(k, ℓ) = \{ \begin{matrix}2m & if t = 1 & or 2 \\ m & if t = 3, 4 & or 6 \\ 0 & if t = 5 & or t > 6. \end{matrix}

2.2 The main theorem

Theorem 2.1.(i) Let A be a finite subgroup of Cr(k). Then v_{ℓ}(A) ⩽ M(k, ℓ).

(ii) Conversely, if n is any integer ⩾ 0 which is ⩽ M(k, ℓ) then Cr(k) contains a subgroup of order ℓ^{n}.

(In other words, M(k, ℓ) is the upper bound of the v_{ℓ}(A).)

The special case where A is cyclic of order ℓ gives :

Corollary 2.2 ([DI 08]). The following properties are equivalent :

a) Cr(k) contains an element of order ℓ

b) φ(t) ≤ 2, i.e. t = 1, 2, 3, 4 or 6.

Indeed, b) is equivalent to M(k, ℓ) > 0.

2.3 Small fields

Let us say that k is small if it has the following properties :

(2.3.1) m(k, ℓ) < ∞ for every ℓ ≠ char(k)
(2.3.2) t(k, ℓ) → ∞ when ℓ → ∞.

Proposition 2.3. A field which is finitely generated over Q or F_{p} is small.

Proof. The formulae given in §1.3 and §1.4 show that both F_{p} and Q are small. If k′/k is a finite extension, one has

[k′ : k].t(k′, ℓ) ⩾ t(k, ℓ) and m(k′, ℓ) ⩽ m(k, ℓ) + log_{ℓ}([k′ : k]),

which shows that k small ⇒ k′ small. If k′ is a regular extension of k , then

t(k′, ℓ) = t(k, ℓ) and m′(k′, ℓ) = m(k, ℓ),

which also shows that k small ⇒ k′ small. The proposition follows.

Assume now that k is small. We may then define an integer M(k) by the following formula

(2.3.3) M(k) = ∏_{ℓ} ℓ^{M(k,ℓ)},

where ℓ runs through the prime numbers distinct from char(k). The formula makes sense since M(k, ℓ) is finite for every ℓ and is 0 for every ℓ but a finite number. With this notation, Th. 2.1 can be reformulated as :

Theorem 2.4. If k is small, then the finite subgroups of Cr(k) of order prime to char(k) have bounded order, and the l.c.m. of their orders is the integer M(k) defined above.

Note that this applies in particular when k is finitely generated over its prime subfield.

2.4 Example : the case k = Q

By combining 1.3 and 2.1, one gets

M(Q, ℓ) = \{ \begin{matrix}7 & for & ℓ = 2, \\ 3 & for & ℓ = 3, \\ 1 & for & ℓ = 5, 7 \\ 0 & for & ℓ > 7. \end{matrix}

This can be summed up by :

Theorem 2.5. M(Q) = 2^{7}.3^{3}.5.7.

2.5 Example : the case of a finite field

Theorem 2.6. If k is a finite field with q elements, we have

M(k) = \{ \begin{matrix}3.(q^{4} − 1)(q^{6} − 1) & if q ≡ 4 or 7 (mod 9) \\ (q^{4} − 1)(q^{6} − 1) & otherwise. \end{matrix}

Proof. Denote by M′(k, ℓ) the ℓ-adic valuation of the right side of the formulae above.

If ℓ is not equal to 3, M′(k, ℓ) is equal to

v_{ℓ}(q^{4} − 1) + v_{ℓ}(q^{6} − 1)

and we have to check that M′(k, ℓ) is equal to M(k, ℓ).

Consider first the case ℓ = 2. It follows from the definition of m that v_{2}(q^{2} − 1) = m + 1, and hence v_{2}(q^{4} − 1) = m + 2 and v_{2}(q^{6} − 1) = m + 1.

This gives M′(k, ℓ) = 2m + 3 = M(k, ℓ).

If ℓ > 3, the invariant t is the smallest integer > 0 such that q^{t} = 1 (mod l). If t = 5 or t > 6, this shows that M′(k, ℓ) = 0.

If t = 3 or 6, q^{4} −1 is not divisible by ℓ and q^{6} −1 is divisible by ℓ; moreover, one has v_{ℓ}(q^{6} − 1) = m. This gives M′(k, ℓ) = m = M(k, ℓ). Similarly, when t = 4, the only factor divisible by ℓ is q^{4} − 1 and its ℓ-adic valuation is m. When t = 1 or 2, both factors are divisible by ℓ and their ℓ-adic valuation is m.

The argument for ℓ = 3 is similar : we have

v_{3}(q^{4} − 1) = m and v_{3}(q^{6} − 1) = m + 1.

The congruence q ≡ 4 or 7 (mod 9) means that t = m = 1.

For instance : M(F_{2}) = 3^{3}.5.7; M(F_{3}) = 2^{7}.5.7.13; M(F_{4}) = 3^{4}.5^{2}.7.13.17;

M(F_{5}) = 2^{7}.3^{3}.7.13.31; M(F_{7}) = 2^{9}.3^{4}.5^{2}.19.43.

2.6 Example : the p-adic field Q_{p}

For ℓ ≠ p, the t, m invariants of Q_{p} are the same as those of F_{ℓ} , and for ℓ = p they are the same as those of Q.

This shows that Q_{p} is “ small ”, and a simple computation gives

M(Q_{p}) = c(p).(p^{4} − 1)(p^{6} − 1),

with c(2) = 2^{7}; c(3) = 3^{3}; c(5) = 5; c(7) = 3.7; c(p) = 3 if p > 7 and p ≡ 4 or 7 (mod 9); c(p) = 1 otherwise.

For instance : M(Q_{2}) = 2^{7}.3^{3}.5.7; M(Q_{3}) = 2^{7}.3^{3}.5.7.13; M(Q_{5}) = 2^{7}.3^{3}.5.7.13.31;

M(Q_{7}) = 2^{9}.3^{4}.5^{2}.7.19.43; M(Q_{11}) = 2^{7}.3^{3}.5^{2}.7.19.37.61.

2.7 Remarks

1. The statement of Th.2.6 is reminiscent of the formula which gives the order of G(k), where G is a split semisimple group and |k| = q. In such a formula, the factors have the shape (q^{d} −1), where d is an invariant degree of the Weyl group, and the number of factors is equal to the rank of G. Here also the number of factors is equal to the rank of Cr, which is 2. The exponents 4 and 6 are less easy to interpret. In the proofs below, they occur as the maximal orders of the torsion elements of the “ Weyl group ” of Cr, which is GL_{2}(Z). See also §6.

2. Even though Th.2.6 is a very special case of Th.2.1, it contains almost as much information as the general case. More precisely, we could deduce Th.2.1.(i)

[which is the hard part] from Th.2.6 by the Minkowski method of reduction (mod p) explained in [Se 07, §6.5].

3. In the opposite direction, if we know Th.2.1.(i) for fields of characteristic 0 (in the slightly more precise form given in §4.1), we can get it for fields of characteristic p > 0 by lifting over the ring of Witt vectors ; this is possible : all the cohomological obstructions vanish.

4. For large fields, the invariant m can be ∞. If t is not 1, 2, 3, 4 or 6, Cor.2.2 tells us that Cr(k) is ℓ-torsion-free. But if t is one of these five numbers, the above theorems tell us nothing. Still, as in [Se 07, §14, Th.12 and Th.13] one can prove the following :

a) If t = 3, 4 or 6, then Cr(k) contains a subgroup isomorphic to Q_{ℓ}/Z_{ℓ} and does not contain Q_{ℓ}/Z_{ℓ} × Q_{ℓ}/Z_{ℓ}.

b) If t = 1 or 2, then Cr(k) contains a subgroup isomorphic to Q_{ℓ}/Z_{ℓ}×Q_{ℓ}/Z_{ℓ} and does not contain a product of three copies of Q_{ℓ}/Z_{ℓ}.

§3 Proof of Theorem 2.1.(ii)

We have to construct large ℓ-subgroups of Cr(k). It turns out that we only need two constructions, one for the very special case ℓ = 3, t = 1, m = 1, and one for all the other cases.

3.1 The special case ℓ = 3, t = 1, m = 1

We need to construct a subgroup of Cr(k) of order 3^{4}. To do so we use the Fermat cubic surface S given by the homogeneous equation

x^{3} + y^{3} + z^{3} + t^{3} = 0.

It is a smooth surface, since p ≠ 3. The fact that t = 1 means that k contains a primitive cubic root of unity. This implies that the 27 lines of S are defined over k, and hence S is k-rational : its function field is isomorphic to K = k(X, Y). Let A be the group of automorphisms of S generated by the two elements

(x, y, z, t) ↦ (rx, y, z, t) and (x, y, z, t) ↦ (y, z, x, t)

where r is a primitive 3-rd root of unity.

We have |A| = 3^{4} and A is a subgroup of Aut(S), hence a subgroup of Cr(k).

3.2 The generic case

Here is the general construction :

One starts with a 2-dimensional torus T over k , with an ℓ-group C acting faithfully on it. Let B be an ℓ-subgroup of T (k). Assume that B is stable under C, and let A be the semi-direct product A = B.C. If we make B act on the variety T by translations, we get an action of A, which is faithful. This gives an embedding of A in Aut(k(T)), where k(T) is the function field of T. By a theorem of Voskresinskii (see [Vo 98, §4.9]) k(T) is isomorphic to K = k(X, Y). We thus get an embedding of A in Cr(k). Note that B is toral, i.e. is contained in the k-rational points of a maximal torus of Cr.

It remains to explain how to choose T, B and C. We shall define T by giving the action of Γ_{k} = Gal(k_{s}/k) on its character group ; this amounts to giving an homomorphism Γ_{k} → GL_{2}(Z).

3.2.1 The case ℓ = 2

Let n be an integer ⩽ m. If z(n) is a primitive 2^{n}-root of unity, k contains z(n) + z(n)^{−1}. The field extension k(z(n))/k has degree 1 or 2, hence defines a character Γ_{k} → 1, −1. Let T_{1} be the 1-dimensional torus associated with this character. If k(z(n)) = k , T_{1} is the split torus G_{m} and we have T_{1}(k) = k^{∗}. If k(z(n)) is quadratic over k, T_{1}(k) is the subgroup of k(z(n))^{∗} made up of the elements of norm 1. In both cases, T_{1}(k) contains z(n). We now take for T the torus T_{1} × T_{1} and for B the subgroup of elements of T of order dividing 2^{n}. We have v_{2}(B) = 2n. We take for C the group of automorphisms generated by (x, y) ↦ (x^{−1}, y) and (x, y) ↦ (y, x) ; the group C is isomorphic to the dihedral group D_{4} ; its order is 8. We then have v_{2}(A) = v_{2}(B) + v_{2}(C) = 2n + 3, as wanted.

(Alternate construction : the group Cr_{1}(k) = PGL_{2}(k) contains a dihedral subgroup D of order 2^{n+1} ; by using the natural embedding of (Cr_{1}(k)×Cr_{1}(k)).2 in Cr(k) we obtain a subgroup of Cr(k) isomorphic to (D × D).2, hence of order 2^{2n+3}.)

3.2.2 The case ℓ > 2

We start similarly with an integer n ⩽ m. We may assume that the invariant t is equal to 1, 2, 3, 4 or 6 ; if not we could take A = 1. Call C_{t} the Galois group of k(z)/k , cf.§1. It is a cyclic group of order t. Choose an embedding of C_{t} in GL_{2}(Z), with the condition that, if t = 2, then the image of C_{t} is {1, −1}. The composition map

r : Γ_{k} → Gal(k(z)/k) = C_{t} → GL_{2}(Z)

defines a 2-dimensional torus T.

The group B is the subgroup T (k)[l^{n}] of T (k) made up of elements of order dividing l^{n} . We take C equal to 1, except when l = 3 where we choose it of order 3 (this is possible since t = 1 or 2 for ℓ = 3, and the group of k-automorphisms of T is isomorphic to GL_{2}(Z)). We thus have :

v_{ℓ}(A) = v_{ℓ}(B) if ℓ > 3 and v_{ℓ}(A) = 1 + v_{ℓ}(B) if ℓ = 3.

It remains to estimate v_{ℓ}(B). Namely :

(3.2.3) v_{ℓ}(B) = 2n if t = 1 or 2

This is clear if t = 1 because in that case T is a split torus of dimension 2, and k contains z(n).

If t = 2, then T = T_{1} × T_{1}, where T_{1} is associated with the quadratic character Γ_{k} → Gal(k(z)/k). We may identify T_{1}(k) with the elements of norm 1 of k(z), and this shows that z(n) is an element of T_{1}(k) of order 2^{n}. We thus get v_{ℓ}(B) = 2n.

(3.2.4) v_{ℓ}(|B|) ⩾ n if t = 3, 4 or 6

We use the description of T given in [Se 07, §5.3] : let L be the field k(z). It is a cyclic extension of k of degree t. Let s be a generator of C_{t} = Gal(L/k). Let T_{L} = R_{L/k}(G_{m}) be the torus ‘ multiplicative group of L ” ; we have dim T_{L} = t, and s acts on T_{L}. We have s^{t} − 1 = 0 in End(T_{L}). Let F (X) be the cyclotomic polynomial of index t, i.e.

\begin{matrix}F (X) = X^{2} + X + 1 & if t = 3 \\ F (X) = X^{2} + 1 & if t = 4 \\ F (X) = X^{2} − X + 1 & if t = 6. \end{matrix}

This polynomial divides X^{t} − 1 ; let G(X) be the quotient (X^{t} − 1)/F (X), and let u be the endomorphism of T_{1} defined by u = G(s). One checks (loc.cit.) that the image T of u : T_{1} → T_{1} is a 2-dimensional torus, and s defines an automorphism s_{T} of T of order t, satisfying the equation F (s_{T}) = 0. This shows that T is the same as the torus also called T above. Moreover, it is easy to check that the element z(n) of T_{1}(k) is sent by u into an element of T (k) of order l^{n} . This shows that v_{ℓ}(B) ⩾ n.

[When t = 3, we could have defined T as the kernel of the norm map N : T_{1} → G_{m}. There is a similar definition for t = 4, but the case t = 6 is less easy to describe concretely.]

This concludes the proof of the “existence part” of Th.2.1.

§4 Proof of Theorem 2.1.(i)

4.1 Generalization

In Th.2.1.(i), the hypothesis made on the ℓ-group A is that it is contained in Cr(k). This is equivalent to saying that A is contained in Aut(S), where S is a k-rational surface, cf. e.g. [DI 07,Lemma 6]. We now want to relax this hypothesis : we will merely assume that S is a surface which is “ geometrically rational ”, i.e. becomes rational over \overline{k}; for instance S can be any smooth cubic surface in P_{3}. In other words, we will be interested in field extensions L of k with the property :

(4.1.1) \overline{k} ⊗ L is \overline{k}-isomorphic to \overline{k}(X, Y).

We shall say that a group A has “ property Cr_{k} ” if it can be embedded in Aut(L), for some L having property (4.1.1). The bound given in Th.2.1.(i) is valid for such groups. More precisely :

Theorem 4.1. If a finite ℓ-group A has property Cr_{k} , then v_{ℓ}(A) ⩽ M(k, ℓ), where M(k, ℓ) is as in §2.1.

This is what we shall prove. Note that we may assume that k is perfect since replacing k by its perfect closure does not change the invariants t, m and M(k, l).

[As mentioned in §2.7, we could also assume that k is finite, or, if we preferred to, that char(k) = 0. Unfortunately, none of these reductions is really helpful.]

4.2 Reduction to special cases

We start from an ℓ-group A having property Cr_{k}. As explained above, this means that we can embed A in Aut(S), where S is a smooth projective ksurface, which is geometrically rational. Now, the basic tool is the “ minimal model theorem ” (proved in [DI 07, §2]) which allows us to assume that S is of one of the following two types :

a) (conic bundle case) There is a morphism f : S → C, where C is a smooth genus zero curve, such that the generic fiber of f is a smooth curve of genus 0. Moreover, A acts on C and f is compatible with that action.

b) (del Pezzo) S is a del Pezzo surface, i.e. its anticanonical class −K_{S} is ample.

In case b), the degree deg(S) is defined as K_{S}.K_{S} (self-intersection) ; one has 1 ⩽ deg(S) ⩽ 9.

We shall look successively at these different cases. In the second case, we shall use without further reference the standard properties of the del Pezzo surfaces ; one can find them for instance in [De 80], [Do 07], [DI 07], [Ko 96], [Ma 66] and [Ma 86].

Remark. In some of these references, the ground field is assumed to be of characteristic 0, but there is very little difference in characteristic p > 0 ; moreover, as pointed out above, the characteristic 0 case implies the characteristic p case, thanks to the fact that |A| is prime to char(k).

4.3 The conic bundle case

Let f : S → C be as in a) above, and let A_{o} be the subgroup of Aut(C) given by the action of A on C. The group Aut(C) is a k-form of PGL_{2}. By using (for instance) [Se 07, Th.5] we get :

v_{ℓ}(A_{o}) ⩽ \{ \begin{matrix}m + 1 & if & l = 2, \\ m & if & l > 2 and t = 1 or 2, \\ 0 & if & t > 2. \end{matrix}

Let B be the kernel of A → A_{o}. The group B is a subgroup of the group of automorphisms of the generic fiber of f. This fiber is a genus 0 curve over the function field k_{C} of C. Since k_{C} is a regular extension of k, the t and m invariants of k_{C} are the same as those of k. We then get for v_{ℓ}(B) the same bounds as for v_{ℓ}(A_{o}), and by adding up this gives :

v_{ℓ}(A) ⩽ \{ \begin{matrix}2m + 2 & if & ℓ = 2 \\ 2m & if & ℓ > 2 and t = 1 or 2 \\ 0 & if & t > 2. \end{matrix}

In each case, this gives a bound which is at most equal to the number M(k, ℓ) defined in §2.1.

4.4 The del Pezzo case : degree 9

Here S is \overline{k}-isomorphic to the projective plane P_{2} ; in other words, S is a Severi-Brauer variety of dimension 2. The group Aut S is an inner k-form of PGL_{3}. By using [Se 07, §6.2] one finds :

v_{ℓ}(A) ⩽ \{ \begin{matrix}2m + 1 & if & ℓ = 2 \\ 2m + 1 & if & ℓ = 3, t = 1 \\ ≤ m + 1 & if & ℓ = 3, t = 2 \\ ≤ 2m & if & ℓ > 3, t = 1 \\ ≤ m & if & ℓ > 3, t = 2 or 3 \\ = 0 & if & t > 3. \end{matrix}

Here again, these bounds are ⩽ M(k, ℓ).

4.5 The del Pezzo case : degree 8

This case splits into two subcases :

a) S is the blow up of P_{2} at one rational point. In that case A acts faithfully on P_{2} and we apply 4.4.

b) S is a smooth quadric of P_{3}. The connected component Aut^{o}(S) of Aut(S) has index 2. It is a k-form of PGL_{2} × PGL_{2}. If we denote by A_{o} the intersection of A with Aut^{o}(S), we obtain, by [Se 07, Th.5], the bounds :

v_{ℓ}(A_{0}) ⩽ \{ \begin{matrix}2m + 2 & if & ℓ = 2 \\ 2m & if & ℓ > 2 and t = 1 or 2 \\ m & if & t = 3, 4 or 6 \\ 0 & if & t = 5 or t > 6. \end{matrix}

Since v_{ℓ}(A) = v_{ℓ}(A_{o}) if ℓ > 2 and v_{ℓ}(A) ⩽ v_{ℓ}(A_{o}) + 1 if ℓ = 2, we obtain a bound for v_{ℓ}(A) which is ⩽ M(k, ℓ).

Remarks. 1) Note the case ℓ = 2, where the M(k, ℓ) bound 2m + 3 can be attained.

2) In the case t = 6, the bound v_{ℓ}(A_{o}) ⩽ m given above can be replaced by v_{ℓ}(A_{o}) = 0, but this is not important for what we are doing here.

4.6 The del Pezzo case : degree 7

This is a trivial case ; there are 3 exceptional curves on S (over \overline{k}), and only one of them meets the other two. It is thus stable under A, and by blowing it down, one is reduced to the degree 8 case. [This case does not occur if one insists, as in [DI 08], that the rank of Pic(S)^{A} be equal to 1.]

4.7 The del Pezzo case : degree 6

Here the surface S has 6 exceptional curves (over \overline{k}), and the corresponding graph L is an hexagon. There is a natural homomorphism

g : Aut(S) → Aut(L)

and its kernel T is a 2-dimensional torus. Put A_{o} = A ∩ T (k). The index of A_{o} in A is a divisor of 12. By [Se 07, Th.4], we have

v_{ℓ}(A_{o}) ⩽ \{ \begin{matrix}2m & if & t = 1 or 2 & (i.e. if φ(t) = 1) \\ m & if & t = 3, 4 or 6 & (i.e. if φ(t) = 2) \\ 0 & if & t = 5 or t > 6. \end{matrix}

Hence :

v_{ℓ}(A) ⩽ \{ \begin{matrix}2m + 2 & if & ℓ = 2 \\ 2m + 1 & if & ℓ = 3 \\ 2m & if & ℓ > 3 and t = 1 or 2 \\ m & if & t = 3, 4 or 6 \\ 0 & if & t = 5 or t > 6. \end{matrix}

These bounds are ⩽ M(k, ℓ).

Remarks. 1) Note the case t = 6, where the bound m can actually be attained.

2) In the case t = 4, the bound v_{ℓ}(A) ⩽ m given above can be replaced by v_{ℓ}(A) = 0.

4.8 The del Pezzo case : degree 5

As above, let L be the graph of the exceptional curves of S. Since deg(S) ⩽ 5, the natural map Aut(S) → Aut(L) is injective. We can thus identify A with its image A_{L} in Aut(L). In the case deg(S) = 5, Aut(L) is isomorphic to the symmetric group S_{5}. In particular we have

v_{ℓ}(A) ⩽ \{ \begin{matrix}3 & if & ℓ = 2 \\ 1 & if & ℓ = 3 or 5 \\ 0 & if & ℓ > 5, \end{matrix}

and we conclude as before.

4.9 The del Pezzo case : degree 4

This case is similar to the preceding one. Here Aut(L) is isomorphic to the group 2^{4}.S_{5} = Weyl(D_{5}) ; its order is 2^{7}.3.5. We get the same bounds as above, except for ℓ = 2 where we find v_{ℓ}(A) ⩽ 7, which is ⩽ M(k, 2) [recall that M(k, 2) = 2m + 3 and that m ⩾ 2 for ℓ = 2].

4.10 The del Pezzo case : degree 3

Here S is a smooth cubic surface, and A embeds in Weyl(E_{6}), a group of order 2^{7}.3^{4}.5. This gives a bound for v_{ℓ}(A) which gives what we want, except when ℓ = 3. In the case ℓ = 3, it gives v_{ℓ}(A) ⩽ 4, but T h.2.1 claims v_{ℓ}(A) ⩽ 3 unless k contains a primitive cubic root of unity. We thus have to prove the following lemma :

Lemma 4.2 - Assume that |A| = 3^{4} , that A acts faithfully on a smooth cubic surface S over k, and that char(k) ≠ 3. Then k contains a primitive cubic root of unity.

Proof. The structure of A is known since A is isomorphic to a 3-Sylow subgroup of Weyl(E_{6}). In particular the center Z(A) of A is cyclic of order 3 and is contained in the commutator group de A. Since A acts on S, it acts on the sections of the anticanonical sheaf of S; we get in this way a faithful linear representation r : A → GL_{4}(k). Over \overline{k}, r splits as r = r_{1} + r_{3} where r_{1} is 1dimensional and r_{3} is irreducible and 3-dimensional. If z is a non trivial element of Z(A), the eigenvalues of z are {1, r, r, r} where r is a primitive third root of unity. This shows that r belongs to k.

4.11 The del Pezzo case : degree 2

Here A embeds in Weyl(E_{7}), a group of order 2^{10}.3^{4}.5.7. This gives a bound for v(A), but this bound is not good enough. However, the surface S is a 2sheeted covering of P_{2} (the map S → P_{2} being the anticanonical map) and we get a homomorphism g : A → PGL_{3}(k) whose kernel has order 1 or 2. We then find the same bounds for v_{ℓ}(A) as in §4.2, except that, for ℓ = 2, the bound is 2m + 2 instead of 2m + 1.

4.12 The del Pezzo case : degree 1

We use the linear series | − 2K_{S}|. It gives a map g : S → P_{3} whose image is a quadratic cone Q, cf. e.g. [De 80, p.68]. This realizes S as a quadratic covering of Q. If B denotes the automorphism group of Q defined by A, we have v_{ℓ}(A) = v_{ℓ}(B) if ℓ > 2 and v_{ℓ}(A) ⩽ v_{ℓ}(B) + 1 if ℓ = 2. But B is isomorphic to a subgroup of k^{∗} × Aut(C), where C is a curve of genus 0. This implies

v_{ℓ}(B) ⩽ \{ \begin{matrix}m + m + 1 & if & ℓ = 2 \\ m + m & if & t = 1 \\ 0 + m & if & t = 2, l > 2 \\ 0 + 0 & if & t > 2. \end{matrix}

The corresponding bound for v_{ℓ}(A) is ⩽ M(k, ℓ).

This concludes the proof of Th.4.1 and hence of Th.2.1.

§5 Structure and conjugacy properties of ℓ-subgroups of Cr(k)

5.1 The ℓ-subgroups of Cr(k)

The main theorem (Th.2.1) only gives information on the order of an ℓsubgroup A of Cr(k), assuming as usual that ℓ ≠ char(k). As for the structure of A, we have :

Theorem 5.1. (i) If ℓ > 3, A is abelian of rank ⩽ 2 (i.e. can be generated by two elements).

(ii). If ℓ = 3 (resp. ℓ = 2) A contains an abelian normal subgroup of rank ⩽ 2 with index ⩽ 3 (resp. with index ⩽ 8).

Proof. Most of this is a consequence of the results of [DI 07] ; see also [Bl 06] and [Be07]. The only case which does not seem to be explicitly in [DI 07] is the case ℓ = 2, when A is contained in Aut(S), where S is a conic bundle. Suppose we are in that case and let f : S → C and A_{o}, B be as in 4.3, so that we have an exact sequence 1 → B → A → A_{o} → 1, with A_{o} ⊂ Aut(C), and B ⊂ Aut(F) where F is the generic fiber of f (which is a genus zero curve over the function field k(C) of C). We use the following lemma :

Lemma 5.2. Let a ∈ A and b ∈ B be such that a normalizes the cyclic group ⟨b⟩ generated by b. Then aba^{−1} is equal to b or to b^{−1}.

Proof of the lemma. Let n be the order of b. If n = 1 or 2, there is nothing to prove. Assume n > 2. By extending scalars, we may also assume that k contains the primitive n-th roots of unity. Since b is an automorphism of F of order n, it fixes two rational points of F which one can distinguish by the eigenvalue of b on their tangent space : one of them gives a primitive n-th root of unity z, and the other one gives z′ = z^{−1}. [Equivalently, b fixes two sections of f : S → C.] The pair (z, z′) is canonically associated with b. Hence the pair associated with aba^{−1} is also (z, z′). On the other hand, if aba^{−1} = b^{i} with i ∈ Z/nZ, then the pair associated to a^{i} is (z^{i}, z^{′i}). This shows that z^{i} is equal to either z or z^{−1}, hence i ≡ 1 or −1 (mod n). The result follows.

End of the proof of Theorem 5.1 in the case ℓ = 2. Since B is a finite 2-subgroup of a k(C)-form of PGL_{2}, it is either cyclic or dihedral. In both cases, it contains a characteristic subgroup B_{1} of index 1 or 2 which is cyclic. Similarly, A has a cyclic subgroup A_{1} which is of index 1 or 2. Let a ∈ A be such that its image in A_{o} generates A_{1}. If b is a generator of B_{1}, Lemma 5.2 shows that a^{2} commutes with b. Let ⟨b, a^{2}⟩ be the abelian subgroup of A generated by b and a^{2}. It is normal in A, and the inclusions ⟨b, a^{2}⟩ ⊂ ⟨b, a⟩ ⊂ B.⟨a⟩ ⊂ A show that its index in A is at most 8.

Remark. Similar arguments can be applied to prove a Jordan-style result on the finite subgroups of Cr(k), namely :

Theorem 5.3. There exists an integer J > 1, independent of the field k, such that every finite subgroup G of Cr(k), of order prime to char(k), contains an abelian normal subgroup A of rank ⩽ 2, whose index in G divides J.

The proof follows the same pattern : the conic bundle case is handled via Lemma 5.2 and the del Pezzo case via the fact that G has a subgroup of bounded index which is contained in a reductive group of rank ⩽ 2, so that one can apply the usual form of Jordan’s theorem to that group. As for the value of J, a crude computation shows that one can take J = 2^{10}.3^{4}.5^{2}.7 ; the exponents of 2 and 3 can be somewhat lowered, but those of 5 and 7 cannot since Cr(C) contains A_{5} × A_{5} and PSL_{2}(F_{7}).

5.2 The cases t = 3, 4, 6

More precise results on the structure of A depend on the value of the invariant t = t(k, ℓ). Recall that t = 1, 2, 3, 4 or 6 if A ≠ 1, cf. Cor.2.2. We shall only consider the cases t = 3, 4 or 6 which are the easiest. See [DI 08, §4] for a (more difficult) conjugation theorem which applies when t = 1 or 2. Recall (cf. §3.2) that A is said to be toral if there exists a 2-dimensional subtorus T of Cr (in the sense of [De 70]) such that A is contained in T (k). We have :

Theorem 5.4. Assume that t = 3, 4 or 6. Then :

(a) A is cyclic of order ℓ^{n} with n ⩽ m.

(b) A is toral, except possibly if |A| = 5.

(c) If A′ is a subgroup of Cr(k) of the same order as A, then A′ is conjugate to A in Cr(k), except possibly if |A| = 5.

Note that the hypothesis t = 3, 4 or 6 implies ℓ ⩾ 5. Moreover, if ℓ = 5, then t = 4 and, if ℓ = 7, then t = 3 or 6.

Proof of (a) and (b). We follow the same method as above, i.e. we view A as a subgroup of Aut(S), where S is either a conic bundle or a del Pezzo surface. The bounds given in §4.3 show that A = 1 if S is a conic bundle (this is why this case is easier than the case t = 1 or 2). Hence we may assume that S is a del Pezzo surface. Let d be its degree. We have an exact sequence :

1 → G(k) → Aut(S) → E → 1, where G = Aut(S)^{o} is a connected linear group of rank ⩽ 2 and E is a subgroup of a Weyl group W depending on d (e.g. W = Weyl(E_{8}) if d = 1).

Consider first the case ℓ > 7. The order of W is not divisible by ℓ; hence A is contained in G(k). Since A is commutative, there exists a maximal torus T of G such that A is contained in the normalizer N of T, cf. e.g. [Se 07, §3.3] ; since ℓ > 3, the order of N/T is prime to ℓ, hence A is contained in T (k) and this implies dim(T) ⩾ 2 by [Se 07, §4.1]. This proves (b), and (a) follows from Lemma 5.5 below.

Suppose now that ℓ = 5 or 7, and let n = v_{ℓ}(A). If n = 1 and ℓ = 5 , there is nothing to prove. If n = 1 and ℓ = 7, then (a) is obvious and (b) is proved in [DI 08, prop.3] (indeed Dolgachev and Iskovskikh prove (b) when v_{ℓ}(A) = 1, and they also prove (c) for ℓ = 7). We may thus assume that n > 1. If d ⩽ 5, then G = 1 and A embeds in E; but E does not contain any subgroup of order ℓ^{2} (see the tables in [DI 07] and [Bl 06]) ; hence this case does not occur. If d > 5, then the order of E is prime to ℓ, hence A is contained in G(k) and the proof above applies.

Proof of (c). By (b), we have A ⊂ T (k) and A′ ⊂ T′(k) where T and T′ are 2dimensional subtori of Cr. By Lemma 5.5 below, these tori are isomorphic ; by a standard argument (see e.g. [De 70, §6] this implies that T and T′ are conjugate by an element of Cr(k) ; moreover A (resp. A′) is the unique subgroup of order ℓ^{n} of T (k) (resp. of T′(k)). Hence A and A′ are conjugate in Cr(k).

Remark. The case |A| = 5 is indeed exceptional : there are examples of such A’s which are not toral, cf. [Be 07], [Bl 06], [DI 07].

5.3 A uniqueness result for 2-dimensional tori

We keep the assumption that t = 3, 4 or 6. We have seen in §3.2.2 that there exists a 2-dimensional k-torus T such that T (k) contains an element of order ℓ.

Lemma 5.5. (a) Such a torus is unique, up to k-isomorphism.

(b) If n ⩽ m = m(k, ℓ), then T (k)[ℓ^{n}] is cyclic of order ℓ^{n}.

Proof of (a). Let L = Hom_{k_{s}}(G_{m}, T) be the group of cocharacters of T. It is a free Z-module of rank 2, with an action of Γ_{k} = Gal(k_{s}/k). If we identify L with Z^{2}, this action gives a homomorphism r : Γ_{k} → GL_{2}(Z) which is well defined up to conjugation. Let G be the image of r. Since G is a finite subgroup of GL_{2}(Z), its order divides 24, and hence is prime to ℓ.

The Γ_{k}-module T (k_{s})[ℓ] of the ℓ-division points of T (k_{s}) is canonically isomorphic to L/ℓL ⊗ μ_{ℓ}, where μ_{ℓ} is the group of ℓ-th roots of unity in k_{s}. This means that L/ℓL contains a rank-1 submodule I which is isomorphic to the dual μ_{ℓ}^{∗} of μ_{ℓ}. The action of G on L/ℓL is semisimple since |G| is prime to ℓ. Hence there exists a rank-1 submodule J of L/ℓL such that L/ℓL = I ⨁ J. By a well-known lemma of Minkowski (see e.g. [Se 07, Lemma 1]), the action of G on L/ℓL is faithful. This shows that G is commutative. Moreover, the character giving the action of Γ_{k} on I has an image which is cyclic of order t. Since t = 3, 4 or 6, this shows that G contains an element of order 3 or 4. One checks that these properties imply G ⊂ SL_{2}(Z) i.e. det(r) = 1, hence the Γ_{k}-modules I and J are dual of each other, i.e. J ≃ μ_{ℓ}. We thus have L/ℓL ≃ μ_{l} ⊕ μ_{ℓ}^{∗} . We may then identify r with the homomorphism Γ_{k} → C_{t} → GL_{2}(Z), where C_{t} is the Galois group of k(μ_{ℓ}/k) and C_{t} → GL_{2}(Z) is an inclusion. Since any two such inclusions only differ by an inner automorphism of GL_{2}(Z), this shows that the Γ_{k}-module L is unique, up to isomorphism ; hence the same is true for T. Proof of (b). Assertion (b) follows from the description of T given in §3.2.2. It can also be checked by writing explicitly the Γ_{k}-module L/ℓ^{n}L; when n ⩽ m this module is isomorphic to the direct sum of μ_{ℓ^{n}} and its dual.

Remarks.

1). If n > m we have T (k)[ℓ^{n}] = T (k)[ℓ^{m}]. This can be seen, either by a direct computation of ℓ-adic representations, or by looking at §3.2.2

2) When t = 1 or 2, it is natural to ask for a 2-dimensional torus T such that T (k) contains Z/ℓZ ⊕ Z/ℓZ. Such a torus exists, as we have seen in §3.2. If ℓ > 2, it is unique, up to isomorphism. There is a similar result for ℓ = 2, if one asks not merely that T (k) contains Z/2Z × Z/2Z but that it contains Z/4Z × Z/4Z.

§6 The Cremona groups of rank > 2

For any r > 0 the Cremona group Cr_{r}(k) of rank r is defined as the group Aut_{k} k(T_{1}, ..., T_{r}) where (T_{1}, ..., T_{r}) are r indeterminates. When r > 2 not much seems to be known on the finite subgroups of Cr_{r}(k), even in the classical case k = C. For instance : 6.0. Does there exist a finite group which is not embeddable in Cr_{3}(C) ?

This looks very likely, but I do not see how to prove it. Still, it is natural to ask for much more, e.g. :

6.1 (Jordan bound, cf. Th.5.5). Does there exist an integer N(r) > 0 , depending only on r, such that, for every finite subgroup G of Cr_{r}(k) of order prime to char(k), there exists an abelian normal subgroup A of G, of rank ⩽ r, whose index divides N(r) ?

Note that this would imply that, for ℓ large enough (depending on r), every finite ℓ-subgroup of Cr_{r}(k) is abelian of rank ⩽ r. 6.2 (cf. [Se 07, §6.9] ). Is it true that r ⩾ φ(t) if Cr_{r}(k) contains an element of order ℓ? 6.3. Let G ⊂ Cr_{r}(k) be as in 6.1, and assume that k is small (cf. §2.3).Is it true that |G| is bounded by a constant depending only on r and k?

If the answer to 6.3 is “ yes ” we may define M_{r}(k) as the l.c.m. of all such |G|’s, and ask for an estimate of M_{r}(k). For instance, in the case r = 3 :

6.4. Is it true that M_{3}(k) is equal to M_{1}(k).M_{2}(k) ?

If k is finite with q elements, this means (cf. §2.5) :

6.5. Is it true that

M_{3}(k) = \{ \begin{matrix}3.(q^{2} − 1)(q^{4} − 1)(q^{6} − 1) & if q ≡ 4 or 7 (mod 9) \\ (q^{2} − 1)(q^{4} − 1)(q^{6} − 1) & otherwise ? \end{matrix}

For larger r’s the polynomial (X^{2} − 1)(X^{4} − 1)(X^{6} − 1) of 6.5 should be replaced by the polynomial P_{r}(X) defined by the formula

P_{r}(X) = ∏_{d} Φ_{d}(X)^{[r/φ(d)]} ,

where Φ_{d}(X) is the d-th cyclotomic polynomial (Φ_{1}(X) = X − 1, Φ_{2}(X) = X + 1, Φ_{3}(X) = X^{2} + X + 1, Φ_{4}(X) = X^{2} + 1, ...).

Examples. P_{4}(X) = (X^{6}−1)(X^{8}−1)(X^{10}−1)(X^{12}−1); P_{5}(X) = (X^{2}−1)P_{4}(X).

With this notation, the natural question to ask seems to be : 6.6. Is it true that there exists an integer c(r) > 0 such that M_{r}(F_{q}) divides c(r).P_{r}(q) for every q?

Unfortunately, I do not see any way to attack these questions ; the method used for rank 2 is based on the very explicit knowledge of the “minimal models”, and this is not available for higher ranks. Other methods are needed.

Acknowlegment. I wish to thank A.Beauville for a series of e-mails in 2003−2005 which helped me to correct the naive ideas I had on the Cremona group.

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