Fields as endomorphism rings (m-parameter argument)

In this section, we give an algorithm (the so called m-parameter argument) that con-structs all r.i.m.f. or s.i.m.f. supergroupsGof an irreducible matrix groupU if End(U) is a field. In particular, this includes irreducible cyclic matrix groupsU.

By the previous section, the problem is to reduce the number of forms that one has to consider. To do so, we need all possible prime divisors of |G|. If we have no other assumptions on G, we can always fall back on the Minkowski bound:

Lemma 2.2.9 (Minkowski’s bound, [Min87]) The least common multiple of the orders of all finite subgroups of GL_n(Q) is given by

Y

p

pb(p−1)ⁿ c⁺bp(p−1)ⁿ c^+j_p2(p−1)^{n k+...}

where the product is taken over all primes p≤n+ 1.

Further, the m-parameter argument needs a set of primes ˜Π(End(U),|G|) depending on End(U) and |G| as follows:

Definition 2.2.10 Let K be an algebraic number field.

(a) Letσ_i: K →R (1≤i≤d) be the real embeddings of K. If we fix the order of the σ_i, we get a group homomorphism

s: K^∗ →F^m2 , x7→(x₁, . . . , x_d) wherex_i =

(0 if σ_i(x)>0, 1 if σ_i(x)<0 . (b) Fork ∈Z define Π(k) to be the set of all primes dividing k.

(c) We define a finite set of primes Π(K) such that

(1) In each class of Cl(Z^K) there exists an integral ideal which contains Q

p∈Π(K)

p^ap with some a_p ∈N0.

(2) The groups(K^∗) is generated bys(x₁), . . . , s(x_`) for somex_i ∈ZKsatisfying Π(Nr_K/Q(x_i))⊆Π(K).

(d) For k ∈ Z set ˜Π(K, k) := Π(k)∪ S

F≤K

Π(F) where the union is taken over all subfieldsF of K. Note that this set is not unique.

We follow [Neb95, Satz III.2, p. 16] to give an algorithm which constructs all r.i.m.f.

or s.i.m.f. supergroups of a given irreducible matrix group of the same dimension.

Theorem 2.2.11 Let G < GL_n(Q) be finite and irreducible. Let L be a ZG-lattice.

Assume that C := C_Q^n×n(G) is either commutative or a positive definite quater-nion algebra. Then there exists a F ∈ F_>0(G) such that F is primitive on L and Π(det(L, F))⊆Π(K)∪Π(|G|) where K denotes the maximal real subfield of Z(C).

Proof: IfC is commutative, a proof is given in [Neb95, Satz III.2, p. 16]. So we may assume that C is a positive definite quaternion algebra. Choose any F ∈ F_>0(G) which is primitive on L. Suppose that there exists some prime p /∈ Π(K) such that p | det(L, F) but p - |G|. It suffices to show that there exists some c ∈ K such that cF is integral on L and the primes dividing det(L, cF) are contained in (Π(|G| · det(L, F))∪Π(K)))\ {p}.

p is maximal in (G)p (see [Rei03, Theorems 41.1 and 41.7]). Therefore EndΛp(Lp) is maximal in Cp and p is not ramified in Cp. Hence Cp'L`

By property (1) of the definition of Π(K), there exists some ZK-ideal a_i whose norm is only divisible by primes in Π(K) and some y_i ∈ K such that p_i·a_i = y_iZK. Then p_i = hp, y_ii since this identity holds locally everywhere. In particular y_i ∈ Z^K and Π(Nr_K/Q(y_i)) ⊆ {p} ∪Π(K). By property (2) of the definition of Π(K), there exists some x∈K such that Nr_K/Q(x)∈Π(K) and y:=x·Q`

i=1y^−k_i ⁱ is totally positive.

LetF⁰ :=y·F. Thenyiεi and pεi are both primitive elements ofKpi. Hence (Lp, F⁰) is self-dual. It might happen thatF⁰ is no longer integral on L. But then there exists some k ∈N such that kF⁰ is primitive on L and the primes dividing k are divisors of

|G|.

From this theorem we finally obtain

Corollary 2.2.12 (m-parameter argument, [Neb95, Korollar III.3, p. 17]) Let U ≤ G < GL_n(Q) be finite subgroups such that C := End(U) is a either a field or a positive definite quaternion algebra. Suppose L ∈ Z(G) ⊆ Z(U). Then there exists some F ∈ F_>0(G) that is primitive on L with Π(det(L, F))⊆ Π(K,˜ |G|) where K denotes the maximal real subfield of the center of C.

Proof: The maximal totally real subfield K⁰ of the center of End(G) is contained in K. By the theorem above, there exists some F ∈ F_>0(G) such that F is primitive on

L and Π(det(L, F))⊆Π(K⁰,|G|)⊆Π(K,˜ |G|).

The following rather technical remark shows how this corollary will be used later.

Remark 2.2.13 LetU <GL_m(Q) be finite such thatC := End(U) is a field. Denote byK the maximal totally real subfield ofC. Further suppose thatL₁, . . . , L_srepresent the isomorphism classes of U-invariant lattices. Finally fix F_i ∈ F_>0(U) and letR_i = End_ZU(L_i)⊆ZC (in most cases R_i =ZC and there exists at least oneiwhere equality holds). Denote by R⁺_i :=R_i∩K. The following algorithm finds (up to conjugacy) all finite supergroups G of U of order dividing a given `∈N:

For 1≤i≤s let

Ni :={x∈N_GLm(Q)(U)|Lix=Li and xFix^trF_i⁻¹ ∈R^∗_i} and

Pi :={aR⁺_i |a∈K>0, (Li, aFi) is normalized and Π(det(Li, aFi))⊆Π(K, `)}˜ . Then the group Ni acts on C,K and Ri via conjugation. Moreover, Pi consists of full orbits under this action. Let S_i be a set of representatives of these orbits.

Finally let U_i be a coset of (R⁺_i )^∗_>0/Nr_C/K(R_i^∗) and let

S :={(Li, uaFi)|u∈Ui, aR_i⁺∈Si, a∈K>0,1≤i≤s}.

Then every finite supergroup of U of order dividing ` is conjugate in N_GLm(Q)(U) to some group that fixes one of the lattices in the finite setS.

In particular, the r.i.m.f. supergroups of U are elements of {Aut(L, F) |(L, F) ∈ S}

and the s.i.m.f. supergroups ofU are elements of{Aut_Kj(L, F)|(L, F)∈ S,1≤j ≤r}

where K₁, . . . , K_r denote the minimal totally complex subfields ofC.

Proof: Let G < GL_m(Q) be a finite supergroup of U with |G| dividing `. By Corol-lary 2.2.12, the group Gfixes some (L<sup>0</sup>, F<sup>0</sup>)∈ Z(U)× F<sub>>0</sub>(U) such that F<sup>0</sup> is integral onL<sup>0</sup> and Π(det(L<sup>0</sup>, F<sup>0</sup>))⊆Π(K, `). Applying the process described in Definition 2.2.4˜ yields a normalized lattice ( ˜L⁰,F˜⁰) with Π(det( ˜L⁰,F˜⁰))⊆Π(K, `).˜

Now ˜L⁰ = Lic for some 1 ≤ i ≤ s and c ∈ C. After replacing G by G^c−1, G fixes (L_i, F) where F := cF˜⁰c^tr. So there exists some aR⁺_i ∈ P_i such that F = aF_i (and thus a ∈ K_>0). By definition, there exists some x ∈ N_i such that a^xR⁺_i ∈ S_i. After replacing G by G^x−1, it fixes Lix⁻¹ = Li and xaFix^tr =a^x−1(xFix^trF_i⁻¹)Fi = ˜aFi for some ˜a ∈ K_>0 with aR⁺_i = ˜aR⁺_i . Now a or ˜a are defined by the ideal aR⁺_i only up to some element of (R⁺_i )^∗_>0. For y ∈ R^∗_i it follows from Lemma 2.1.9 that G^y−1 fixes (Liy⁻¹, y˜aFiy^tr) = (Li,Nr_C/K(y)˜aFi).

So we have shown that G is conjugate (in N_GLm(Q)(U)) to some group that fixes a lattice in the set S.

The result now follows, if we can show thatS is finite. The setUi is finite by Dirichlet’s unit theorem (note that if K 6=C then (R⁺_i )² ≤Nr_C/K(R^∗_i)). The number of isomor-phism classes of ZU-invariant lattices is finite by the Jordan-Zassenhaus theorem.

So it remains to prove that P_i is finite: FixbR⁺_i ∈P_i and let J_i := Ann_R⁺

i (L^#,bF_i ⁱ/L_i).

If x ∈ K_>0 such that xbF_i is integral on L_i, then L_ixJ_i ⊆ L^#,bF_i ⁱJ_i ⊆ L_i. Thus

x∈J⁻¹.

To simplify the definition of the N_i in the previous remark, one can use the following Remark 2.2.14 Assume the situation of the previous remark.

(a) If F_i is integral on L_i and det(L_i, F_i) = 1 then

{F ∈ F>0(U)|(Li, F) is integral}={cFi |c∈Ri∩K>0}. (b) If {F ∈ F_>0(U)|(L_i, F) is integral}={cF_i |c∈R_i∩K_>0} then

Ni =N_GLm(Q)(U)∩GL(L) = {x∈N_GLm(Q)(U)|Lix=Li}. Proof:

(a) Suppose c ∈ K_>0 such that cF_i is integral on L_i. Then xcF_iy^tr ∈ Z for all x, y ∈L_i. HenceL_ic⊆L^#,F_i ⁱ =L_i and therefore c∈R_i.

(b) Let x ∈N_GLm(Q)(U)∩GL(L). Then xF_ix^tr ∈ F_>0(U). Hence there exists some c ∈ K_>0 such that xF_ix^tr = cF_i. Now xF x^tr is integral on L_ix⁻¹ = L_i. This showsc∈R⁺_i . From det(c) = 1 it follows that Nr_K/Q(c) = 1 and thusc∈(R⁺_i )^∗.

This proves x∈N_i.

Note that there does not always exist some F_i such that the condition of (b) holds.