The Margulis–Platonov conjecture over number fields

1 Introduction

Let k be a number field and let G be an absolutely almost simple simply connected algebraic k-group. For a place v of k, write k_v for the completion. The local group G(k_v) has its usual topology. Let

and let \delta_A:G(k)\to H_A be the diagonal homomorphism. The set A is finite. The product H_A is given its product topology; if A is empty, it is the trivial group.

The Margulis–Platonov conjecture asserts that the noncentral normal subgroups of G(k) are accounted for by this finite collection of compact local groups. Normal subgroups in this assertion are abstract: no topology or finite-generation condition is imposed on them. We prove the conjecture in this form.

Thus the result resolves the Margulis–Platonov conjecture positively over number fields, including globally anisotropic groups. In the case A=\varnothing, the conclusion is N=G(k), not merely that N has finite index or is dense in a local topology.

1.1 Background and significance

The problem joins two kinds of rigidity. At isotropic completions, generation by unipotent subgroups and simplicity modulo the center leave little room for normal quotients. At anisotropic nonarchimedean completions, compactness allows finite quotients and their open kernels. The assertion is that there is no additional abstract normal-subgroup obstruction invisible to these local factors.

The conjecture grew from Kneser’s quaternionic simplicity question and Platonov’s local-to-global simplicity proposal. Margulis formulated the description by open subgroups of the anisotropic finite local product [20, Russian original, Section 2.4.8]. Strong approximation and Margulis’s normal subgroup theorem supply the arithmetic framework and finite-index reduction [25, 21]. The isotropic case follows from the Kneser–Tits theorem over global fields; Gille’s account includes the final outer-E_6 cases, using work of Garibaldi and Chernousov–Timoshenko [14, Theorem 8.1]. For anisotropic groups, Chernousov proved projective simplicity of the rational points when the absolute rank is at least two and the group splits over a quadratic extension [5, Theorem 1]. Tomanov and Sury established substantial classical and low-index unitary cases [41, 40].

For inner type A, the arithmetic work of Platonov–Rapinchuk and Raghunathan [24, 28] preceded the reduction of Rapinchuk–Potapchik to finite simple quotients of the multiplicative group of a division algebra [31]. Segev obtained a commuting-graph obstruction, and Segev–Seitz verified the required finite-simple-group properties, completing this case [35, 36]. Rapinchuk–Segev–Seitz later proved that every finite quotient of the multiplicative group of a finite-dimensional division algebra is solvable [32]; Rapinchuk gave a shorter proof of the inner-A case using two-generation of finite simple groups [29]. The established cases are surveyed in [27, Sections 3.4–3.5]. They include isotropic groups and all inner forms of type A, as well as the other types outside the remaining anisotropic outer A, trialitarian D_4, and E_6 cases. We use that reduction in Theorem 2.1 and supply the remaining arguments here. In particular, the previously established inner-type-A theorem is available over every number field that occurs in a restriction-of-scalars subgroup.

The conjecture also enters the congruence subgroup problem: it is a normal-subgroup input to several descriptions of congruence kernels. That relationship does not justify using a theorem which assumes the conjecture to prove it. Our central-extension argument instead uses the unconditional finite-nonarchimedean injectivity in the metaplectic-kernel computation of Prasad and Rapinchuk [26, Proposition 2.6]. The distinction between metaplectic vanishing and congruence-kernel conclusions under the normal-subgroup hypothesis is also explicit in [30, Section 2].

1.2 A positive-density congruence consequence

Theorem 1.1 supplies the Margulis–Platonov hypothesis in Rapinchuk’s positive-density theorem, giving the following consequence.

For an inner form of the split group, M=k, and the density condition is positive Dirichlet density of the nonarchimedean places in S. The corollary is restricted to the stated positive-density range: it does not address Serre’s finite-S conjecture, arbitrary infinite S, or centrality or finiteness of congruence kernels outside that range.

1.3 The finite obstruction

We first prove that a noncentral normal subgroup N has finite index. Choose a positive integer e that annihilates the finite group G(k)/N, and let

Here G(k)^e is the subgroup generated by powers, not just the set of powers. The group P_0 is open and normal in H_A. It remains to show that the finite group V_0/R is trivial. This is the distinction between the desired abstract equality and the closure statement furnished by approximation.

Section 2 develops two tools for this finite obstruction. First, it constructs representatives of a fixed coset modulo R satisfying specified local openings and avoiding a prescribed codimension-two reduction locus at almost all other places. The parameters are products of conjugates of torus powers. Scalar invariance permits an affine sieve while retaining control at the omitted approximation place; possible single poles are treated explicitly. Second, a signed permutation lattice calculation gives the Hasse principle and weak approximation for the norm torus needed to prescribe commuting elements with exact determinants.

1.4 Odd-degree division algebras

The first unresolved case is a special unitary group S=\SU(D,*), where D is an odd-degree central division algebra over a quadratic extension L/k and * is a unitary involution. Let U=\U(D,*) be its full unitary group. Section 3 proves that a nontrivial finite defect for S(k) yields one of two objects on U(k): a finite abelian character nonzero on S(k), or a homomorphism to a finite nonabelian simple group which is surjective on the defect preimage V_0. The commuting approximation and central-extension arguments are needed to obtain this dichotomy on the full unitary group. The class-preserving extension step adapts the inner-type argument of Rapinchuk–Potapchik [31, Section 2]; here the commuting approximation must also prescribe determinants. A quotient of a unitary subgroup need not extend to the multiplicative group of its algebra.

Section 4 rules out the first object. An algebraic difference function on D has enough exact elementary invariances to be constant on the required orbits. The exact additive span of U(k) in D, together with the known inner-type-A theorem and a real-signature interpolation argument, forces any finite abelian character to vanish on S(k).

Sections 5 and 6 rule out the second object. A finite simple quotient assigns a color to each rational unitary element. The distribution of these colors is not assumed continuous. We construct a translation-invariant probability law on their finite patterns whose individual color distributions are uniform. Fractional unitary transformations turn additive recurrence into a centralizer double-coset condition. A finite-simple-group lemma, stated in Section 5 and proved in Section 6, excludes that condition. Elementary transformations then force an impossible invariance under all left color translations.

The construction of the uniform-marginal law is an important part of the proof: additive averaging by itself need not preserve all colors. Conformal weights on rational points of a flag variety use Langlands’s Eisenstein-series convergence theorem [18]. Finite-volume harmonicity and Howe–Moore matrix-coefficient decay [15, 6] then provide the required marginals before additive averaging is performed. The later positive-density argument uses the multidimensional theorem of Furstenberg–Katznelson [12]. The uniform-marginal construction, the coset-preserving sieve, and the explicit norm-torus calculation are developed before the final case reductions so that their hypotheses and conclusions remain visible.

1.5 Completion of the proof

Section 7 proves the other outer-type-A cases by induction on reduced degree, simultaneously over all number fields. The ternary split-algebra case uses quadratic norm equations on a Châtelet surface. Even degree is reduced by a quaternion factorization and a smaller unitary centralizer, with a separate degree-four determinant correction. The approximation construction controls the finite places of the subgroups as they vary.

Section 8 treats D_4 by finding, in each relevant rational coset, a regular element whose torus admits a rational inverter. It is a product of two involutions, each already controlled inside a type-A_2 subgroup. Section 9 reduces a generic E_6 element to a simply connected D_4 subgroup and the cover of a root-involution centralizer of type A_1A_5. The last step is a torsor under a connected simply connected group, so its local factorizations can be globalized. The proof ends by returning to the original abstract subgroup N.

2 Finite quotients and approximation

The first objective is to isolate what local approximation does not already prove. Every noncentral normal subgroup has finite index, but its closure at the anisotropic finite places need not a priori recover the subgroup. We express this possible discrepancy as a finite quotient. We then develop two tools for eliminating it: an approximation procedure that retains the abstract rational coset, and a Hasse principle with weak approximation for a particular norm torus.

2.1 Conventions and established inputs

Throughout, k is a number field, V_k is its set of places, and G is an absolutely almost simple simply connected k-group. Write

The diagonal homomorphism to this product is denoted by \delta. Thus G_A=H_A and \delta=\delta_A in the notation of the introduction. An empty product is the trivial group. The set A is finite: outside finitely many places the group has a reductive model and is unramified and quasi-split, hence has positive relative rank. At a nonarchimedean place an absolutely simple anisotropic group has type A.

For a finite set of places B, a subscript B on a rational element denotes its tuple of local images. Products over finitely many places have their ordinary product topology. In an adelic product we use the restricted product with respect to fixed integral compact open subgroups outside a finite set. Weak approximation means density at finitely many completions; strong approximation means density in the specified restricted adelic product.

For an integer e\geq 1, the notation

always denotes the abstract subgroup generated by the powers. No closure is included in this definition. Degrees of central simple algebras are reduced degrees; for example, \SL_1(M_r(\Delta)) has type A_{r\deg\Delta-1}. Restrictions of scalars will be treated over their defining extension fields when applying an induction or an approximation theorem.

We use the strong approximation theorem for simply connected absolutely almost simple groups: if G(k_w) is noncompact, then G(k) is dense in the adelic product away from w [25, Theorem 7.12]. We also use the lattice theorem for semisimple S-arithmetic groups [25, Theorem 5.7(1)], the normal subgroup theorem for irreducible S-arithmetic lattices of total rank at least two, and the local Kneser–Tits and simplicity theorems. In the last assertion, for a nonarchimedean v with positive rank, G(k_v) is perfect and G(k_v)/Z(G(k_v)) is abstractly simple. These results are used in their number-field forms [25, Chapters VII and IX][21, Chapter VIII, Theorem A].

We refer to the assertion of Theorem 1.1 for a particular group over a particular number field as (MP). We require the following established cases.

The formulation, including the results for multiplicative groups of division algebras that enter the inner type A case, is recalled in [27, Sections 3.4–3.5]. For the isotropic completion see [14, Theorem 8.1]; for inner type A see [31, 35, 36], with the subsequent strengthening in [32]. Thus, when we address one of the remaining types, we may assume that the group is anisotropic over its ground field.

We will also use the following form of the Hasse principle. A torsor under a k-group H means a principal homogeneous space for H over k.

The local and global vanishing assertions are the Kneser–Harder–Chernousov Hasse principle; we use the standard number-field statements in [25, Theorems 6.4 and 6.6]. For the last assertion, a rational point identifies X with H. Decompose H as a product of restrictions of scalars of simply connected absolutely almost simple groups. For each factor, choose an auxiliary split finite place outside the given approximation set and apply strong approximation away from it. This gives the stated weak approximation, also for products and restrictions of scalars.

We use the usual local and global theory of central simple algebras and involutions, including local Brauer invariants, the embedding criterion for fields, the Hasse–Schilling norm theorem, and the classification of hermitian forms [25, 33, 16]. When an etale algebra is a product of fields, norm and local arguments are understood factor by factor. All fields have characteristic zero unless they are explicitly residue fields. In root computations the roots of a simply laced system have squared length two. A superscript 1 on a multiplicative group denotes the kernel of the indicated norm.

2.2 The finite quotient not detected by closure

The following consequence records exactly how an already established case of (MP) will be used for subgroups. In particular it does not require a finite homomorphism to be continuous in advance.

2.3 An affine avoidance lemma

The closure calculation permits finitely many local conditions. The next argument supplies an additional condition at almost all finite places. It will be applied to an affine space of parameters, not to integral points of a torus. Codimension-two avoidance in affine space is an instance of arithmetic purity of strong approximation; see [4, Proposition 3.6] and [43, arXiv version, Lemma 1.1]. We give the fixed-direction version with an unbounded parameter at the omitted place needed below.

2.4 Approximation by abstract power factors

We now use anisotropy to make the unrestricted direction in Lemma 2.5 harmless at v_0. Assume G is k-anisotropic. Fix a maximal k-torus T_0\subset G, a finite Galois extension P/k splitting T_0 and any fixed auxiliary data, and a regular cocharacter \lambda:\bbG_{m,P}\to T_{0,P}. The map

is defined over k. Its restriction to the diagonal \bbG_m has cocharacter \sum_{\sigma\in\Gal(P/k)}\lambda^\sigma. This is a k-cocharacter and hence is zero by anisotropy. In particular

For x\in G(k), or for x in a fixed rational algebraic coset X=x_*G inside a larger linear algebraic group, consider

Its values lie in the same right coset xR, with R=G(k)^e. Its parameter domain is the open torus (\Res_{P/k}\bbG_m)^r in the affine k-space E=(\Res_{P/k}\bbA^1)^r. Over P, its boundary consists of the hyperplanes a_{j,\sigma}=0, one for each absolute coordinate.

We call a reduction a single-pole reduction if exactly one of these absolute coordinates vanishes. Away from the other hyperplanes, Equation 2.3 then takes the form

where the side factors are regular group- or coset-valued functions. At a good finite place with this reduction, Frobenius fixes the unique vanishing coordinate. Its action on the embeddings of the Galois extension P/k is regular. Thus the place splits completely in P, t has positive valuation, and the side factors are integral. The local analysis of Equation 2.4 may therefore be carried out in split coordinates.

Here is the precise criterion used in later sections. Its hypothesis about the single-pole configurations is substantive and will be verified in each application. A consecutive list of factors in Equation 2.3 is called a basic list if its product map is dominant and has surjective differential somewhere.

The proposition allows poles, but only in a controlled form at places that split completely in P. In particular, it makes no assertion that the torus parameters can be integral units everywhere. The later applications will specify D, prove the nonemptiness of every single-pole test, and check its local consequence.

2.5 Cohomology detected on cyclic subgroups

Our final preliminary tool solves simultaneous norm equations with prescribed local approximations. We first separate the arithmetic duality input from the finite-group calculation.

For a discrete Galois module M, write

The notation \Sha_\omega^i(k,M) means classes whose localizations vanish at all but finitely many places. For a group H, the degree-one notation has its pointed-set meaning. Galois cohomology and its cochains are continuous throughout. A lattice is a finitely generated free abelian group with a continuous Galois action, which necessarily has finite image.

To calculate this kernel, it is convenient to use extensions by lattices. The following lemma will also apply when the finite group is not a direct product.

2.6 The simultaneous norm torus

We have obtained two distinct kinds of control. The approximation procedure chooses a representative inside a fixed abstract coset while controlling its local centralizers. The norm-torus lemma then solves the simultaneous norm equations in those centralizers. The next section uses this combination to construct commuting rational elements with specified determinants.

3 Finite quotients of an odd-degree unitary group

We now consider the first remaining family: special unitary groups defined by division algebras of odd reduced degree. The finite defect constructed in Proposition 2.3 is a quotient of a finite-index normal subgroup of the special unitary group. The purpose of this section is to use a nontrivial simple quotient of that defect to construct a homomorphism defined on the full unitary group. The abelian and nonabelian alternatives will then be excluded separately.

Let L/k be a quadratic extension of number fields, let D be a central simple L-algebra of reduced degree n, and let * be a unitary involution on D. Write

where the bar is the nontrivial automorphism of L/k. The determinant homomorphism on U is the reduced norm:

Fix i\in L^\times with \bar i=-i. On a dense open subset of the k-vector space of Hermitian elements, the Cayley map

is a birational parametrization of U. In particular, U(k) satisfies weak approximation: first perturb finitely many local targets into this open chart and then approximate their inverse Cayley coordinates.

We recall the relevant local descriptions of these groups [25, 33, 16]. At a nonsplit nonarchimedean place of L/k, the algebra D splits. Indeed, the existence of the unitary involution makes its Brauer corestriction zero [16, Theorem 3.1(2)], and corestriction for a local field extension preserves the local Brauer invariant. A Hermitian form of dimension at least three over a quadratic extension of nonarchimedean local fields is isotropic. Thus, when n\geq3, the finite places where S has rank zero are split places of L/k. At each such place v, choose one of the two components of D\otimes_k k_v and denote it by \mathcal D_v. It is a central division algebra of degree n over k_v, the involution exchanges the two components, and

Under this identification, the local determinant is the reduced norm on \mathcal D_v.

For the rest of this section assume that D is a division algebra and that n\geq3 is odd. Then S is k-anisotropic. Let A be its finite set of anisotropic nonarchimedean places and use the notation of Proposition 2.3:

where e\geq1, S_A=\prod_{v\in A}S(k_v), and the power subgroup is abstract. Finally, put A_0=\det U(k)\subset L^1. An empty product over A is the trivial group throughout.

The first step is an approximation statement that retains both commutativity and exact determinants. Its second part is also required to retain a specified coset modulo the abstract subgroup R.

3.1 Commuting approximation

3.2 A simple quotient preserved by the full unitary group

The passage from class preservation to a quotient of the ambient group follows the inner-type argument of Rapinchuk–Potapchik [31, Corollary 2.3 and Theorem 2.4]. Here Proposition 3.2 supplies the required unitary input. Conjugation by U(k) preserves R, since R is characteristic in S(k). It also preserves P_0 and V_0 by passage to local closures. In fact the resulting action on V preserves each conjugacy class. To see this, take \gamma\in V_0 and h\in U(k). By Proposition 3.2(ii), choose x'\in\gamma R commuting with t, with \det t=\det h and t_A sufficiently close to h_A that t h^{-1}\in V_0. In V, conjugation by t fixes the image of x', whereas t h^{-1} induces an inner automorphism. Conjugation by h consequently carries \gamma R to a conjugate of itself.

If V\ne1, choose a maximal proper normal subgroup of V and denote the resulting simple quotient by B. Its kernel is invariant under the U(k) action because a normal subgroup is a union of conjugacy classes. The induced automorphisms of B are class-preserving. If B is nonabelian, the theorem of Feit and Seitz [10, Theorem C] says that every such automorphism is inner. Since a nonabelian simple group has trivial center, the identification \Inn(B)=B gives

Its restriction to V_0 is the quotient map to B, and R acts trivially. This is alternative (ii) of Proposition 3.1.

It remains to treat the case that B is cyclic of prime order. Let R' be its kernel upstairs, so that

Class preservation on this abelian quotient means that the action is trivial. Thus R' is normal in U(k) and B is central in U(k)/R'. The rest of the section uses a nonzero character of this central subgroup to construct a finite character of U(k) that is nonzero on S(k).

3.3 Local abelianization and the metaplectic input

We need two local facts: the continuous characters of the compact groups S(k_v), and a splitting theorem for their finite product. The first fact also identifies the closed subgroup generated by commutators of the full local multiplicative group.

Here is the precise metaplectic consequence we use. Let I=\bbR/\bbZ, written additively. If G is a simply connected absolutely almost simple group over a number field, not of type A_1, and A' is a finite set of nonarchimedean places where G is anisotropic, then the restriction map

This is the specialization V_2=\varnothing of [26, Proposition 2.6, p. 114]. If A' is nonempty, the local classification forces the absolute type to be A, so the type-D exclusion inherited from Proposition 2.4 there is harmless. If A' is empty, the source is H^2_{\mathrm m}(1,I)=0. The source uses measurable cohomology for locally compact groups and abstract cohomology on the rational group. In degree two these classify, respectively, topological central circle extensions with Borel sections and abstract central extensions; see [27, Section 3.2]. Thus a topological central circle extension of the displayed finite local product that splits over the abstract rational group has a continuous homomorphic section. Indeed, the zero measurable class gives a Borel homomorphic section, and Borel homomorphisms between these locally compact second-countable groups are continuous.

We apply only Equation (3.7) with G=S and A'=A. Here n\geq3 excludes type A_1, and every support place is nonarchimedean and anisotropic. In particular, this invocation involves neither a comparison with real-place metaplectic kernels nor an identification of a congruence kernel requiring the normal-subgroup assertion we are proving.

3.4 A circle extension of the full local determinant group

Give A_0 the discrete topology and define

Its determinant kernel is S_A. The group Q is locally compact and second countable: each of its countably many determinant fibers is a translate of the compact group S_A. There is a natural homomorphism

The subgroup P_0\subset S_A is open and normal in Q. Normality follows first under the dense group \rho(U(k)) and then under Q. In fact \rho(U(k)) is dense in every determinant fiber by weak approximation for S. The same argument, applied modulo the open subgroup P_0, gives an isomorphism of discrete groups

The central quotient in Equation (3.5) provides the discrete central extension

Pull it back along Equation (3.9) and push it out by an injective character \iota:B\hookrightarrow I. We obtain a locally compact second-countable central extension

The pullback description gives continuous local sections, and it also gives a specified abstract lift

Over P_0 there is a canonical continuous homomorphic section c:P_0\to\widetilde Q, obtained by taking the identity element in the discrete quotient U(k)/R'. For u\in V_0, these descriptions give

where the last factor denotes the central element of \widetilde Q corresponding to \iota(uR')\in I.

Restriction of Equation (3.10) to S_A splits abstractly over S(k) by \ell. Equation (3.7) therefore gives a continuous homomorphic section

Our remaining task is to modify j so that this splitting extends from S_A to all of Q. We first make j(S_A) normal and then eliminate the commutator pairing on the determinant quotient A_0.

3.5 Making the splitting equivariant

For q\in Q, choose a lift \widetilde q and define the discrepancy character d_q:S_A\to I by

We again regard values in I as central factors. This is a continuous character of s, independent of the lift of q. It is zero for q\in S_A, since there a lift differs from j(q) by a central element. Composition of conjugations shows that the d_q form a cocycle for the conjugation action. More explicitly,

The cocycle and its action factor through Q/S_A=A_0.

By Lemma 3.3, its coefficient group is the finite group

On the v-component the action of a\in A_0 is the power of \sigma_v specified by \ord_v(a_v). The corresponding component of the discrepancy is zero whenever \ord_v(a_v)=0. Indeed, choose a representative of a in Q whose v-component is an unramified unit of reduced norm a_v. Such a unit exists by surjectivity of unramified norms on units. It commutes with every Teichmüller lift of T_v, viewed as an element of S_A supported at v. Lemma 3.4 then makes Equation (3.12) zero on these lifts, which determine the character.

It follows that both this component of the cocycle and its action factor through the single valuation map A_0\to\bbZ. This map is onto: weak approximation in U(k) allows its v-component to be close to a local element of reduced-norm valuation one. Let d_v be the discrepancy for such a generator. It vanishes on T_v^{\sigma_v}. Indeed, the Teichmüller lifts of these fixed residue elements are central in \mathcal D_v^\times, so the same commuting-lifts test applies for any representative of the generator.

For a finite abelian group T with automorphism \sigma, the image of

is exactly the characters annihilating T^\sigma. This follows by duality from the exact sequence for \sigma-1 on T, or directly by extending a character from its image to I. Consequently d_v is a coboundary. Choose a correcting character in each component and replace j(s) by j(s)\lambda(s) for the resulting continuous character \lambda:S_A\to I. Equation (3.12) then has zero discrepancy for every q. Thus we may, and do, assume that

so that j(S_A) is normal in \widetilde Q.

3.6 Removing the determinant commutator pairing

The quotient by this normal subgroup is a central extension

Since A_0 is abelian, commutators in this extension give an alternating biadditive pairing, written multiplicatively in its arguments,

By Lemma 3.4, \omega(a,b)=0 whenever, at every v\in A, the two determinants a_v,b_v are norms of elements of one common commutative subfield of \mathcal D_v.

In particular, if every a_v is an nth power, represent it by a central scalar at each place; it commutes with any representative of b_v. Thus \omega factors through the finite group

The map A_0\to K_A is onto. Local reduced norms are surjective, and weak approximation in U(k) realizes any prescribed classes because the local power subgroups are open. Mixed terms between two distinct factors of Equation (3.15) vanish: use central representatives for each locally trivial power class. We may therefore treat the pairing one place at a time.

Fix v, suppress it from the notation, and let \pi be a uniformizer. Unit-unit terms vanish because units are norms from the unramified maximal subfield. Alternation kills uniformizer-uniformizer terms, so the local pairing is specified by the character

This character vanishes on 1+\mathfrak p_v. To prove it first take c\in k_v of valuation one. The Eisenstein extension defined by \theta^n=c has degree n, is totally ramified, and embeds in \mathcal D_v by the local invariant criterion. Since n is odd,

The common-subfield test gives \omega_v(c,1+c)=0; writing c as a uniformizer times a unit and using unit-unit vanishing gives \omega_v(\pi,1+c)=0. If \ord_v(d)\geq2, choose any such c and write

Both numerator and denominator have the form 1+c' with \ord_v(c')=1. This proves vanishing on all principal units. The remaining parameter is therefore a character of \bbF_q^\times killed by n.

We still have freedom to change j by characters of S_A invariant under Q. At the fixed place these are the \sigma-invariant characters of T_v. Such a change preserves Equation (3.13). Its effect on the pairing is evaluation, up to the fixed sign convention for commutators, on the residue of the ordinary local commutator of determinant representatives. For a prime element \Pi and an unramified unit with residue r, that residue is

The residue norm of r is the residue of the unit determinant u. Replacing r by another norm preimage changes this expression by an element of (\sigma-1)T_v. Hence we have a well-defined map

This map is onto. To see this explicitly, put

and choose a generator g of \bbF_{q^n}^\times. The group T_v is generated by g^{q-1}. Since (h,n)=1,

The image of the generator g^M of \bbF_q^\times in Equation (3.16) is represented, in the generator g^{q-1}, by the exponent

It is a generator because (h,d)=1. Dualizing this surjection shows that invariant characters of T_v produce exactly all residue unit characters killed by n. They can therefore cancel the remaining parameter u\mapsto\omega_v(\pi,u). A unit adjustment between \Nrd(\Pi) and the chosen \pi makes no difference, since unit-unit terms already vanish.

Perform this adjustment at every place of A. The new section still has normal image, and the commuting-lifts test still holds; in particular, all the preceding reductions of the pairing remain valid. Its only possible parameters have now been cancelled, so \omega=0. The extension in Equation (3.14) is therefore abelian. The circle group is divisible and hence injective as an abelian group, so this extension splits. A homomorphism from the discrete group A_0 is automatically continuous.

Let C_0 be the inverse image in \widetilde Q of the image of such a section of Equation (3.14). Then C_0\cap I=1 and p(C_0)=Q: for any lift of an element of Q, its central coordinate in the quotient can be removed using I. Consequently p:C_0\to Q is a group isomorphism. Locally on each determinant fiber its inverse is a translate of j, so the inverse is continuous. We have proved that Equation (3.10) has a continuous homomorphic section

3.7 Extracting a nonzero finite character

Compare the continuous section in Equation (3.17) with the specified abstract lift of rational points. Their difference is a character

It has finite image on S(k). Indeed, for u\in R' the canonical formula in Equation (3.11) reads \ell(u)=c(u_A). The difference of c and s|_{P_0} is a continuous character of the profinite group P_0. Its image in the circle is finite: a quotient of a profinite group is profinite, whereas a closed subgroup of the circle is either finite or the whole circle. Thus \chi_0(R') is finite. Since R' has finite index in S(k), the image \chi_0(S(k)) is a finite union of cosets of that finite subgroup and is finite.

It is also nonzero. Choose \beta\ne1 in B and u\in V_0 representing it. Because R is dense in P_0 and u_A\in P_0, there is a sequence r_j\in R such that

All ur_j still represent \beta modulo R'. If \chi_0 vanished on S(k), Equations (3.11) and (3.18) would give

Both continuous sections tend to the identity, which would imply \iota(\beta)=0, contrary to injectivity of \iota.

Finally we make the character finite on all of U(k) without changing its restriction to S(k). The multiplicative group of a number field is free abelian modulo its finite roots of unity: use the ideal map and the finite generation of the unit group. Therefore its subgroup A_0 has the form

where T is finite and J may be infinite. Choose u_j\in U(k) with \det u_j=a_j, and choose a character \psi:A_0\to I satisfying \psi(a_j)=\chi_0(u_j) and \psi|_T=0. Then

vanishes on all the chosen u_j and equals \chi_0 on S(k). If m kills the finite group \chi_0(S(k)) and t kills T, then mt kills \chi(U(k)): remove the free determinant part using the u_j, and the tth power of the remaining element lies in S(k). The subgroup of the circle killed by mt is finite. Thus \chi is a finite character and remains nonzero on S(k).

When A=\varnothing, the same construction has Q=A_0 and trivial S_A. Commuting approximation makes the circle extension abelian immediately, and all subsequent arguments apply unchanged. We have proved alternative (i) in the remaining case, completing the proof of Proposition 3.1.

The finite defect can therefore be nontrivial only if the full unitary group admits one of the two homomorphisms in that proposition. The character argument in Section 4 and the color argument in Sections 5–6 exclude these alternatives, retaining the division and odd-degree hypotheses fixed here.

4 Excluding finite abelian characters

Throughout this section, L/k is quadratic, D is a central division algebra over L of odd degree n\geq3, and * is a unitary involution. We retain U=\U(D,*) and S=\SU(D,*) from Section 3. Our aim is the following assertion, which eliminates the abelian alternative of Proposition 3.1.

Fix such a homomorphism \chi:U(k)\to B, and write B additively. The proof first converts \chi into a difference function on D. Exact elementary transformations make this difference independent of its arguments. A real-signature argument then converts that algebraic identity into continuity on S(k), after which local commutators finish the proof. No continuity of \chi is assumed.

4.1 A noncommutative difference identity

Let \mathcal D be the set of nonzero b\in D for which b+b^*=a^*a for some a\in D. Define

These definitions also allow a=0, in which case b is skew and m_b=1. Multiplication using a^*a=b+b^* gives m_bm_b^*=1. If a'{}^*a'=a^*a\ne0, then a'=va with v\in U(k), and the two resulting elements m_b are conjugate by v. Thus f is well-defined. Moreover,

because ax can be used in the definition on the left.

For the moment fix d\in D^\times such that the reduced spectra of d and d^* are disjoint. For (a,c)\in D^2\setminus\{(0,0)\}, there is a unique e\in D satisfying

Indeed the equation

has a unique solution: after splitting the algebra its linear part is the Sylvester operator, whose eigenvalues are the differences of the two disjoint spectra. Taking adjoints shows that c^*c-e^* is another solution of this same equation, giving the first identity in Equation 4.3 and then the second. The solution is nonzero, for e=0 would imply a=c=0. Consequently

is defined on all nonzero pairs.

4.2 Producing exact elementary transformations

Suppose now that d lies in a *-stable maximal subfield E\subset D which is Galois over L. Write \bar z=z^* for z\in E, and suppose that the n conjugates t_1,\ldots,t_n of d are distinct and disjoint from \bar t_1,\ldots,\bar t_n. Skolem–Noether gives a decomposition

Let E^1=\{z\in E^\times:z\bar z=1\}. The pair move in Equation 4.4 and independent left multiplications by elements of E^1 preserve the n two-dimensional E-blocks in D\oplus D.

In the invariance group take the subgroup H generated by diagonal E^1-multiplications on either coordinate and by the simultaneous matrices

Indeed changing the sign of the second output in Equation 4.4 gives B_{-1}, and conjugating by diagonal phases gives all B_h. Each determinant is 1-t_j\ne0.

To turn a single nonzero shear coefficient into an arbitrary additive change in D, we need the following exact statement, not merely local density.

4.3 Scalar normalization and a cyclic maximal field

Let \Omega be the nonsplit real places of k. At v\in\Omega write the involution as x^*=H_v^{-1}\bar x^{\,t}H_v on M_n(\mathbb C). For a nonzero Hermitian element h\in D, call its signature admissible when the signature of H_vh is that of H_v up to overall sign, for every v\in\Omega. This does not depend on the matrix realization. Define \mathcal D^\# to consist of the nonzero skew elements and the b\in D for which h=b+b^*\ne0 has admissible signature. Division makes every such h invertible.

Fix this field E and an integer m\ge1 annihilating B. Call d\in T(L)^m admissible if its conjugates over L are distinct and disjoint from their bars. These conditions specify a nonempty k-Zariski open subset of \Res_{L/k}T: after extending scalars, the two determinant-one eigenvalue lists can be chosen independently, avoiding equality within or between the lists. In particular admissible elements exist, since the rational points of T are Zariski dense and the power map is dominant. Let \mathcal C be their D^\times-conjugates. This set is closed under inversion.

4.4 Keeping products in the real signature domain

The subgroup J\subset\SL_1(D)(L) generated by \mathcal C is noncentral and normal. The established inner-type case of the Margulis–Platonov theorem, applied over the number field L, gives an open neighborhood W of 1 in

whose rational inverse image is contained in J. Here A_D is the set of finite places of L where D\otimes_LL_w is division. This use of the known inner case is independent of the outer case being proved [36, Theorem 2][27, Sections 3.4–3.5]. Although every element of J is a word in the allowed steps, an arbitrary word need not preserve the real signature domain at its intermediate products. The next lemma supplies the required control.

4.5 Continuity and the final vanishing

5 Excluding nonabelian simple colors

We retain the odd-degree division-algebra hypotheses and notation of Section 3. Thus L/k is quadratic, D is a central division L-algebra of odd degree n\geq3, the involution * induces the nontrivial automorphism of L/k, and S=\SU(D,*)\subset U=\U(D,*). Fix i\in L^\times with \bar i=-i. For R=S(k)^e let P_0 and V_0 have the meaning of Proposition 2.3. We now eliminate the nonabelian alternative in Proposition 3.1.

Suppose, for a contradiction, that such a homomorphism exists. We call its values colors. It is surjective, so central scalars in U(k) have color 1: their images centralize \mathcal F. The finite-group input is the following assertion. Its statement is independent of the arithmetic constructions below.

The proof of Lemma 5.2 occupies Section 6. We use exactly its displayed double-coset condition. Our task in this section is to force that condition from sufficiently many additive returns of colors, and then to show that avoiding it would make membership in \mathcal S constant on \mathcal F.

5.1 The compact space of color configurations

Let \widetilde G be the simply connected special unitary group of the hyperbolic binary hermitian form over (D,*) with coefficients 1,-1. Let X be its projective variety of isotropic right D-lines. Since D is division, the form has relative rank one, and the stabilizer of such a line is a minimal k-parabolic of \widetilde G. Every rational isotropic line has a unique representative (u,1) with u\in U(k): the second entry cannot vanish, and division makes it invertible. This identifies X(k) with U(k).

There is also an additive parametrization. Put

Here J is a k-vector space, and

Indeed x-i is nonzero for x=x^*, and u-1 is invertible for every u\ne1 in U(k). The corresponding column on the additive chart is (x+i,x-i). At a completion k_v, these two parametrizations give open charts isomorphic to U(k_v) and J(k_v), respectively. Their complements are proper algebraic subvarieties and therefore have measure zero for any smooth measure on X(k_v). We shall use this assertion at finitely many completions at a time.

Let \mathcal H be the countable group of rational transformations of X induced by rational form similitudes. It contains \widetilde G(k), the left and right rotations

and the additive translations T_b(x)=x+b for b\in J(k). Write \tau_s=T_s when s\in k\subset J(k). For p\in X(k) define a configuration

We call this configuration a palette, and let \mathcal C\subset\mathcal F^{\mathcal H} be the closure of the rational palettes. The product topology makes \mathcal C compact and metrizable. The action is

Every pattern on finitely many coordinates of a palette in \mathcal C occurs at a rational point, because the colors form a finite discrete space. In particular, all finite-coordinate identities satisfied by rational palettes hold throughout \mathcal C.

5.2 A translation-invariant law with uniform marginals

The next proposition supplies additive recurrence without losing any color. Arbitrary additive averages of rational palettes need not have uniform marginals, so we first construct a measure with controlled conditional laws over the local flag varieties.

5.3 Fractional transformations and simultaneous color returns

Fix the conjugacy-invariant subset \mathcal S supplied by Lemma 5.2. We shall relate the colors of w and \tau_s w to returns under a quadratic additive translation. For w\in U(k) and s\in k, put

These expressions are defined for every s: i\notin k, and

The elements C and C^* commute, and CC^*=t\bar t is central. Division and d\ne0 imply that 1-C and 1-C^* are invertible. It follows that R_C is unitary and commutes with w. Hence z is unitary and commutes with w. Direct Cayley substitution gives

The scalar factor is central and has norm one, so \phi(z)=\phi(\tau_s(w)).

For such a C, the fractional transformation

is induced by a form similitude: its matrix \left(\begin{smallmatrix}1&C\\C^*&1\end{smallmatrix}\right) has multiplier 1-t\bar t. The denominator is invertible for unitary u; otherwise it vanishes, which would force t\bar t=1. Writing u=wa shows that

This calculation does not commute u with w.

Starting from a variable u\in U(k), define

Apply Equation 5.10 first with unitary parameter w^{-1} and scalar \bar t. It gives [wv,wuR_C]=1; conjugating by w^{-1} gives [vw,uz]=1. Applying the same equation with scalar -t gives [w^{-1}v,z^{-1}u']=1. Thus, writing

we have

The transformation u\mapsto u' is an additive translation. Indeed its fractional matrix is

After dividing by d and applying it to the column (x+i,x-i), the result is translation by 2i(C-C^*)/d. Expanding the latter gives

The matrix calculation includes the point \infty of the additive chart. Neither w nor the variable point is required to have a finite Cayley coordinate.

Suppose that, for this fixed translation T_{b_w(s)}, every color occurs as X_1=X_1' at some rational point; different colors may use different points. Put B=Z^{-1}X_1. Equations 5.11 then imply

Since WZ=ZW, direct multiplication gives

As X_1 ranges over \mathcal F, so does B, and this is exactly Equation 5.1. Consequently

When s=0, the translation is the identity and Z=W, so this implication has no degenerate exception.

5.4 Uniform finite witnesses for recurrence

The translation-invariant law of Proposition 5.3 will now force simultaneous returns. Uniformity is essential: it lets us pass the resulting restrictions from rational palettes to their closure. A subset D_0 of the additive group k is called thick if it contains a translate of every finite subset of k.

Apply the lemma to independent copies of (\mathcal C,\mu) indexed by a\in\mathcal F, and to the event

Its probability is the fixed number \delta=|\mathcal F|^{-|\mathcal F|}>0. Writing b_w(s)=sA_w+s^2B_w as in Equation 5.12, use the k^2-action

on this product space. Positive return of E yields every recurrent color in some palette. Each required two-coordinate pattern then occurs rationally, so Equation 5.13 applies. For part (i), and any u_0\in k, use instead the \bbZ^2-action

We have obtained two statements uniform in w:

1. (a)

   for every u_0\in k, some s\in\{u_0,2u_0,\ldots,ru_0\} gives simultaneous recurrence of all colors;

2. (b)

   for every fixed thick D_0\subset k, a fixed finite subset of D_0 contains such an s for every w.

The second finite subset is not asserted to be uniform over thick sets.

5.5 Additive invariance of membership in the finite subset

Fix c\in\mathcal C and a prefix h\in\mathcal H, and set

If x\notin C_0, then C_0-x cannot contain \{u_0,2u_0,\ldots,ru_0\} for any u_0\in k. Indeed, match the finitely many relevant coordinates by a rational palette and apply statement (a) to its rational point w=u(\tau_xhp). This contradicts Equation 5.13. Nor can C_0-x be thick: apply statement (b) to that particular thick set and match the finitely many witness coordinates by a rational palette. The same contradiction results.

We record explicitly the combinatorial consequence:

For this purpose the upper Banach density of C\subset k is the supremum of M(1_C) over translation-invariant means M on bounded functions on the discrete additive group k. Such means exist because this group is abelian. If this density is zero, the density of C along every Folner sequence tends to zero: a nonzero upper limit would yield an invariant mean with positive value by taking a weak limit of the averaging functionals.

First, a run of r consecutive points of C_0 in a direction u_0 extends to the adjacent point on either side. Otherwise that adjacent point, lying outside C_0, would have the forbidden run immediately ahead in direction u_0 or -u_0. Iteration fills the entire integer progression.

Suppose now that C_0 has positive upper Banach density. To prove it thick, fix any finite list g_1,\ldots,g_d\in k, and choose an invariant mean M with M(1_{C_0})>\eta>0. The finite multidimensional Szemeredi theorem supplies a box size N_0 such that every subset of \{1,\ldots,N_0\}^d of density at least \eta contains an affine homothetic copy of \{0,\ldots,r-1\}^d, with positive integer dilation \ell\leq N_0. This is the finite-box form of the density theorem of [12, Theorem B]. Put L_0=\operatorname{lcm}(1,\ldots,N_0) and average, with respect to M, the density of the pullback of C_0 under

Translation invariance makes this average M(1_{C_0})>\eta, so one translate has pullback density at least \eta. Injectivity of this box map is not required. The resulting homothetic r-box lies in C_0, with steps \ell g_j/L_0. Extend progressions successively along each coordinate axis. This fills the whole integer grid with these steps. Because \ell divides L_0, that grid contains a translate of \{0,g_1,\ldots,g_d\}. Thus C_0 is thick. If any x were outside C_0, then C_0-x would also be thick, contrary to the preceding restriction. This proves Equation 5.15.

Now let the palette be random with law \mu. For a fixed Folner sequence, Equation 5.15 makes the densities of C_0 converge pointwise to 1_{\{C_0=k\}}. Uniform coordinate marginals and dominated convergence give

For each s, the event C_0=k is contained in s\in C_0, and the latter has exactly this same probability. Therefore

Countability makes these identities simultaneous for all s,h. The palette rules also give, for every prefix h, invariance of this indicator under left rotation by any element of \ker\phi and under conjugation by any element of U(k), since \mathcal S is conjugacy-invariant. These are exact finite-coordinate identities, so they hold throughout \mathcal C.

We have reached a concrete restriction on a common full-measure set of palettes: membership of a color in \mathcal S is unchanged under scalar additive translations, kernel left rotations, and all unitary conjugations, at every prefix. It remains to show that these transformations already force invariance under all left colors.

5.6 Generating left rotations by squares

Let \mathcal K be the subgroup of \mathcal H generated by the three classes of transformations just listed. Equation 5.16 and the simultaneous prefix identities show that

on one set of full measure. This assertion is obtained by composing finite-coordinate identities; it does not assume that \mu is invariant under the whole group \mathcal K.

Fix \xi\in U(k). The commutative algebra E=L(\xi) is a field, since D is division, and it is stable under * because \xi^*=\xi^{-1}. Let E_0 be its fixed field, so E=LE_0 and [E:E_0]=2. Write E^1 for its norm-one group to E_0. Choose n_0\in E^1\cap\ker\phi whose finite Cayley coordinate x_0\in E_0 generates E_0 over k. Such a choice is possible: the rational norm-one torus is Zariski dense by its Cayley parametrization, the power map on that torus is dominant, and powers of an exponent killing \mathcal F lie in \ker\phi. The requirement that x_0 be finite and primitive is a nonempty Zariski-open condition.

Since there are only finitely many colors, choose t_0\in k for which the set

is infinite. For each s\in I, the word

belongs to \mathcal K. Both indicated left rotations have kernel parameters. The word successively sends 1 to n_0, to \tau_{t_0}n_0, to \tau_sn_0, to n_0, and back to 1. It is represented by a fractional matrix over E; fixing 1 in the unitary chart means fixing infinity in the additive chart. Its latter matrix is therefore upper triangular, say

On the commutative E_0-chart its affine multiplier is

To check the multiplier, use q'(x)/q(x)=-2i/(x^2-i^2) in Equation 5.18. The derivative at u=1 is ((x_0+s)^2-i^2)/((x_0+t_0)^2-i^2); the affine multiplier at infinity is its reciprocal. All denominators are nonzero because x_0+s\in E_0 whereas i\notin E_0.

This commutative calculation produces transformations on the full noncommutative chart, not merely on E_0. Conjugating T_{s'} by the upper triangular matrix gives the matrix

since s' is central. Thus \mathcal K contains translations by every k-multiple of a_s, and hence by their k-linear span.

That span is all of E_0. Indeed suppose a nonzero k-linear functional on E_0 annihilated all the a_s with s\in I. Extend scalars to an algebraic closure and write this functional as a linear combination of the distinct embeddings of E_0, which may be extended while fixing L. Put x_j for the corresponding distinct conjugates of x_0. We obtain a linear combination, zero for infinitely many s, of the rational functions

It must vanish identically. These functions are linearly independent. Their poles are -x_j+i and -x_j-i. Poles from distinct denominators can coincide only when the corresponding centers differ by 2i or -2i. The resulting finite components form chains: each center has at most one successor and predecessor, and characteristic zero excludes cycles. An endpoint pole belongs to just one denominator, so its coefficient in any linear relation is zero. Remove that denominator and repeat along the chain. All numerators in Equation 5.20 are nonzero, so every coefficient vanishes, the required contradiction. Consequently \mathcal K contains all E_0-translations.

Left rotation by -1 belongs to \mathcal K, because -1 is a central kernel scalar. In the additive chart it is x\mapsto i^2/x, represented by an antidiagonal matrix over k. It conjugates all upper E_0-unipotents to all lower E_0-unipotents. These generate \SL_2(E_0), so \mathcal K contains the corresponding fractional transformations of the full chart. In particular it contains the transformation whose matrix in the unitary chart is \diag(\xi,\xi^{-1}): its conjugate by the Cayley change of basis is

Its action is u\mapsto\xi u\xi. Composing with the allowed conjugation u\mapsto\xi u\xi^{-1} gives u\mapsto\xi^2u. Hence

6 The finite simple-group obstruction

We prove Lemma 5.2. The argument uses the classification of finite simple groups [1, Classification Theorem, p. 737]. Two elementary tests handle the alternating, linear, and sporadic families. A counting argument in a full unitary group handles the other classical families, and small maximal tori handle the exceptional families.

Throughout this section, \mathcal F is a finite nonabelian simple group, C(B)=C_{\mathcal F}(B), and

For commuting W,Z\in\mathcal F, the condition to be excluded is

The choice of the nonempty proper conjugacy-invariant set \mathcal S will be specified in each family. In every case we show that Equation (6.1) is incompatible with W\in\mathcal S and Z\notin\mathcal S.

6.1 Two elementary tests

First take

This set is nonempty, since a group of exponent two is abelian, and it does not contain the identity. For commuting W\in\mathcal S and Z\notin\mathcal S, put v=ZW. Then Z^2=1, v^2=W^2\ne1, and Equation (6.1) becomes

Suppose that \mathcal F acts on a set and that some point p satisfies v^2p\ne p. If C(B) fixes p and vp pointwise, writing v=a v^{-1}b with a,b\in C(B) gives vp=a v^{-1}p. Applying a^{-1} gives vp=v^{-1}p, a contradiction. We call this the point-pair test. We will also use its incidence version: if C(B) fixes a line p and preserves a subspace L, while vp\subset L and v^{-1}p\not\subset L, the same equation is impossible.

The second test uses a self-centralizing abelian subgroup.

6.2 Alternating, linear, and sporadic groups

For alternating and projective special linear groups we use Equation (6.2). In A_n, choose p with v^2p\ne p. There is an even permutation B fixing exactly p,vp and having distinct odd cycle lengths greater than one on the remaining points, except when n=6 or 8. Indeed, for odd n use the single cycle of length n-2, and for even n\ge10 use lengths 3,n-5. Every permutation centralizing these moving cycles is even on their support. Therefore a centralizer element in A_n cannot interchange the two fixed points. The point-pair test applies. The two exceptions are A_6\simeq\PSL_2(9) and A_8\simeq\PSL_4(2).

Now let \mathcal F=\PSL_l(q), l\ge3. Choose a line p such that v^2p\ne p. The three lines p,vp,v^{-1}p are distinct. A hyperplane L can be chosen to contain vp and contain neither p nor v^{-1}p: in the quotient by vp, choose a linear functional nonzero on both remaining nonzero vectors. The union of two proper hyperplanes in the dual is not the entire dual, also over \bbF_2.

On the direct sum p\oplus L, take B to be the identity on p and an irreducible norm-one field multiplication on L. Such an element exists in the cyclic group of order (q^{l-1}-1)/(q-1): a generator cannot lie in a proper subfield, since that group’s order exceeds q^{(l-1)/2}-1 when l-1\ge2. A projective centralizer of B is an actual centralizer. Indeed, scalar multiplication must preserve its unique base-field eigenvalue, which is 1, and hence the multiplier is 1. The centralizer fixes p and preserves L, contradicting the incidence version of Equation (6.3).

For simple \PSL_2(q), the pointwise stabilizer of two points of the projective line is a split torus. It contains an element whose centralizer fixes both points, unless q=5: its order is q-1 for even q and (q-1)/2 for odd q, and a noninvolution in it is regular with that torus as centralizer. The case q=5 is A_5. Choosing the torus for p,vp proves the assertion in all these cases.

For the sporadic groups other than M_{12}, and also for the Tits group, Table 1 gives a prime c for which a cyclic subgroup of order c is the centralizer of each of its nonidentity elements. The centralizer orders, class numbers, and group orders are those in the Atlas and the online ATLAS of Finite Group Representations [9, 45]. In each row, an element x in the class labeled c\mathrm A in that group’s conjugacy-class table has C(x)=\langle x\rangle of order c. Since c is prime, every nonidentity power of x has the same centralizer. The displayed k is the exact class number. In every row the group order exceeds kc^4, so Lemma 6.1 applies. For example, the smallest margin is 10\cdot5^4=6250<7920=|M_{11}|. One may also check the other inequalities using k\le200, except that k\le25 suffices for M_{22},M_{23},J_1.

For M_{12} use its sharply 5-transitive action of degree 12 and the Atlas class 8B, of cycle shape 1^2\,2\,8 [9, 45]; in the online database these are the M_{12} degree-12 representation and its class 8B. An element centralizing B\in8B restricts to a cyclic shift on its eight-point orbit. This restriction is injective: its kernel fixes eight points, and an element fixing five points is the identity. The centralizer is therefore exactly \langle B\rangle, and fixes the two fixed points individually. Two-transitivity moves this pair to p,vp, so the point-pair test applies with Equation (6.2).

6.3 An auxiliary unitary count

The remaining classical groups need a torus argument on a large irreducible block. The relevant obstruction takes place in the full unitary isometry group, which need not itself be simple.

6.4 Symplectic and orthogonal groups

Let \mathcal F be a projective symplectic or orthogonal simple group on its natural space of dimension 2l or 2l+1. We first treat l\ge3, using the standard low-rank identifications and treating \PSp_4 separately below. Choose an odd prime

requiring p<l in plus orthogonal type. For l\ge4, set m=\lfloor l/2\rfloor\ge2. Bertrand’s postulate gives m<p<2m, hence l/2<p<l, and this prime is odd. For l=3 take p=3, apart from plus type, which is \PSL_4 and has already been treated. In odd-dimensional orthogonal type we may assume odd characteristic, since in even characteristic the group has the corresponding symplectic realization.

We use a nondegenerate block E of dimension 2p carrying a torus of order q^p+1. In orthogonal type this block has minus type; the complement can be chosen with the sign needed for the ambient form. For minus type p=l the block is the whole space, whereas the requirement p<l in plus type leaves a complement of the necessary sign. The torus is the norm-one subgroup of \bbF_{q^{2p}}^\times over \bbF_{q^p}, acting on E by field multiplication.

Define \mathcal S to consist of those projective elements whose square, on the natural space, has an irreducible self-dual block of dimension 2p. This condition is independent of the choice of a scalar lift and is conjugacy-invariant. The block is unique, because 4p>2l+1, and is nondegenerate: its dual would otherwise require a second block of the same degree. The set excludes identity and is nonempty. Indeed, if t generates the torus of order q^p+1, use t^2 on the block and identity on the complement. Squaring meets the orthogonal component and spinor conditions, whose quotients have exponent two. Moreover t^4 still has degree 2p. To see this, the quotient of the norm-one torus by its intersection with \bbF_{q^2}^{\times} has odd order

Thus t^4\notin\bbF_{q^2}. The only other maximal proper subfield is \bbF_{q^p}, where a norm-one element is \pm1 and hence already lies in \bbF_{q^2}.

Suppose now that W\in\mathcal S, that Z\notin\mathcal S commutes with W, and that Equation (6.1) holds. Lift W,Z to natural isometries, using \Omega in orthogonal type. Their possible projective commutation multiplier has order at most two. They therefore commute actually with W^2 and preserve its unique marked block E. On E they act as field multiplications w,z\in\bbF_{q^{2p}}^\times of norm one. Since Z\notin\mathcal S, z^2 lies in a proper subfield. The norm-one condition puts z^2 in \bbF_{q^2}, and then z itself is in \bbF_{q^2}: its degree over that field divides both p and 2.

There is a full unitary group H=\U_p(q) acting on E and containing its field torus T. One way to see this directly is to write the invariant form in field coordinates. Its bilinear form has the shape

Take the trace first to \bbF_{q^2}. If the resulting form is anti-Hermitian and the characteristic is odd, multiply it by \delta\in\bbF_{q^2}^{\times} with \delta^q=-\delta; this makes it Hermitian without changing its isometry group. In characteristic two an anti-Hermitian form is already Hermitian. This gives the required unitary structure. In the quadratic case the form is

In characteristic two this expression follows from the polarization and W^2-invariance: two invariant quadratic forms with the same polarization differ by the square of an invariant linear form, and an irreducible block of degree greater than one has no such nonzero form. These formulas show that H preserves the original form on E. The element z is a scalar of this unitary group.

For each h\in H, choose B acting on E by a square of a generator of T^h and as identity on the complement. This is an allowed ambient element. Even if h is not in \Omega, conjugation by an isometry preserves \Omega, so the orthogonal component condition is unchanged. Both B and BZ^2 have a unique marked square block E: multiplication of a full-degree element by a base-field scalar preserves its degree over \bbF_{q^2}, and the norm-one condition then gives full degree 2p over \bbF_q.

Each projective centralizer of B or BZ^2 commutes actually with its square, preserves E, and restricts there into T^h. This does not require the action on the complement to be semisimple; the complement is too small to have an irreducible factor of degree 2p. Restricting Equation (6.1) to E and absorbing its scalar multipliers and the scalar z into T^h therefore gives

The element w has degree p over \bbF_{q^2}. This contradicts Lemma 6.2.

6.5 Projective unitary groups and rank two symplectic groups

Let \mathcal F=\PSU_l(q), l\ge3, excluding the nonsimple pair (l,q)=(3,2). Choose an odd prime l/2<p\le l. Define \mathcal S by requiring the square of a natural lift to have an irreducible block of degree p over \bbF_{q^2}. As above this block is unique and nondegenerate. If l>p, the determinant of a chosen torus element on the block can be corrected on its nonzero dimensional orthogonal complement. If l=p, use instead the determinant-one torus of order

Its scalar intersection has order d=(p,q+1). Except for (p,q)=(3,2), c>d: for p\ge5, c\ge q^{p-2}(q-1)+1\ge2^{p-2}+1>p, and for p=3,q\ge3, c\ge7>3. As c is odd, a generator and its square are both nonscalar and hence have degree p. This proves that \mathcal S is nonempty; it again excludes identity.

We check the projective-centralizer issue before applying the block argument. Suppose gAg^{-1}=\lambda A, where A\in\SU_l(q) has the marked square block. Multiplication by a base-field scalar preserves irreducible degrees, so g preserves the unique block. The determinant on the block gives \lambda^p=1, and the total determinant gives \lambda^l=1. If l>p, then p<l<2p, so these conditions force \lambda=1. If l=p and \lambda\ne1, multiplication by \lambda cycles all p distinct eigenvalues. Their product is w^p, since p is odd, and determinant one gives w^p=1. But \lambda has order p in the scalar group, so p\mid q+1; all pth roots of unity then lie in \bbF_{q^2}, contradicting the full degree of w. Thus these projective centralizers are actual centralizers on the block.

For commuting W\in\mathcal S and Z\notin\mathcal S, the restrictions are consequently field scalars w,z, with z\in\bbF_{q^2} by the same odd-degree argument as before. For every conjugate T^h in the full block unitary group choose B with irreducible square on that torus, correcting its determinant on the complement if necessary. The square of BZ^2 retains full degree. Applying the preceding projective-centralizer argument to B and BZ^2, Equation (6.1) restricts to the relation prohibited by Lemma 6.2.

For \PSp_4(q) take \mathcal S to be the set whose square is irreducible on the natural four-dimensional space. It is nonempty by the torus of order q^2+1 and excludes identity. Lifting a commuting pair W\in\mathcal S, Z\notin\mathcal S gives norm-one field scalars w,z\in\bbF_{q^4}^\times. Since z^2\in\bbF_{q^2}, the norm-one condition gives z^2=\pm1. Thus Z^2=1 projectively, and v=ZW still has an irreducible square.

Choose a nonzero vector x such that x,vx span an isotropic plane L. Such an x exists in the field description of the alternating form: the condition \Tr_{\bbF_{q^4}/\bbF_q}(\kappa x(vx)^*)=0 is one homogeneous \bbF_q-linear condition on xx^*\in\bbF_{q^2}. It has a nonzero solution, and the field norm x\mapsto xx^* is surjective. Irreducibility ensures that x,vx are independent and that v^{-1}x\notin L; otherwise v would satisfy a quadratic polynomial.

Choose a regular unipotent B whose unique line-plane flag is \langle x\rangle\subset L. Its existence in every characteristic can be seen in a symplectic flag basis from the matrix

for the alternating form with nonzero upper anti-diagonal entries 1,1. This matrix preserves the form and has one Jordan block, also in characteristic two. Its centralizer preserves \ker(B-1)=\langle x\rangle and \ker((B-1)^2)=L. A projective centralizer is actual, since a scalar multiple of a unipotent matrix can have the same eigenvalues only when the scalar is one. The incidence test now contradicts Equation (6.3). The nonsimple group \Sp_4(2) is not needed; its simple derived group is A_6.

6.6 Small tori in exceptional groups

We have proved the obstruction for all classical simple groups. For most exceptional families we apply Lemma 6.1 to a rational maximal torus. We use the usual simply connected algebraic realization followed by its central quotient. Rational tori are obtained by Weyl-group twisting, and their orders are determinants of the Frobenius action minus one on their character lattices; see [39] and [19, Propositions 25.1–25.3]. Semisimple centralizers in a simply connected semisimple group are connected [38, Section 2.10]. The tori below and their self-centralizing property in the simple quotient occur in [36, Corollary 8.2 and Lemma 8.7]. We verify that property here by a root-lattice estimate before applying Lemma 6.1. Their orders outside type G_2 are also recorded in [13, Section 3]; the two G_2 orders follow directly from its order-three and order-six Weyl actions.

Write \Phi_j for the jth cyclotomic polynomial. Use the tori in Table 2, before passing to the central quotient. For G_2(q) choose the sign avoiding a factor 3, and for q=3 choose the smaller order 7.

These actions are irreducible over \bbQ. For G_2,F_4,E_8 use a Coxeter element or an appropriate power. In trialitarian D_4, let b be the central node, a an outer node, and \gamma the diagram cycle of the outer nodes. The action s_a s_b\gamma has characteristic polynomial X^4-X^2+1. For E_6, the regular order-nine Weyl element in Springer’s tables [37, Section 5.4, Table 1] has a primitive ninth-root eigenvalue. Its rational characteristic polynomial, of degree six, is therefore \Phi_9. Its negative belongs to the nontrivial diagram coset and gives the twisted case.

After passing to the central quotient, dividing by (3,q-\epsilon) in type E_6, every prime divisor of these torus orders is a primitive cyclotomic prime. Their respective lower bounds are

Indeed a prime dividing \Phi_m(q) either has multiplicative order m for q modulo that prime, or is an exceptional prime dividing m. The displayed polynomials have no such exceptional factor except for 3 in \Phi_9(\epsilon q); when present this factor has valuation one and is exactly removed by the center. All resulting torus orders are odd and greater than one.

We next prove that every nonidentity element of each resulting torus has centralizer exactly that torus. It suffices to do so for an element t of prime order r upstairs, with r one of these primitive primes. If a root \alpha vanished on t, Frobenius invariance would make every root in its twisted Weyl orbit vanish on t. By irreducibility that orbit spans the rational character space. Choose a linearly independent subset. The lattice it generates has full rank in the weight lattice, and its index must be divisible by r, since evaluation at t is a nontrivial character of order r trivial on this sublattice.

Normalize long roots to have squared length two. Hadamard’s inequality bounds these lattice indices by

in the respective rows of Table 2. The respective weight-lattice covolumes are

The second value holds for both D_4 and F_4. Each bound is smaller than the corresponding r. Thus t is regular. Connectedness of its algebraic centralizer makes that centralizer exactly the torus.

Any nonidentity element of a torus in the simple quotient has a nontrivial prime-order power of this kind. The argument survives projective centralization: one can lift that power with order r coprime to the center, and a conjugate differing from it by a central element must have that central factor equal to one, by preservation of its order. Therefore its projective centralizer is precisely the torus in the quotient. The hypotheses on C in Lemma 6.1 are now verified.

For the numerical inequality, we use

where l is the absolute rank. This follows, with a smaller constant, from [11, Theorem 1.1, p. 3024] for the simply connected fixed-point groups; taking a quotient cannot increase the number of conjugacy classes. The standard group order formulas [19, Tables 24.1–24.2] give

In the two small G_2 cases the direct comparisons are

For q\ge5 the chosen G_2 torus has order c\le1.24q^2, and 0.7q^{14}>100q^2(1.24q^2)^4 already at q=5, with increasing ratio thereafter. In trialitarian type c<q^4, so it is enough that 0.7q^8>100, true at q=2. For F_4,E_6,E_8 use respectively c<q^4,c<2q^6,c<2q^8; the displayed lower bounds exceed 100q^l c^4 already at q=2, and thereafter with increasing ratio. This proves the desired inequality in every row, so Lemma 6.1 completes these families.

6.7 Type E_7 and the Suzuki–Ree groups

For E_7 we transfer the torus obstruction from a suitable E_6 subgroup. In the simply connected algebraic group choose a maximal-rank subgroup M geometrically a Levi subgroup with derived group D_M=[M,M] of type E_6 and one-dimensional center. Both rational forms E_6 and {}^2E_6 can be obtained: the ambient Weyl group contains -1, which normalizes this subsystem and induces its nontrivial diagram option, so rational Weyl twisting gives the second form. Choose the sign \epsilon such that (3,q-\epsilon)=1.

Conjugation on the derived group gives a map \pi from M(q) to adjoint {}^{\epsilon}E_6(q). Its image is the simple group of that type. Indeed the simply connected derived subgroup already maps onto it: both the finite center and the Frobenius cohomology of its order-three algebraic center vanish under the chosen coprimality condition. Denote this finite simple image by \mathcal F_0, and let C_0\subset\mathcal F_0 be the torus used above. The kernel K_M=\ker\pi is central, with order dividing a product of q-\epsilon and powers of 3. It has no prime factor in common with |C_0|. Since \pi(D_M(q))=\mathcal F_0, every element of M(q) differs from an element of D_M(q) by an element of K_M. Thus

In particular, for every primitive prime r dividing |C_0|, this quotient has order prime to r, and every element of r-power order in M(q) belongs to D_M(q).

Define \mathcal S in projective E_7(q) to be the union of the conjugates of the images of those elements of M(q) whose projection belongs to C_0\setminus\{1\}. The preimage in M of the algebraic torus containing C_0 is a maximal torus of M, and also of E_7. Thus all these elements are semisimple. The set is nonempty by surjectivity, and it excludes identity: the ambient central elements project trivially to adjoint E_6.

We need their primitive-prime powers to be regular not only in E_6 but in E_7. Such a power lies in D_M(q) by the quotient calculation just given. Every E_7 root restricts to a nonzero weight on this E_6. To check nonvanishing, the fundamental weight perpendicular to the E_6 subsystem has squared norm 3/2; a root proportional to it would have rational proportionality factor of square 4/3, impossible. The restriction has length at most \sqrt2. Its orbit under the irreducible twisted Weyl action spans the six-dimensional weight space. The same index argument as in Equation (6.10) bounds the resulting weight-lattice index by 8\sqrt3<19, smaller than the primitive prime. Hence these powers are regular in E_7 as well.

Now conjugate W\in\mathcal S into the specified torus in M(q). Every element commuting with it projectively belongs to this torus: the center of simply connected E_7 has order at most two, so the primitive-prime power is actually centralized, and its connected centralizer is the torus. In particular a commuting Z\notin\mathcal S lies in M(q) and projects to 1 in \mathcal F_0; a nonidentity projection would lie in C_0 and put Z in \mathcal S.

For each conjugate of a nonidentity element of C_0 in \mathcal F_0, choose a lift B in M(q). Both B and BZ^2 have that nonidentity projection, so the preceding regularity argument applies to each. Their projective centralizers are toral and project into the corresponding conjugate of C_0. Projecting Equation (6.1) to \mathcal F_0 therefore puts the nonidentity projection w of W in

The counting proof of Lemma 6.1, already verified for this E_6 group, excludes this. This proves the obstruction in E_7.

For the simple Suzuki and rank-one Ree groups, use Equation (6.2) and their doubly transitive rank-one BN-pair action. A pair stabilizer has a torus of order q-1, containing a prime-order element of order greater than three. In Suzuki type q=2^{2a+1}\ge8, and q-1 is odd and not divisible by three. In Ree type q=3^{2a+1}\ge27, the number q-1 has 2-adic valuation one and an odd factor greater than one, not divisible by three.

Such a prime-order element t is regular. On the standard Frobenius-stable pair torus, the exceptional Frobenius F acts on roots by F^*\alpha=\ell^j\alpha', where \ell is the characteristic and \alpha' is a root of the opposite length; the factor \ell^j includes the field-Frobenius part. If \alpha(t)=1, then F(t)=t gives \alpha'(t)^{\ell^j}=1. Since the order of t is prime to \ell, also \alpha'(t)=1. The two roots are nonparallel, because a reduced root system has no parallel roots of different lengths. The index of the lattice they span in the weight lattice is at most 2\sqrt2 in B_2 and at most 2 in G_2, by the same length and covolume calculation as above. Neither index can be divisible by the chosen prime. Its centralizer is therefore the pair torus, which fixes both points individually. Choose the pair to be p,vp and apply the point-pair test.

Finally consider {}^2F_4(q), q=2^{2a+1}\ge8. Let \gamma be the normalized diagram reversal in F_4. The twisted Weyl action s_1s_2\gamma has square s_1s_2s_4s_3, a Coxeter element. The resulting finite torus C is contained in the Frobenius-square torus of order \Phi_{12}(q). Its order is one of the two factors

with the signs chosen together; these are Malle’s two cyclic tori, as recorded in [13, Theorem 8], and their orders multiply to \Phi_{12}(q). Equivalently, the determinant formula bounds its order between (\sqrt q-1)^4 and 4q^2. It is odd and greater than one. Every prime divisor is at least thirteen, and the F_4 root-lattice argument proves that every nonidentity element has centralizer precisely C. The standard group order gives |{}^2F_4(q)|>q^{26}/2. Also k({}^2F_4(q))\le100q^4; the sharper bound q^2+4q+17 is given in [11, Table 1, p. 3049]. Thus

and Lemma 6.1 applies.

7 The remaining unitary groups

We now pass from the odd-degree division case to arbitrary outer forms of type A. The induction is on the reduced degree, simultaneously over all number fields. Thus a special unitary group of an algebra of degree m over a finite extension of k is an admissible induction input whenever m is smaller than the degree currently under consideration. Restriction of scalars does not change this convention.

Write L/k for a quadratic extension, D for a central simple L-algebra of degree n, and * for a unitary involution on D. Set S=\SU(D,*) and U=\U(D,*). We continue to write i\in L^\times for a fixed element with i^*=-i. By the established isotropic cases and Proposition 2.3, it suffices to assume that S is k-anisotropic and prove V_0/R=1, where R=S(k)^e is an abstract power subgroup. In this section a group element is said to be killed when its image in the finite group S(k)/R is trivial. Lemma 2.4 will be used for smaller special groups and their restrictions of scalars. Its local power criterion applies to their images in S as well.

The degree-two case is an established inner type A_1 case of Theorem 2.1. For odd division degree, Proposition 3.1 and Theorems 4.1 and 5.1 give V_0/R=1. Three arguments remain: a ternary split-algebra case, the odd matrix-algebra case, and the even-degree induction step.

7.1 Uniform local power roots

Some of the smaller subgroups below vary with the element being factored. The following observation allows their local neighborhoods to be chosen in the fixed ambient algebra before those subgroups are constructed.

7.2 Ternary Hermitian forms

7.3 Odd matrix degree

Reduction to Hermitian dimension two is an established strategy in the unitary problem; see Tomanov [42, p. 895]. Here the induction parameter is the reduced degree over all number fields. Suppose n is odd and D=M_l(\Delta_0), where \Delta_0 is a division algebra of degree d and l>1. Then l\geq3 and n=ld. The classification of unitary involutions realizes S as the special group of a Hermitian l-space over a unitary division algebra [16, Theorem 4.2(2)]. Anisotropy implies that every subspace is nondegenerate.

The group is generated by its special transformations on two-dimensional \Delta_0-subspaces. To see this, choose a nonzero vector x and an isometry u. In a plane containing x,ux, Witt’s extension theorem gives an isometry sending x to ux. Its determinant can be adjusted on the orthogonal line in that plane. Here the unitary group of a nondegenerate \Delta_0-line maps onto L^1 by reduced determinant. At a nonsplit finite place, the existence of a unitary involution makes the Brauer corestriction of the algebra zero. For a quadratic extension of nonarchimedean local fields, corestriction preserves the local Brauer invariant, so the algebra itself is split. The line therefore becomes an ordinary Hermitian d-space, and its unitary determinant is onto by varying one orthogonal coordinate. At a nonsplit real place the extension is \bbC/\bbR, the algebra is again split, and the same argument works for every Hermitian signature. At split finite places use reduced norm surjectivity. At split real places the division degree is odd, so there is no quaternionic sign restriction. Globally, for d>1, each determinant fiber is a torsor under the simply connected special group of that line; Theorem 2.2 gives its rational point. For d=1 surjectivity is immediate. Dividing by the resulting special plane move fixes x, and repetition in its orthogonal complement proves the generation assertion.

If d>1, these plane groups have reduced degree 2d<n. They have no finite rank-zero places. At a split place this follows from their two-dimensional module over the local division algebra; at a nonsplit finite place the algebra is split and the Hermitian dimension 2d\geq6 is isotropic. The induction hypothesis therefore kills their rational points. If d=1, put each plane move inside a three-dimensional subspace and apply Proposition 7.3. This proves the odd matrix-algebra case.

7.4 A quadratic Hermitian subfield in even degree

Assume now that n=2m with m\geq2. We first construct a fixed quadratic subfield S_0\subset D, disjoint from L, on which * is the identity. The centralizer of S_0 will supply the smaller special unitary group in the induction.

Fix this s and put

The field LS_0 is biquadratic over k, with third quadratic field J_0. The algebra C_s has degree m over LS_0, and its involution is unitary over S_0. We shall factor a representative u into a quaternion norm-one element and an element of \U(C_s).

7.5 The factorization and its local norm conditions

For u\in S, define

These are identities in D. For example, expanding 4pp^* gives 2+usu^{-1}s^{-1}+sus^{-1}u^{-1}. We have p\in C_s, and both s and s' invert t by conjugation. Hence h commutes with both of them.

Over a splitting field, use the two s-eigenspaces to write

The two diagonal blocks of h have the same characteristic polynomial \Psi. On the open where the blocks are invertible, the first is

The analogous expression for the second block is conjugate to this one, since the two products of the invertible off-diagonal block maps are conjugate. The equality of characteristic polynomials then holds everywhere by polynomial continuation. Exchange of the s-labels preserves \Psi, and h=h^* preserves it under the unitary involution; thus \Psi\in k[T].

Set d_0=\Nrd_{C_s}(p^*). Then d_0\in J_0: conjugation of LS_0/S_0 takes it to \Nrd_{C_s}(p), and exchange of the two s-labels does the same, by the complementary-minor identities for u and \det u=1. Their product therefore fixes d_0, which is the condition that it belong to J_0.

We will choose u so that F=k(h) is a field of degree m and h(1-h)\ne0. When m\geq3, we also require that its normal closure have group S_m even after adjoining LS_0. Taking determinants of pp^*=h gives

In D, the elements

anticommute and satisfy

Indeed ss'+s's=2r(2h-1), from which both assertions follow by expansion. Since both squares are nonzero, the quaternion algebra they define over F is central simple, and its homomorphism into D is injective. Denote its image by Q. The involution acts on Q as quaternion conjugation. Moreover t=(is')j^{-1}, so z\in Q and its quaternion norm is h. Scalar extension gives LQ=C_D(F), a central simple algebra of degree two over LF. Consequently \Res_{F/k}\SL_1(Q) maps into S.

Suppose that we find a\in J_0F satisfying

The subfield J_0F is contained in both C_s and Q, and * acts on it as the involution over F. We obtain

The last membership follows because wq=u and w is unitary of ambient determinant one. For m\geq3, the second norm condition also gives \Nrd_{C_s}(q)=1, so q\in\SU(C_s). We now construct u,a with the local control needed to kill these factors.

Choose a finite set B containing all archimedean and fixed exceptional places, the places below the rank-zero places of \SU(C_s) over S_0, and the anisotropic finite places of S. At these places ask that u be close to 1. Additional places split in LS_0 allow us to prescribe all needed Frobenius cycle types of F. The block matrices

have determinant one and are arbitrarily close to 1 for small \varepsilon. Taking X with any prescribed separable residue factorization gives the desired local unramified algebra for F. These are open conditions on u. Prescribing representatives of every conjugacy class of S_m at places split in LS_0 ensures that the Galois subgroup over LS_0 is S_m: a proper subgroup misses a conjugacy class, by the derangement theorem for its coset action. For m=2 a single irreducible quadratic residue condition gives the required disjointness. Include these auxiliary places in B.

Near 1, the norm equations have local solutions near 1. Start with the analytic square root a_0=h^{1/2}\in F\otimes k_v; its norm from J_0F to F is h. For m\geq3, the ratio

lies in J_0^1, by Equation (7.5), and is close to 1. Choose its m-th root c\in J_0^1 close to 1 and replace a_0 by ca_0. This solves both norm equations. All reduced eigenvalues involved are ambient eigenvalues of matrices near 1, so a,w,q can be made uniformly small in the sense of Lemma 7.2.

7.6 Controlling every other place

It remains to ensure local solvability of Equation (7.7) and splitting of Q away from the finitely many places at which the factors are made small. To apply Proposition 2.6, we first specify the allowed local conditions.

Let \Omega\subset S be the fixed Zariski open on which \Psi is separable and h(1-h) is invertible. For each place outside B, let \mathcal U_v be the union of S(k_v)\setminus\Omega(k_v), the locus in \Omega(k_v) on which Q is split and the required norm equations are soluble, and a sufficiently small generic identity opening I_v\subset\Omega(k_v) on which the analytic-root construction supplies norm solutions with both factors w,q local e-th powers. The preceding analytic construction makes I_v nonempty. Thus a point of I_v is allowed even when its quaternion algebra is not split.

The integral and pole tests below will give the split-Q condition for points in \Omega. Including the complement of \Omega in \mathcal U_v lets these tests meet the universal integral-point condition of Proposition 2.6 without adding a discriminant divisor to its codimension-two exclusion. We will ensure that the final rational representative lies in \Omega by one of the auxiliary local openings already prescribed. After the tests, we specify which exceptional places use the small opening I_v and which use the split-Q condition.

At integral reduction, consider the four block-determinant hypersurfaces

Exclude their pairwise intersections. If all four determinants are nonzero and m\geq3, also require that \Psi have at least one simple root over the algebraic closure of the residue field. If exactly one block determinant vanishes, require, for m\geq3, that 0 be a simple root of \Psi for a diagonal block, or 1 a simple root for an off-diagonal block. The latter roots follow respectively from the block and complementary-minor determinant identities for h and 1-h.

All excluded loci have geometric codimension at least two. This can be checked first on \GL_{2m}: the four block determinants define distinct irreducible hypersurfaces. On the open where they are all invertible, the map u\mapsto B'(D')^{-1}CA^{-1} is a submersion, as is seen by varying B' with the other three blocks fixed. The transformation M\mapsto(1-M)^{-1} in Equation (7.4) preserves eigenvalue multiplicities on this open. For m\geq3, the locus of monic degree-m polynomials with no simple root has dimension at most \lfloor m/2\rfloor in coefficient space and thus codimension at least two. The characteristic-polynomial map on matrices is flat: each fiber has dimension m^2-m, by the decomposition into finitely many Jordan types and their centralizer dimensions. Thus the same codimension bound holds for its inverse image. On each single block hypersurface the requisite simple root holds on a nonempty open, as one sees from independent two-by-two transformations between paired eigenlines, with only one of the relevant scalar blocks vanishing. Finally, scalar multiplication changes none of the zero or root conditions, so passage from \GL_{2m} to \SL_{2m} preserves these codimensions. The union is Galois stable and spreads to the chosen integral models after finitely many exceptions.

Consider first an integral place outside the fixed exceptions where J_0 is inert. Frobenius interchanges the two diagonal block divisors and also the two off-diagonal block divisors. To check this, exchange of the s-labels swaps each pair, and the unitary diagram action does the same: in adapted dual bases it is inverse transpose, and complementary minors swap the paired determinants. Exactly one of these actions occurs when the third quadratic field J_0 is inert. A rational residue point therefore cannot lie on exactly one divisor. All four determinants are nonzero. Hence h and 1-h are units in every local component of F. The quadratic algebra J_0F/F is unramified or split, so these units are norms. In particular Equation (7.6) shows that Q splits, and one can choose unit preimages of h for the first norm equation.

For m\geq3 the simple residue root of \Psi supplies an unramified local factor E/k_v of F. It need not be a degree-one factor. Write K=k_v and J=J_0\otimes_k K, an unramified quadratic field in the case under consideration. If a_0 denotes the chosen unit preimages of h, the required correction is

by Equation (7.5). Hilbert 90 writes \eta=y/\bar y with y\in J^\times. Since J/K is unramified, multiplication by a power of a K-uniformizer makes y a unit without changing this quotient. The unramified etale algebra JE/J has surjective norm on units. Choose a unit x\in JE with N_{JE/J}(x)=y, and put \beta=x/x^*. Then

Extend \beta by 1 on the other factors of JF and replace a_0 by a_0\beta. This corrects the determinant while preserving the norm to F, for either parity of [E:K].

If J_0 splits, then Q splits automatically. Write a=(a_+,a_-) and d_0=(d_+,d_-). Choose N_{F/K}(a_+)=d_+ and set a_-=h/a_+; compatibility gives N_{F/K}(a_-)=d_-. Thus one component must have a prescribed norm from F. When all block determinants are nonzero, the prescribed value is a unit, so an unramified factor supplies it by ordinary unit norm surjectivity. If one determinant vanishes, the prescribed simple residue root is literally 0 or 1, so Hensel’s lemma supplies a degree-one factor of F. Its norm map supplies any prescribed value, including a nonunit. This proves all the local assertions at the permitted integral reductions. When m=2, no determinant norm is being imposed, so the splitting and first-norm assertions already suffice.

We must check the single poles permitted by Proposition 2.6. Enlarge its fixed splitting field to split L,S_0,D and all the fixed data. At a single pole the place splits in this field, and

with integral side factors. For m\geq3 impose total splitting of \Psi by requiring finitely many nonzero leading minors. Explicitly, if c_b is its b-th elementary coefficient, exterior-power expansion of the compression in Equation (7.4) gives

For completeness, the least exponent uses the first b exponent indices in the positive exterior power and the last b in its inverse. Its coefficient is the product of the corresponding opposite minors in the two rank-m projections

where P_+ is projection onto one s-eigenspace. All other terms have larger valuation. The indicated minors are nonzero for a generic conjugate of a rank-m projection; require them all to be units on the pole hyperplane. These are nonempty simultaneous open conditions. Their choices can be made in the staged order required by Proposition 2.6. With all other torus parameters equal to 1, the boundary of the factor with conjugator k_j has g_L=xk_j and g_R=k_j^{-1}. First choose the basic conjugators so that every right-projection minor for k_j^{-1} is nonzero, along with the basic-list dominance conditions. Then choose x in the intersection of the nonempty left-minor opens for the finitely many xk_j. All coordinate conjugates give only finitely many such dense open conditions. For an appended factor, with x already fixed, its conjugator k can be chosen in the intersection of the right-minor open and the inverse translate by x of the left-minor open. This intersection is again dense open and meets the required local approximation openings. Setting each new parameter to 1 preserves all the old boundary tests. This proves nonemptiness of every single-pole test without changing the earlier choices.

The consecutive slopes in Equation (7.10) are strictly increasing. Every Newton-polygon segment has horizontal length one, so \Psi splits completely over k_v. Since J_0 also splits here, all required norm equations are soluble and Q splits. For m=2, splitting of J_0 alone suffices.

The tests establish the required membership in \mathcal U_v at permitted integral reductions and single poles. It remains to retain global genericity and to choose the openings at the finite exceptional places. The final rational representative lies in \Omega because one of the existing auxiliary local openings is contained in \Omega(k_v). In Equation (7.9), choose X with separable characteristic polynomial and with both X and I+\varepsilon^2X invertible. Then h_+=I+\varepsilon^2X has distinct eigenvalues different from 0,1, and these conditions persist in its local opening. This argument uses the local matrix itself, not the reduction of \Psi, which can have repeated roots when \varepsilon is small. After the initial representative x is chosen, let B_1 be its finite set of denominator places outside B. At these places specifically choose the opening I_v\subset\mathcal U_v. This requires no splitting assumption; the block model in Equation (7.9) was needed only to prescribe Frobenius types at the auxiliary split places. The finitely many later model exceptions are assigned unit-parameter reductions with all four block determinants nonzero and separable \Psi; such opens are nonempty over sufficiently large residue fields, with small fields already included among the fixed exceptions, and thus use the split-Q branch as well. Consequently the small branch is used only at the controlled finite set B^+=B\cup B_1. The construction produces a representative of the required coset for which Q is split outside B^+. At every place Equation (7.7) is soluble with the prescribed near-identity solutions at B^+.

For m\geq3, the simultaneous equations form a torsor under the norm torus of Lemma 2.9, with quadratic field J_0 and degree-m field F. The full symmetric action over LS_0 implies the joint S_m\times C_2 hypothesis for F,J_0. That lemma gives a global a with the local approximations. When m=2, the quadratic Hasse norm theorem over F [22, Chapter VIII, Theorem 3.1] and weak approximation on its norm torsor give a. In either case the quaternion factor w in Equation (7.8) is killed: all its rank-zero places lie over the controlled set, where it is a local e-th power by Lemma 7.2. If m\geq3, the factor q lies in the smaller group \SU(C_s) over S_0 and is small at every rank-zero place of that fixed group. Induction and Lemma 2.4 kill q as well.

7.7 The determinant correction in degree four

It remains to treat m=2. The preceding construction has killed w and produced q\in\U(C_s)\cap S, small at the fixed set B, but its determinant in C_s may not be 1. Write

Both conjugations of LS_0 over S_0 and over L invert r_0: the first by unitarity, the second by ambient determinant one. Thus r_0\in J_0^1. Taking q sufficiently close to 1 at B ensures r_0\ne-1. Put

where the bar is the nontrivial automorphism of J_0/k. The element d is close to 1 at B.

We will remove this determinant by two elements in smaller binary special groups. The following construction gives the freedom needed to keep their varying rank-zero places under control. For any finite set of places outside the fixed exceptions, one can choose a quadratic field F_i=k(f_i)\subset C_s, disjoint from LS_0, with f_i^*=f_i, such that F_i splits at these places and the binary group \SU(C_D(F_i),*) is split over both local components of F_i. Here and below B includes all fixed exceptions of the structures defined from s.

To prove the construction, let B_s be the quaternion algebra over S_0 obtained by descent from C_s: its defining semilinear involution is the composition of * and standard quaternion conjugation. These involutions commute, and B_s\otimes_{S_0}LS_0=C_s. For a trace-zero element y\in B_s, the element f=i^{-1}y satisfies f^*=f. Prescribe f^2=c\in k^\times, so y^2=i^2c. A quadratic etale algebra embeds into B_s if and only if it is a field at every ramified place of B_s, by the local embedding criterion and Brauer invariants. We may impose this by choosing i^2c nonsquare there. At a finite place of k the kernel of the square-class map to a quadratic component of S_0 has at most two elements; in the split case it has one. Thus the forbidden square classes do not exhaust the local square classes. At real places the requirement is simply a sign. These ramified places are above B. At the prescribed places outside B require c to be a square, and at one further place split completely in LS_0 require it to be a nonsquare. Weak approximation supplies c. The last condition ensures that F_i is a field disjoint from LS_0.

The embedding of S_0(\sqrt{i^2c}) into B_s can approximate any prescribed local embeddings: they are conjugate, and weak approximation in the open affine variety B_s^\times approximates the conjugators. At a specified place split in L, ordinary split block embeddings make the two binary centralizers split. At a good unramified nonsplit place where S_0 splits, decompose both two-dimensional s-blocks into orthogonal lines with unit norm values. Give each f_i-eigenspace one line from each block. Its Hermitian determinant is a unit, and hence a norm, so each binary space is hyperbolic. If S_0 does not split there, it locally equals L; the two s-blocks are dual totally isotropic spaces. Choose paired line decompositions and assign one hyperbolic plane to each f_i-eigenvalue. In each case f_i is Hermitian and traceless in C_s, and the desired local property persists on an open set of embeddings. This proves the construction.

For such a field F_i, the compositum LS_0F_i is a maximal subfield of C_s, and there is an inclusion

Indeed the determinant in the binary algebra C_D(F_i) is the norm from LS_0F_i to LF_i. On J_0F_i its nontrivial automorphism exchanges the s-labels, which is the norm-one condition displayed on the left. The determinant of the same element in C_s is its norm from J_0F_i to J_0.

Choose F_1 first. Outside B, all but finitely many places have both of the following properties: d is a local norm from J_0F_1 to J_0 at every place above the given k-place, and the binary group in Equation (7.11) is isotropic at every place of F_1 above it. This follows from unramified unit norm surjectivity and good reduction. Call these the places covered by F_1. Choose F_2 by the preceding construction to split, with split binary groups in both components, at every remaining place outside B. Every such remaining place is covered by F_2, since the local norm from a split quadratic algebra is surjective.

Solve

This equation satisfies the Hasse principle and weak approximation on its locus where the variables are invertible. Indeed putting y_2=x_2^{-1} turns it into the nonzero locus of the nonsingular four-variable quadratic equation

Hasse–Minkowski [22, Chapter VIII, Theorem 3.5(b)] gives a rational isotropic vector whenever there are local ones. The corresponding projective quadric is rational once it has a point, and its affine cone minus the vertex has weak approximation. Removing the two norm-zero loci preserves the needed approximation to local points in this open set.

The requisite local points can be chosen with additional control. At B take x_1=\sqrt d and x_2=1, near 1; the square root is in the local J_0-algebra. Outside B, use one covering field for the entire k-place to carry the norm d, and take the other variable to be 1 at every component above that place. In particular, at any rank-zero place of one of the two binary groups, its own variable can be taken to be 1: that field cannot be the covering field there. These rank-zero places form a finite set. Use weak approximation in Equation (7.12) at their underlying places and at B, obtaining global invertible x_1,x_2 with these local approximations.

Set b_i=x_i/(x_i^*). By Equation (7.11), each b_i belongs to the indicated binary special group, of reduced degree two over F_i, and

At every rank-zero place of its binary group, b_i is as close to 1 as necessary. The A_1 case, local power lifting, and Lemma 2.4 therefore kill b_1,b_2. The element (b_1b_2)^{-1}q has determinant one in C_s and remains close to 1 at B. Its special group over S_0 has degree two and all its rank-zero places lie above B, so it too is killed. Together with the already killed factor w, this kills u.

The even induction step is complete. The degree-two base, the odd-degree division result, the odd matrix argument, and this step prove V_0/R=1 in every reduced degree over every number field. Proposition 2.3 now proves Theorem 7.1.

8 Groups of type D_4

For type D_4 there are no anisotropic nonarchimedean factors. Our objective is therefore to show that every finite quotient of the rational group is trivial. The argument first kills rational noncentral involutions and then represents every power-subgroup coset by a product of two such involutions. The second step requires a maximal torus on which inversion lifts rationally, not just geometrically.

The isotropic case is among the established cases recalled in Section 2. We may thus assume throughout this section that G is k-anisotropic. Fix a positive integer e, put R=G(k)^e, and write q:G(k)\to\mathcal Q=G(k)/R. By Proposition 2.3, this is a finite quotient and its cosets have the approximation properties used below. It will suffice to prove that \mathcal Q=1.

8.1 Root involutions and a quadratic splitting field

We record explicitly the lattice calculations used both here and in Section 9. With roots of squared length two, the coroot lattice of the simply connected group is

The two-torsion of a split maximal torus is Q_4/2Q_4. Its four central elements are the classes having all coordinates of the same parity. The other twelve classes are represented by the twelve pairs of opposite roots. These representatives are distinct: if two roots differ by an element of 2Q_4, their coordinate parities agree, and the defining even-sum condition then leaves only equality or opposition. The Weyl group is transitive on the roots, so all noncentral involutions form one geometric conjugacy class.

For n=\alpha^\vee(-1), the roots of C_G(n) are precisely the four orthogonal axes represented, after a Weyl conjugacy, by

The semisimple centralizer is connected, by the connected-centralizer theorem for simply connected groups [38, Section 2.10, p. 53]. The lattice spanned by these four roots has index two in Q_4, and

Consequently the centralizer and its simply connected covering have the geometric description

The product of the four standard reflection representatives acts by -1 on the torus and has square one in this quotient. Every other representative of inversion has the same square: for a in the torus and an inverter x, one has (ax)^2=a\,xax^{-1}x^2=x^2.

8.2 Local representatives of inversion

The next two lemmas give the local conditions imposed on a representative of an arbitrary coset of R. At a prescribed place we can choose a suitable open set freely. At the other places we will use a uniform condition on integral reduction.

8.3 A torus with no local-to-global obstruction

Write P_4=\Hom(Q_4,\bbZ) for the weight lattice and W=W(D_4) for its Weyl group. The following calculation explains why forcing the full Weyl group into a torus’s Galois action is sufficient.

We now construct the torus to which this lemma will apply. Fix a splitting field and a pinning of G there. At finitely many sufficiently large finite places that split completely in this field, prescribe unramified maximal tori with Frobenius in each conjugacy class of W. Such tori exist by Weyl twisting over the residue field and smooth lifting. Each has rational inversion by the no-wall argument in Lemma 8.4. Choose regular elements in them and impose small regular neighborhoods with the same local tori.

The labeling here matters. At each chosen place fix a completion embedding of the splitting field and use the fixed pinning. After a global identification of a new maximal torus with the pinned torus by inner conjugation, these local identifications differ by Weyl elements, not arbitrary diagram automorphisms. The image of the Galois group fixing the chosen splitting field thus meets every W-conjugacy class inside W. This image is all of W. Indeed a proper subgroup of a finite group cannot meet every conjugacy class: its transitive action on cosets would have no element without a fixed point, contradicting the average fixed-point count of one. The full character action \Gamma therefore contains W.

At each place, define the allowed set as the union of the elements that are not regular semisimple and the regular semisimple elements whose centralizer torus admits rational inversion. The auxiliary regular neighborhoods just chosen force the resulting rational element to be regular semisimple, hence regular semisimple at every completion. For this element, membership in the allowed sets therefore gives local rational inversion everywhere.

Apply Proposition 2.6 to any prescribed coset of R, with these auxiliary conditions and the openings from Lemma 8.3 at the other fixed exceptional places. We spell out the two uniform conditions needed for that proposition.

At integral good places, exclude the closed set where the semisimple part lies on two independent root walls. In a maximal torus these intersections have codimension at least two, as does their image under the finite quotient by W. The conjugacy-quotient map has fibers of dimension \dim G-\rank G, so its inverse image still has codimension at least two. To see the fiber dimension directly, a fiber is a union of conjugates of su, where s is fixed semisimple and u ranges over the unipotent variety of C_G(s); the latter has dimension \dim C_G(s)-\rank G. These descriptions and codimension bounds spread to the integral models after excluding finitely many places [38, Theorem 6.11, p. 64]. At every remaining integral place, Lemma 8.4 applies if the element is regular semisimple; otherwise it is in the allowed set by definition.

At a single pole, the place splits in the fixed splitting field used in Proposition 2.6. Use a split eight-dimensional vector representation. Choose the cocharacter to take distinct values on its eight weights, also after every relevant diagram translate. This requires avoidance of only finitely many hyperplanes in the cocharacter lattice. Order the exponents as d_1<\cdots<d_8. In the single-pole normal form g_L\lambda(t)^e g_R, require all initial principal minors of g_Rg_L in this order to be nonzero upon reduction. These conditions are nonempty open conditions: they hold at the identity. For each pole factor, at least one of the two disjoint complete basic parameter lists lies entirely before or after that factor. After cyclically combining the side factors as g_Rg_L, this list occurs between fixed invertible translates. Its product map is dominant, so it meets the simultaneous principal-minor open set. This argument includes poles in the first or last factor; no boundary factor needs parameters on both sides. The jth characteristic coefficient then has a unique least-valuation term, of valuation e(d_1+\cdots+d_j)\ord_v(t). The Newton polygon has eight segments of horizontal length one with distinct integral slopes. Hensel lifting therefore splits the characteristic polynomial into distinct linear factors over k_v. The centralizer torus is split: the vector representation has finite kernel, so its weights span the character space over \bbQ, on which Galois now acts trivially. Split tori have rational inversion representatives by the lattice calculation above. The finitely many extra model or denominator exceptions are treated by the local openings, as provided by Proposition 2.6.

We have proved that every coset of R has a regular semisimple representative u for which T=C_G(u) has everywhere locally rational inversion and character action containing W.

9 Groups of type E_6

The last case reduces a generic rational element to a product from a subgroup of type D_4 and the centralizer of a root involution. That centralizer has types A_1 and A_5. The geometric product decomposition alone is insufficient: we will identify its rational descent obstruction as a torsor under a simply connected semisimple group, and arrange the required local points.

Again A=\varnothing, and only the k-anisotropic case needs proof. Fix e\ge1, set R=G(k)^e, and write q:G(k)\to\mathcal Q=G(k)/R. Our goal is to kill each coset of this finite quotient.

9.1 A rational root involution

Fix n as in the lemma, put H=C_G(n), and denote its simply connected cover over k by \pi:\widetilde H\to H. Its almost simple factors have types A_1 and A_5, for which (MP) is now known.

9.2 The generic D_4 subgroup

9.3 The full pullback and the factorization torsor

We next justify the rational descent of the preceding geometric factorization. It is important here to identify the entire pullback subgroup, not just its identity component.

To use this torsor, we must choose local factorizations that are small in the \widetilde H factor. The next elementary observation ensures that the square root used for this purpose remains in the varying torus T_*.

Fix a faithful representation of G. At any completion, sufficiently near the identity, the matrix analytic square root t\mapsto t^{1/2} is defined and commutes with conjugation. For all the tori T_* in Lemma 9.3, one can choose this neighborhood uniformly so that

Indeed these tori form one geometric conjugacy class of subtori, and t is regular semisimple, so its image in the fixed representation is diagonalizable. In such a diagonalization their character relations are a fixed finitely generated lattice of monomial relations, up to a permutation of the eigenvalues. The square root is defined by a convergent matrix power series and commutes with conjugation. Its eigenvalues are uniformly near one when the original matrix is near one: a bound on the matrix norm bounds the absolute values of all its eigenvalues, without any bound on a diagonalizing matrix. Every defining monomial evaluated on the root eigenvalues has square one, because t\in T_*. After shrinking a single neighborhood using the finite list of relations, each such monomial must be one, proving Equation (9.5). This argument applies also at the places above two, by choosing a smaller analytic neighborhood.

9.4 Completion of the normal-subgroup theorem