The Margulis–Platonov conjecture over global function fields

1 Introduction

Let k be a global function field: a finite extension of \bbF_q(t). Let G be an absolutely almost simple simply connected algebraic group over k. The group G(k) is considered as an abstract group. Its normal subgroups nevertheless reflect the topologies of its local groups. For a place v of k, write k_v for the completion and set

The set A(G) is finite. Each of its factors is compact, and H_A carries the product topology. Let \delta_A:G(k)\longrightarrow H_A be the diagonal homomorphism. The Margulis–Platonov conjecture asserts that every noncentral normal subgroup of G(k) is obtained by pulling back an open normal subgroup of H_A.

If A(G) is empty, the product H_A is trivial, and the theorem says that G(k) has no proper noncentral normal subgroup. In general it identifies all noncentral normal subgroups using finitely many compact local groups. In particular, every finite quotient of G(k) is continuous for the topology induced by these local factors: its finite-index kernel is Zariski dense and therefore noncentral. The substantive issue is this particular local description: finite index alone does not identify which local topology detects a normal subgroup.

1.1 Historical context

The conjecture asks whether the compact groups at the anisotropic places account for every noncentral normal subgroup of the rational-point group. Its origins lie in Kneser’s simplicity question for quaternion norm-one groups and Platonov’s extension of that question to simply connected simple groups. Margulis formulated the description by anisotropic local factors over global fields in [20, Russian original, Section 2.4.8]. The finite-index assertion in his Corollary 2.4.9 is an essential first step. The further problem is to show that each resulting finite quotient comes from the specified local groups. This also explains the conjecture’s role in the congruence subgroup problem, which studies finite-index subgroups of arithmetic groups and needs additional information about their congruence completions [31, pp. 73–76].

For anisotropic inner forms of type A, the problem concerns norm-one groups of division algebras. The quaternion case was completed by Margulis in 1980, as recalled in Tomanov’s historical account [44, p. 895]. Tomanov later proved the conjecture for division algebras of power-of-two index and reduced arbitrary index to odd index [44, Theorem and Corollaries 1–2, p. 896]. Platonov–Rapinchuk and Raghunathan developed the arithmetic control of commutator subgroups [24, 29]; its global-field form is used explicitly in [33, proof of Theorem 6.4]. Rapinchuk–Potapchik reduced the number-field problem to excluding nonabelian finite simple quotients of the multiplicative group of the division algebra [30, Theorem 2.1 and Corollary 2.5]. Segev’s division-algebra argument and Segev–Seitz’s analysis of commuting graphs supplied this exclusion [38, Theorem A][39, Theorems 1–3]. Rapinchuk–Segev–Seitz proved the stronger result that every finite quotient of the multiplicative group of a finite-dimensional division algebra is solvable; their Theorem 6.4 proves the Margulis–Platonov conjecture for inner type A over arbitrary global fields [33]. Rapinchuk subsequently gave a shorter proof using two-generation of finite simple groups [32].

Over a global function field, Harder’s results restrict the anisotropic absolutely almost simple simply connected groups to type A [15][26, p. 914]. The isotropic case follows from the global Kneser–Tits theorem and projective simplicity [13, Theorem 8.1]. Together with the established inner-type-A theorem, these results reduce the present problem to anisotropic outer forms of type A, represented by special unitary groups. The earlier state of the conjecture is discussed in [28, Section 3.5]; general outer-type-A applications were still explicitly conditional on it in [34, Corollary 1.6]. Theorem 1.1 provides the required normal-subgroup description for these groups in every characteristic.

1.2 Relation to the number-field proof

The immediate predecessor is the number-field theorem of [23]. We use its approach through finite quotients of abstract power subgroups, approximation within a prescribed coset, and the distinction between finite characters and nonabelian simple quotients. More specifically, the power-factor construction comes from [23, Proposition 2.6], and the finite lattice calculation for the symmetric norm torus is the argument of [23, Lemmas 2.7–2.9]. The difference-function and exact-transitivity argument for finite characters adapts [23, Section 4]. The commuting approximation and circle-extension reduction follow [23, Section 3.1 and Sections 3.3–3.7]; the invariant probability law follows [23, Section 5.2]. The finite-group obstruction is [23, Lemma 5.2 and Section 6], and the unitary induction follows [23, Section 7]. We reproduce the lattice calculation and the finite-group argument. We also verify an additional subgroup-complement property of the chosen finite-group subsets, which is needed for the present recurrence proof.

Several independent inputs retain their own roles. We use Prasad’s strong approximation theorem over function fields [25]; the local and global cohomological results of Bruhat–Tits and Harder provide the structural reductions [2, 15]. Demarche–Harari’s duality and local-global theorems supply the positive-characteristic arithmetic for tori and homogeneous spaces [6, 7]. The passage from a simple quotient to an action of the full unitary group uses Feit–Seitz’s theorem that class-preserving automorphisms of a finite simple group are inner [9, Theorem C]. The circle-extension argument uses Prasad–Rapinchuk’s metaplectic-kernel computation [27], while the invariant probability law uses Howe–Moore decay in its nonarchimedean form [16][3, Theorem 4.19 and Definition 4.23].

The changes in positive characteristic occur at specific points of this argument. When the exponent is divisible by the characteristic, the power map need not be locally invertible. We adapt the inherited construction by products of powers to this setting, separating the inseparable part of the exponent and using the openness of the relevant local power subgroups. The additive group of the field has finite exponent, so the integer-dilation recurrence used in the number-field proof does not give the required invariance. That proof uses the density theorem of Furstenberg–Katznelson [11, Theorem B] in [23, Section 5.5]. Here we prove a replacement by finite additive Fourier analysis. Odd characteristic uses orthogonality on finite additive subgroups and a quadratic parametrization; characteristic two uses additive polynomials and finite-dimensional coordinates over power subfields. The resulting invariance is then combined with the finite-group obstruction and the unitary induction.

1.3 The argument

The structural reductions leave an anisotropic special unitary group S=\SU(D,*), where D is a central simple algebra of degree n\geq3 over a separable quadratic extension L/k, and * is a unitary involution. Write U=\U(D,*) for the full unitary group. Every noncentral normal subgroup of S(k) has finite index. It therefore contains the subgroup R generated by all eth powers for some positive integer e. For this e, define

The closure is taken in the product of the anisotropic local groups. The group V is finite; it measures the possible difference between an abstract power subgroup and the subgroup prescribed by its local closure. We prove V=1 for every e by induction on n, simultaneously over all global function fields.

Two arithmetic constructions permit us to work with this finite group. First, products of conjugated torus powers let us choose a rational point in an actual prescribed coset of R, while imposing conditions at finitely many places and controlling the remaining places by a codimension-two exclusion. A line in the parameter space avoids that exclusion on reduction. The anisotropy of the torus makes the parameter map invariant under scalar multiplication; this recovers approximation at the place omitted from strong approximation. Second, a particular norm torus has both the Hasse principle and weak approximation. This turns locally solvable simultaneous norm conditions into exact rational solutions. Sections 3 and 4 establish these constructions.

Suppose first that D is a division algebra and n is odd. Commuting approximation makes the action of U(k) on V preserve conjugacy classes. If V\ne1, a simple quotient and a circle-extension argument then produce one of two obstructions: a finite abelian character of U(k) nontrivial on S(k), or a homomorphism from U(k) onto a finite nonabelian simple group whose restriction to V_0 is surjective and which kills R. Section 5 proves this dichotomy.

The simple quotient gives a finite coloring of the Hermitian elements of D through a Cayley parametrization. Local mixing supplies an invariant probability law for all translated colorings, with uniform color marginals. A finite-group argument selects a nonempty proper conjugacy-invariant subset of the target group and constrains returns between colors inside and outside that subset under fractional transformations. Additive recurrence in characteristic p forces the indicator of membership in this subset to be translation invariant. The fractional transformations then generate left rotations through every target color, forcing the chosen subset to be either empty or the whole group. Sections  6 and 7 develop this contradiction; Appendix A proves the finite-group input. Finite abelian characters require a different argument. Exact transitivity on nonzero pairs in D^2 and the additive span of the full unitary group reduce them to the known inner-type-A theorem in Section 8.

It remains to pass from odd division degree to all degrees. Section 9 does this using smaller unitary groups and quaternion norm-one groups. In even degree, a fixed quadratic Hermitian subfield supplies a centralizer of half the degree. A factorization separates an element into a quaternion factor and a factor in that centralizer. The approximation construction controls the anisotropic places of both factors, and the norm-torus theorem makes the factorization rational. Degree four requires a separate two-norm correction. Uniform control of local power subgroups allows this induction even when p divides e. Finally, Section 10 converts V=1 into Theorem 1.1.

2 Arithmetic reduction and exact cosets

The proof will show that certain finite quotients of a special unitary rational group are trivial. We first explain why these quotients suffice, and why every one of their cosets admits the local approximations used later. Throughout, a global function field has its full finite field of constants, and all quadratic field extensions are separable. We use normalized absolute values at its places. A product indexed by the empty set is the trivial group.

2.1 The remaining unitary groups

We use three established results to isolate the case that requires proof. For a simply connected absolutely almost simple isotropic group over a global field, the global Kneser–Tits theorem and Tits’s simplicity theorem give perfection and simplicity modulo the center of the rational group [13, Theorem 8.1][43]. A noncentral normal subgroup therefore has central, hence abelian, quotient and must be the whole rational group. In this case the algebraic group is isotropic at every completion, so there are no anisotropic places. The Margulis–Platonov theorem is also known for every inner form of type A over every global field [33, Theorem 6.4]; see also [32, Proposition 1]. Finally, Harder’s theorem implies that an anisotropic absolutely almost simple group over a global function field has inner or outer type A [15][26, p. 914]. Consequently it remains to consider an anisotropic outer form of type A.

We write such a group as

where D is a central simple algebra over a quadratic field extension L/k and * is an involution inducing the nontrivial automorphism of L/k. The algebra D need not be a division algebra. Degree two gives an inner form of type A_1, already covered by the preceding result. Write \bar a for the conjugate of a\in L and

On U, the notation \det will always mean the reduced norm \Nrd_{D/L}. It takes values in the norm-one torus \Res^1_{L/k}\bbG_m, whose group of rational points is L^1. We retain the notation

The set A is finite: outside finitely many places a reductive model has quasi-split fibers and positive local rank. Each factor of H_A is compact.

2.2 Finite index and the power quotient

We will use the local Kneser–Tits theorem in the following form. If a simply connected absolutely almost simple group over a nonarchimedean local field is isotropic, its group of local points is perfect and is simple modulo its finite center. In particular, a noncentral normal subgroup is the whole group. The perfection clause matters: simplicity modulo the center first gives a central quotient, and perfection then makes that quotient trivial. This local result applies in equal characteristic, including characteristics two and three [43, 2, 13].

For an integer e\geq1, let

The notation S(k)^e means the subgroup generated by powers, not merely the image of the power map. The distinction will be essential when the characteristic divides e.

Our task is to prove V=1 for every e. The quotient V measures precisely the failure of the power subgroup R to contain all rational points allowed by its closure at the anisotropic places. The next proposition provides the approximation in each individual coset of R; passing only to its image in a finite quotient would not suffice. The power-subgroup reduction and this use of exact cosets follow the organization of [23, Section 2.2]; the arithmetic inputs above are their function-field forms.

2.3 Finite quotients of groups already understood

The induction will restrict the finite quotient S(k)/R to smaller special unitary groups, often over finite extensions of k. The precise consequence of their Margulis–Platonov theorem is the following.

3 Approximation in a prescribed power coset

The closure theorem of Section 2 permits approximation at finitely many places within a prescribed coset of the subgroup generated by powers. We shall also need to control a local property at every other place. The construction in this section achieves that control when its failure at integral points has geometric codimension at least two, provided certain explicitly described nonintegral points satisfy the property as well. The output remains in the prescribed abstract coset throughout.

We adapt the line-avoidance and power-factor constructions of [23, Sections 2.3–2.4]. In positive characteristic, a power map need not have a nonzero differential. We separate its inseparable part before making the geometric codimension argument. We also keep track of the order in which the finite sets of exceptional places are chosen; this will allow the construction to be applied when the local groups arising later depend on the rational point being constructed.

3.1 Avoidance in affine space

For a place v, write \mathcal O_v for the valuation ring of k_v, \kappa_v for its residue field, and q_v=|\kappa_v|. An integral model of an affine k-variety means a model over the ring of integers outside a finite set of places. All assertions about reduction below refer to fixed such models. Enlarging the finite set allows us to extend any fixed finite collection of morphisms and equations to these models.

The following elementary form of avoidance leaves one place unrestricted. That freedom will later be removed using a symmetry of the parameter map.

3.2 Products of torus powers

For the remainder of this section, let S=\SU(D,*) be k-anisotropic of degree n\geq3, with notation as in Section 2. Fix e\geq1 and let R=S(k)^e denote the subgroup generated by all e-th powers. We allow the ambient variety to be S itself or a rational translate X=x_*S in U, where x_*\in U(k).

Choose a maximal k-torus T\subset S and a finite Galois extension M/k splitting T and any fixed auxiliary data needed in an application. Write p=\operatorname{char}k. Over M, identify T with the determinant-one diagonal torus in \SL_n. Choose a cocharacter

where the integers w_i are distinct, have sum zero, and are not all congruent modulo p. Such a choice exists for n\geq3: start with a vector in \bbZ^n of sum zero whose residues are not all equal, and add multiples of p of sum zero to make its entries distinct. In particular d\lambda(1) is noncentral in \mathfrak{sl}_n, even if p divides n.

The restriction of scalars of \lambda, followed by the torus norm, defines a k-morphism

Over M the coordinates of its source are indexed by \sigma\in\Gal(M/k), and its formula is \eta((a_\sigma)_\sigma)=\prod_\sigma \lambda^\sigma(a_\sigma). On the scalar copy of \bbG_m, this map has cocharacter \sum_\sigma\lambda^\sigma. That cocharacter is defined over k and must vanish, since S is anisotropic. Consequently

for every field extension K/k.

For x\in X(k) and k_1,\ldots,k_r\in S(k), consider

The parameter domain is the open torus \mathcal T_r=(\Res_{M/k}\bbG_m)^r in the affine space E_r=(\Res_{M/k}\bbA^1)^r. For rational parameters,

Indeed each factor is the e-th power of k_j\eta(a_j)k_j^{-1}\in S(k).

We call either of the two lists in Lemma 3.2 a basic list. Its actual map with exponent e is dominant, although it need not be smooth. We need one further consequence of dominance, uniform in the translating point.

3.3 Boundary conditions and the approximation theorem

The boundary of \mathcal T_r in E_r is a union of hyperplanes after extending scalars to M:

We call a reduction a single-pole reduction if it lies on exactly one of these hyperplanes. At a place unramified in M, Frobenius must fix the unique vanishing coordinate. Its action on the embeddings of the Galois extension M/k is regular; hence this place splits completely in M. Near this boundary hyperplane, away from all the others, (3.3) has the form

where t=a_{j,\sigma} and the two side factors are regular. At a single-pole reduction outside model exceptions, t has positive valuation and the side factors are integral and invertible. Thus the necessary local analysis can be done in split matrices, even though S is anisotropic over k.

In the theorem below, a pole test means finitely many algebraic nonvanishing conditions on H_{j,\sigma}\setminus\bigcup_{(i,\tau)\ne(j,\sigma)}H_{i,\tau}. The conditions may involve x and the conjugators as coefficients. They must be taken with all their Galois conjugates. The theorem states explicitly the nonemptiness requirement that applications must verify.

4 Simultaneous norm equations

Prescribing a determinant inside a unitary torus is a norm equation. Later we shall prescribe two norms of the same element: one to a degree-m field and one to a quadratic field. This section proves that the resulting equations satisfy the Hasse principle and weak approximation when the two fields have the indicated symmetric Galois group. The finite-lattice argument is that of [23, Lemmas 2.7–2.9]; we reproduce it and supply the global-function-field duality input.

For a finite separable extension E/K, write \Res^1_{E/K}\bbG_m for the kernel of its norm map.

We first compute the character lattice and then express its arithmetic obstruction in terms of a finite group. A k-lattice is a finitely generated free abelian group with a continuous action of the absolute Galois group of k; its action factors through a finite quotient. Let P be the joint normal closure in the proposition and let c generate the second factor of \Gamma=\Gal(P/k)=S_m\times C_2. The character map dual to the norm in Equation (4.1) is the diagonal inclusion

Here S_m permutes the m coordinates and c acts by -1. The inclusion is primitive, so its cokernel is a lattice. Consequently the norm is surjective with connected kernel T, and

For any k-lattice M, set

and let \Sha^2_\omega(k,M) denote the subgroup of classes whose localizations vanish at all but finitely many places. All Galois cohomology here uses continuous cochains and discrete coefficients.

The arithmetic problem has now reduced to showing that Equation (4.4) vanishes for the specific lattice in Equation (4.3). We use the interpretation of H^2(\Gamma,M) as extensions of \Gamma by the abelian group M inducing the specified action. Restriction to a subgroup is zero exactly when the extension splits over that subgroup. A central element acting by -1 makes these extensions particularly concrete.

5 Finite quotients in odd division degree

Assume throughout this section that D is a division algebra of odd reduced degree n\geq3 over the quadratic extension L/k. Put S=\SU(D,*) and U=\U(D,*), as in Section 2. The one-dimensional Hermitian space over D is anisotropic, so S is k-anisotropic. The reduced norm gives a surjective homomorphism

with kernel S(k). Recall that, for a fixed integer e\geq1,

Here S(k)^e denotes the subgroup generated by the eth powers. The group V is finite by Proposition 2.3. Our purpose is to turn a nontrivial quotient of V into a finite quotient defined on the full unitary group.

The argument follows the quotient construction of [23, Section 3, especially Propositions 3.1 and 3.2]. We give the details because two parts require care in positive characteristic: commuting elements must be constructed in separable field tori, and the treatment of local principal units must not assume that the subgroup of nth powers is open. We first prove the commuting approximation statement that controls the conjugation action on V.

5.1 Commuting elements with prescribed determinants

At a place v\in A, the extension L/k splits and

where \mathcal D_v is a central division algebra of degree n over k_v. A separable field torus in U(k_v) means the multiplicative group of a maximal separable subfield of \mathcal D_v. This qualification matters when the characteristic divides n.

5.2 The action of the full unitary group

The group-theoretic passage from a class-preserving action to a quotient of the ambient group is adapted from [30, Corollary 2.3, Theorem 2.4, and the proof of Theorem 2.1]. Here Proposition 5.2 supplies the unitary input. Conjugation by U(k) preserves R, which is characteristic in S(k), and consequently preserves P_0 and V_0. We show that its action on V preserves every conjugacy class. Take \gamma\in V_0 and h\in U(k). Proposition 5.2(ii) gives x\in\gamma R commuting with t, where \det t=\det h. Choose the approximation at A sufficiently close that th^{-1}\in V_0. Conjugation by t fixes \gamma R, while conjugation by th^{-1} is inner on V. Hence conjugation by h sends \gamma R to a conjugate of itself.

Suppose now that V\ne1, and choose a simple quotient C of V. Every normal subgroup of V is a union of conjugacy classes, so the kernel of V\to C is preserved by U(k). The induced automorphisms of C are class-preserving. If C is nonabelian, the theorem of Feit–Seitz [9, Theorem C] makes each of these automorphisms inner. As C has trivial center, the action gives a homomorphism

On V_0 it is the original quotient map and it kills R. This proves the second alternative of Proposition 5.1.

It remains to handle the case in which C is cyclic of prime order. Let R' be its kernel upstairs. Then

The rest of the section uses this nonzero central quotient to construct a finite character of U(k) that is nonzero on S(k). We shall use the compact local groups at A to construct and split a circle extension.

5.3 Characters of a local norm-one group

The required local calculation is the division-algebra commutator calculation underlying [36]; we include its filtration proof, using the standard cyclic description of local division algebras [35]. It explains why divisibility of n by the characteristic causes no additional abelian quotient.

5.4 A circle extension associated to the central quotient

Write I=\bbR/\bbZ, additively, and fix an injective character \iota:C\hookrightarrow I. Give L^1 the discrete topology and set

The determinant kernel is S_A=\prod_{v\in A}S(k_v). Thus Q is locally compact and second countable, with countably many open determinant fibers, each a translate of the compact group S_A. The map

has dense image on every fiber by weak approximation for S. It follows that P_0 is open and normal in Q and that

For normality, first conjugate by the dense subgroup \rho(U(k)) and then pass to its closure; the quotient identification follows from density and openness, with kernel V_0.

The central quotient in (5.2) gives the discrete central extension

Pull it back along Q\to Q/P_0\simeq U(k)/V_0, using (5.6), and push its kernel out by \iota. We obtain a locally compact second-countable central extension

The pullback construction gives continuous local sections and an abstract homomorphic lift

It also gives a canonical continuous homomorphic section c:P_0\to\widetilde Q, obtained by using the identity in the discrete group U(k)/R'. For u\in V_0,

We regard values in I as central factors in this multiplicative notation for \widetilde Q. Comparing \ell with a continuous splitting of p will produce a character of U(k). Identity (5.8) will ensure that its image on S(k) is finite and nonzero.

The restriction of (5.7) to S_A has a continuous homomorphic splitting. Here is the metaplectic input with its hypotheses made explicit. Choose an isotropic place v_0 of S outside A, and pull the extension back to S(\bbA^{v_0}) by projection to S_A. The lift \ell|_{S(k)} splits this pullback on rational points. The metaplectic kernel away from v_0 is zero, since S is absolutely almost simple and simply connected and v_0 is a nonarchimedean isotropic place [28, Theorem 3]; see also [27]. The pullback extension consequently has zero measurable cohomology class. A measurable homomorphic section is continuous for these locally compact second-countable groups. Restrict it along the subgroup S_A\subset S(\bbA^{v_0}) supported at A to obtain

This invocation involves only nonarchimedean places and therefore has no omitted-archimedean-place qualification.

We must extend this splitting from S_A to Q. First we will make j(S_A) normal. Its quotient will then be a central extension of the abelian determinant group L^1; we will remove that extension’s commutators by a further change of j.

5.5 Making the splitting equivariant

For q\in Q and any lift \widetilde q, define d_q:S_A\to I by

This is a continuous character, independent of the lift. It is zero for q\in S_A, and composition of conjugations gives

Thus it is a character-valued cocycle factoring through Q/S_A=L^1. Lemma 5.3 identifies its coefficient group as

On T_v, a determinant a acts by \sigma_v^{\ord_v(a_v)}; on its character group, the action in this cocycle is pullback, \lambda\mapsto\lambda\circ\sigma_v^{\ord_v(a_v)}.

The vth component of d_q is zero if \ord_v(a_v)=0. Indeed, since d_q depends only on the determinant, choose a representative whose vth component is a unit in an unramified maximal subfield with reduced norm a_v; norms from that field supply every unit. It commutes with all Teichmüller lifts of T_v, supported at v in S_A. At the other places choose regular semisimple representatives of the specified determinant, so that these components and the identity belong to separable field tori. Lemma 5.4 then makes (5.10) zero on those Teichmüller lifts, which determine the character. The component and its action therefore factor through the single valuation map L^1\to\bbZ at v. This map is onto by weak approximation in L^1.

Let d_v be the discrepancy for valuation one. It annihilates T_v^{\sigma_v}: the Teichmüller lifts of this subgroup are central scalars, and we may choose regular separable representatives of the determinant and again apply Lemma 5.4. For a finite abelian group T with automorphism \sigma,

This is the dual of the kernel-image sequence for \sigma-1; characters of a subgroup extend to I because I is divisible. Choose a correcting character in each local factor, and let \lambda:S_A\to I be their sum. For j_\lambda(s)=j(s)\lambda(s), direct substitution gives the discrepancy

The characters may therefore be chosen to make this discrepancy zero at the generator of each valuation group. The cocycle identity then makes every discrepancy zero. Hence we may assume

In particular, j(S_A) is normal in \widetilde Q. It is also compact, hence closed, because S_A is compact.

5.6 Removing the determinant commutators

The quotient by j(S_A) is a central extension

Commutators define an alternating biadditive pairing \omega:L^1\times L^1\to I, with multiplicative arguments. Lemma 5.4 shows that

whenever, at every v\in A, both a_v and b_v are norms from one common maximal separable subfield of \mathcal D_v.

We reduce this pairing to characters of local residue units. Put

Using unramified maximal subfields in (5.13) gives \omega(U_0,U_0)=0. The valuation map L^1\to\bbZ^A is onto by weak approximation. For each v\in A, choose a maximal totally ramified separable field E_v/k_v inside \mathcal D_v. Its norm group is open: the separable norm map has surjective trace differential, and hence has open image on local points. That norm group contains an element of valuation one. Weak approximation now gives elements \pi_i\in L^1, i\in A, whose valuation vectors are the standard basis vectors and all of whose components at v belong to N_{E_v/k_v}(E_v^\times). The common-norm test gives \omega(\pi_i,\pi_j)=0 for all i,j. Every element of L^1 is a product of an element of U_0 and powers of the \pi_i. Only the pairings \omega(\pi_v,u), u\in U_0, remain to be considered.

For fixed v, this pairing vanishes whenever u_v\in N_{E_v/k_v}(E_v^\times): use E_v at v and unramified maximal subfields at the other places, where both determinants are units. Put

The group H_v is open in the compact unit group, so its index is finite. Weak approximation makes the projection U_0\to\mathcal O_v^\times/H_v onto. The vanishing just proved says that its kernel is killed by u\mapsto\omega(\pi_v,u). Thus this pairing factors through that finite quotient. Composing the quotient character with the projection from \mathcal O_v^\times defines a continuous character \eta_v extending the pairing. Since the norm group contains all nth powers, n\eta_v=0. This argument uses openness of a separable norm group, not openness of the nth-power subgroup.

We claim that \eta_v kills the principal units. Let c\in k_v have valuation one. Perturb the Eisenstein polynomial X^n-c, with its constant coefficient fixed, to separable Eisenstein polynomials f_j tending to it. If the characteristic divides n, one may take f_j=X^n+\varepsilon_jX-c with nonzero \varepsilon_j\to0. If it does not divide n, no perturbation is necessary. Let \alpha_j be a root and E'_j=k_v(\alpha_j). These degree-n separable fields embed in \mathcal D_v, and oddness of n gives

For each j, choose \pi'_v\in L^1 with the same valuation vector as \pi_v, with vth component in N_{E'_j/k_v}(E_j'{}^\times). At the other places choose unit components. Weak approximation and openness of the norm groups permit this choice. Since \pi'_v/\pi_v\in U_0, their pairings with U_0 agree. Choose elements of U_0 whose vth components belong to the same norm group and tend to N_{E'_j/k_v}(1+\alpha_j). At the other places their components, and those of \pi'_v, are norms from unramified maximal subfields. The common-norm test and continuity give

Letting j tend to infinity gives \eta_v(1+c)=0. These units generate 1+\mathfrak p_v: if \ord_v(d)\geq2, then

and both numerator and denominator have the form 1+c' with \ord_v(c')=1. Thus \eta_v is a character of \bbF_{q_v}^\times, killed by n.

We can remove these residual characters without losing (5.11). The permissible changes of j are characters of S_A invariant under Q, whose vth factors are the characters of T_v/(\sigma_v-1)T_v. If a section is multiplied by such a character \lambda, the commutator pairing changes by -\lambda evaluated on the ordinary local commutator of determinant representatives. To see this, write the commutator of any two lifts as j of their base commutator times the old value of \omega; substitution of j\lambda gives the asserted formula.

To compute the effect on \eta_v, represent the determinant (\pi_v)_v by a division prime multiplied by a unit of the fixed unramified maximal subfield. Such a unit adjusts the reduced norm to the specified value because the ratio of the two norms is a unit. It commutes with every other unit of that subfield, so this adjustment does not affect their commutator. For an unramified unit with residue r, the residue of that commutator is therefore \sigma_v(r)/r. Changing the norm preimage r alters it by (\sigma_v-1)T_v, so there is a well-defined homomorphism

It is onto by finite-field Hilbert 90. Its target has order \gcd(n,q_v-1): on the cyclic group T_v the cokernel of \sigma_v-1 has the same order as its kernel, and

The domain in (5.14) is cyclic. Its dual therefore identifies all characters killed by n with characters of that target. Choose the invariant local character that cancels \eta_v, and make these changes for every v\in A. At other places the determinant representatives for a mixed pair can be taken in unramified subfields, so no other local component is introduced. Unit-unit pairings and pairings of the chosen \pi_i still vanish by (5.13), which holds for every equivariant choice of j. All the generating pairings now vanish, so \omega=0.

The extension (5.12) is consequently abelian. The divisible group I is injective as an abelian group, so the extension splits, continuously since its base L^1 is discrete. Take in \widetilde Q the inverse image of the image of such a section. This subgroup meets I trivially and maps onto Q; hence it maps isomorphically to Q. The inverse map is continuous on each open determinant fiber, where it is a translate of j. We have constructed a continuous homomorphic splitting

of the original extension (5.7).

5.7 Extracting a finite character

We finish the proof of Proposition 5.1 by comparing (5.15) with the rational lift. Define \chi_0:U(k)\to I by

Both maps are homomorphisms lifting \rho, and their difference is central, so \chi_0 is a character.

Its image on S(k) is finite. For u\in R', Equation (5.8) reads \ell(u)=c(u_A). Thus \chi_0|_{R'} is the restriction of the continuous character p_0\mapsto c(p_0)s(p_0)^{-1}\in I on the profinite group P_0. Such a character has finite image: its compact image is a profinite quotient, while a closed subgroup of the circle is either finite or the circle itself. Since R' has finite index in S(k), the image \chi_0(S(k)) is also finite.

This image is nonzero. Choose a nonidentity element \beta\in C and a representative u\in V_0. The subgroup R is dense in P_0, so there are r_j\in R with (ur_j)_A\to1. All these elements still represent \beta modulo R'. If \chi_0 vanished on S(k), Equations (5.8) and (5.16) would imply

Both continuous sections tend to the identity. This would give \iota(\beta)=0, contradicting injectivity of \iota.

Finally, we make the character finite on all of U(k) without changing its restriction to S(k). Let \mathbb F_L be the full constant field of L. The divisor map on L^\times has kernel \mathbb F_L^\times and image in the free abelian divisor group. Every subgroup of a free abelian group is free, and an extension of a free abelian group splits. Therefore

for a finite torsion group T and a possibly infinite basis indexed by J. Choose u_j\in U(k) with \det u_j=a_j and a character \psi:L^1\to I satisfying \psi(a_j)=\chi_0(u_j) and \psi|_T=0. Then

vanishes on the chosen u_j and equals \chi_0 on S(k). If m annihilates \chi_0(S(k)) and t annihilates T, then mt annihilates \chi(U(k)): remove the free determinant part by products of the u_j, and the tth power of what remains belongs to S(k). The subgroup of I annihilated by mt is finite. Thus \chi has finite image and is nonzero on S(k), proving the first alternative.

All empty-product cases are included. In particular, if A is empty, then Q=L^1 and S_A=1; Lemma 5.4 makes the circle extension abelian immediately, and the character extraction above is unchanged. This completes the proof of Proposition 5.1.

6 Palettes and simultaneous color returns

Throughout this section and Section 7, D is a division algebra of odd reduced degree n\geq3. Suppose that the second alternative of Proposition 5.1 occurs: there is a homomorphism

where \mathcal F is finite, nonabelian, and simple. We call the values of \phi colors. Every central unitary scalar has color 1, since its image centralizes the surjective image \mathcal F.

We shall rule out this homomorphism by comparing multiplicative colors with additive translations on the Hermitian elements of D. Two facts make the comparison possible. First, a probability law on color configurations will retain every color while being invariant under all additive translations. Second, the following finite-group obstruction will prohibit certain simultaneous returns of those colors.

The following finite-group theorem provides the distinction between colors that the recurrence argument will preserve. Its double-coset assertion is [23, Lemma 5.2 and Section 6]. We include the proof, together with the additional assertion about abelian subgroups, in Appendix A.

Fix a set \mathcal S\subset\mathcal F supplied by Theorem 6.1. We shall use both its double-coset obstruction and its assertion that, on an abelian subgroup meeting \mathcal S, the complement of \mathcal S is a subgroup. In particular, \mathcal S is inverse-stable. The probability construction and the fractional transformations below adapt [23, Sections 5.2–5.4]. Section 7 supplies the additive argument in positive characteristic.

6.1 A compact space of color configurations

Let \widetilde G be the special unitary group of the hyperbolic Hermitian form on D^2 with coefficients 1,-1, and let X be its projective variety of isotropic right D-lines. This is a flag variety for a minimal k-parabolic of \widetilde G: the division hypothesis makes its relative k-rank one. Every rational isotropic line is represented uniquely by a column (u,1) with u\in U(k). Indeed, its second coordinate cannot be zero, and every nonzero element of D is invertible. We henceforth identify X(k) with U(k) in this way.

Choose \theta\in L\setminus k, with \bar\theta=-\theta when \operatorname{char}k\ne2, and put

The space J is an additive k-vector group; its rational points will also be denoted by J. The column (x+\theta,x+\bar\theta) is isotropic, and the Cayley parametrization gives

For example, x+\bar\theta cannot vanish for Hermitian x, and the inverse parametrization is defined at every u\ne1 because u-1 is invertible. Over a completion, the unitary and additive parametrizations are open charts in X(k_v); their complements are proper algebraic subvarieties and have measure zero in its smooth measure class.

Let \mathcal H be the countable group of rational transformations of X induced by form similitudes. Besides \widetilde G(k), it contains

It also contains the central affine maps x\mapsto lx+s, with l\in k^\times and s\in k. We write \tau_s=T_s for scalar translations. For p\in X(k) define

The closure of these configurations in \mathcal F^{\mathcal H} is denoted by \mathcal C; its members are called palettes. It is a compact metrizable space. The action and its relation to rational points are

Every finite coordinate pattern occurring in \mathcal C occurs at a rational point. Consequently identities involving finitely many colors pass to all palettes. For instance,

6.2 An invariant law retaining all colors

A translation-invariant probability is easy to obtain by averaging. What requires proof is that averaging can be arranged to retain every color at every coordinate. We first construct a joint probability on local flags and palettes, whose conditional color distributions are uniform.

We divide the proof into the construction of a joint law, identification of its base measure, and uniformity of its conditional colors.

6.2.1 Rational weights and a joint law

For each place v, choose an Iwasawa compact subgroup \mathcal K_v\subset\widetilde G(k_v) and its invariant probability m_v on X(k_v). The compact group is transitive on this flag variety. At almost all places use the integral models. Set

Each local probability belongs to the smooth measure class. A rational similitude preserves m_v at all but finitely many places: away from its denominators and the places where its multiplier is a nonunit, it normalizes the chosen integral compact. Thus

is a continuous positive function given by a finite product of local densities. With this pushforward convention its cocycle identity is

Assign a positive weight w(p) to each rational point by transporting weight 1 from a fixed base point under \widetilde G(k), multiplying by the forward local volume Jacobians. Rational conjugacy of parabolics gives transitivity. The weight is independent of the transporter: at a rational fixed point the product of tangent determinants is 1 by the product formula. The same argument gives the weight rule for every h\in\mathcal H, including similitudes.

We need convergence of these weights before they can define a probability. The following proof uses the discreteness of the function-field modulus and makes the exponent explicit.

For t>1, give (p,c(p))\in Y\times\mathcal C normalized mass w(p)^t, and denote the resulting probability by \mu_t. The weight rule gives

The reciprocal implicit in the pushforward density is accounted for here: the mass at hp after pushforward is the old mass at p. By compactness take a weak limit \mu^0 as t\downarrow1. Uniform convergence of \rho(h,\cdot)^t for each fixed h gives

6.2.2 The base projection is the product probability

Let \nu be the projection of \mu^0 to Y. We show that \nu=m; in particular, no singular base probability enters the later conditional argument. Fix a sufficiently large nonempty finite set T of places. Subscripts T denote products over T, and superscripts T denote the complementary products. An arithmetic lattice \Gamma\subset\widetilde G_T integral outside T preserves m^T. Consequently (6.5) gives

Define

The cocycle rule and this transformation law imply f(\gamma g)=f(g). The function is positive and continuous on \widetilde G_T.

Choose a probability \eta of full support on \widetilde G_T that is invariant under left multiplication by \mathcal K_T. Averaging first over that compact group shows that

compact transitivity turns the inner average into the integral of a probability density over m_T. By (6.3), it follows that

Apply Jensen’s inequality to the bounded strictly concave function a\mapsto a/(1+a) on the positive ray. Its nonnegative Jensen discrepancy has integral zero on the finite-volume quotient \Gamma\backslash\widetilde G_T, by right invariance of quotient measure. Equality in strict Jensen implies that f(gh) is constant for \eta-almost every h, for almost every g. Full support and continuity then make f constant everywhere. Its value at 1 is 1. This argument requires no integrability assumption on f itself.

These density integrals determine the base probability. Choose a_j\in\widetilde G_T contracting a big cell to a point x_0\in Y_T, using a parabolic cocharacter in each local factor. The complement of the cell has measure zero, so (a_j)_*m_T\to\delta_{x_0}. For b\in C(Y_T) set

For fixed y, the density in this integral makes k_0^{-1}y have law (a_j)_*m_T. This follows by pushing compact Haar measure to Y_T and using \rho_T(k_0a_j,y)=\rho_T(a_j,k_0^{-1}y). Compact equicontinuity therefore gives B_j(y)\to b(y) uniformly. Since f(k_0a_j)=1,

It follows that \nu_T=m_T. Exhausting the places proves \nu=m.

Disintegrate the joint law as

The compact metrizable spaces admit regular conditional probabilities. Equation (6.5) and uniqueness of disintegration give Q_{hy}=h_*Q_y almost everywhere. Since \mathcal H is countable, we may use all these identities simultaneously.

6.2.3 Uniform conditional colors

Let \mathbf p(y) be the c_1-marginal of Q_y, regarded as a probability vector on \mathcal F, and put N=\ker\phi\cap S(k). The palette rules (6.2) imply that \mathbf p is invariant under the separate left and right rotations by every element of N. We first prove invariance under a whole local special unitary group.

Enlarge T to contain a place v_0 at which S is split, as well as the required integral-model exceptions. On the unitary chart over T, the base measure is in the Haar class of U_T. Disintegrate that class along the smooth determinant map, and choose measurable representatives d_\lambda of its fibers. Write

The conditional measure class in g is Haar measure. For almost every determinant \lambda, each component of \mathbf p(gd_\lambda,y^T) descends, by its left invariance, to

The group \Gamma' has finite index in an arithmetic lattice. The quotient has finite measure, obtained from Haar measure on a lattice fundamental domain times m^T; the integral left tail actions preserve the latter. The bounded components of \mathbf p thus belong to L^2(Z_\lambda).

Right invariance by a\in\Gamma' acts on this quotient through

The tail rotations commute with the left action and belong to a compact group of measure-preserving transformations. There are elements a_j\in\Gamma' escaping every compact set at v_0 while remaining bounded at the other factors of S_T. For example, use Proposition 2.4 to find infinitely many elements of R\subseteq N in a fixed compact-open cylinder away from v_0. Adelic discreteness forces an escaping subsequence at v_0. After another subsequence the bounded components and compact tail rotations converge. Conjugating by the fixed d_\lambda does not change these properties.

Decompose L^2(Z_\lambda) into the invariant subspace for the right S(k_{v_0}) action and its orthogonal complement. The local group is the split group \SL_n(k_{v_0}), with its finite center, and local Kneser–Tits identifies it with its root-generated group. Hence Howe–Moore coefficient decay applies on the complement in every characteristic; see [16] and [3, Theorem 4.19]. If v is the projection of a component of \mathbf p to that complement, the operators fixing it have the form \pi(g_j)A_j, where g_j escapes in S(k_{v_0}) and the commuting unitaries A_j converge strongly to a unitary A. Therefore

The local and compact actions used here are strongly continuous, as one first checks on continuous functions with compact support and then by density in L^2. We conclude that \mathbf p is invariant under the full local right group S(k_{v_0}).

Lifting back and using Haar measure and Fubini, \mathbf p consequently depends on u_{v_0} only through its determinant, up to null sets. More explicitly, restrict at v_0 to the conull unitary chart and consider the measurable projection

Give \overline Y the pushforward probability \overline m. The local invariance just proved says that \mathbf p=\overline{\mathbf p}\circ\pi almost everywhere for a bounded measurable function \overline{\mathbf p} on this space. Every rational left rotation from S(k) acts trivially on the first coordinate of \overline Y.

Take g\in V_0. Proposition 2.4 supplies r_j\in R tending to g in the restricted adelic product away from v_0. Their left actions on \overline Y converge to that of g. This convergence passes to bounded measurable functions: outside one fixed finite set the actions preserve the factor probabilities, and on the remaining local factors their densities are uniformly bounded. The determinant factor of \overline m is unchanged. For a bounded continuous cylinder function, convergence follows from local continuity and bounded convergence; approximation in L^1(\overline m) then proves it for \overline{\mathbf p}. Since every r_j belongs to R\subseteq N, invariance passes to the limit and gives \mathbf p(L_g y)=\mathbf p(y) for every g\in V_0, almost everywhere.

Conditional equivariance also gives \mathbf p(L_g y)=\phi(g)\mathbf p(y). Since \phi(V_0)=\mathcal F, the probability vector is uniform. Equivariance identifies the c_h-marginal at y with the c_1-marginal at hy, so all conditional coordinate marginals are uniform.

6.3 A translation obstructed by a change of membership

We now connect this law to Theorem 6.1. Given w\in U(k) and a scalar affine map x\mapsto lx+s, its image in unitary coordinates agrees, up to a central norm-one scalar, with

Here, putting

one takes \bar t=-b/a when a\ne0. Direct substitution in (6.1) proves this formula. Moreover

If a=0, then (l,s)=(-1,-\theta-\bar\theta), and the affine transformation inverts colors. It therefore preserves membership in \mathcal S and will never cause an exceptional membership change.

For a\ne0 set

These definitions agree with (6.8). The elements C,C^* commute and CC^*=t\bar t is central. Division and d\ne0 make the denominators invertible; R_C is unitary and commutes with w. In particular z commutes with w.

Consider the similitude

Its matrix has multiplier d. Its denominator cannot vanish at a unitary element, since that would force t\bar t=1. Writing u=w\alpha gives the useful identity

No commutation between u and w is being assumed.

For a variable u\in U(k) put

Apply (6.9) first with parameters w^{-1},\bar t, and then with parameters w,-t. Conjugating the first resulting commutation gives

On the other hand, matrix multiplication gives the fractional matrix of u\mapsto u' as

Applying it to (x+\theta,x+\bar\theta) and dividing by d shows that it is the additive translation by

The matrix calculation also applies at infinity.

To detect simultaneous returns, take one independent copy of (\mathcal C,\mu) for each color, and in the product space let E_0 be the event that the first coordinate of the copy indexed by a\in\mathcal F equals a. Then

For the diagonal translation action define its return correlation

This is nonnegative and positive definite. The spectral theorem for the discrete abelian group J gives a positive measure \sigma on its compact character group \widehat J such that

The last inequality follows by projecting \mathbf1_{E_0} onto invariant vectors; its projection onto constants alone has squared norm \delta^2.

It remains to record the precise dependence of this translation on (l,s). This will allow us to apply additive Fourier analysis to scalar affine positions without treating multiplicative colors as continuous functions. Put

For fixed w define the k-linear map

Expanding -a\bar b and using the preceding denominator identity gives

Consequently

Both scalar arguments vanish at the exceptional affine parameter, so this formula extends there by b_w=0. Equations (6.13) and (6.14), together with Proposition 6.4, are the inputs to the recurrence argument.

7 Additive recurrence and the exclusion of simple colors

We retain the odd division degree hypothesis and the hypothetical quotient \phi:U(k)\to\mathcal F from Section 6. Our objective is to show that membership in \mathcal S is unchanged by scalar translations on the additive chart. Once this is known, fractional transformations generate left rotations whose colors generate \mathcal F, contradicting the fact that \mathcal S is nonempty and proper. This follows the organization of [23, Sections 5.5–5.6]. The recurrence proofs below are specific to positive characteristic: odd characteristic permits a quadratic Fourier argument, whereas characteristic two requires additive polynomials and the subgroup-complement property of Theorem 6.1.

Write \tau_s for the scalar translation x\mapsto x+s, with s\in k. For a palette c\in\mathcal C, put

We shall produce a palette satisfying

Recall that the nonnegative function K constructed in Section 6 has a spectral representation

Here additive groups have the discrete topology, and their duals are compact groups of characters with values in the complex unit circle.

7.1 Two averaging facts

We first isolate the Fourier argument shared by the two characteristics. An average on a countable set \Omega will mean integration against a finitely supported probability measure on \Omega, denoted by \mathbb E_j. A subset has density zero for these averages if its indicator has average tending to zero.

We shall also use the following elementary form of polynomial avoidance. It applies to finite additive subgroups without any compatibility with the coordinates of the ambient vector space.

7.2 Odd characteristic

Assume first that \operatorname{char}k\ne2. Our choice of Cayley parameter gives \bar\theta=-\theta, so that T=0. Let \mathcal A=k^\times\ltimes k be the scalar affine group, with (l,s) acting by z\mapsto lz+s. Associate to its elements the vectors

For a fixed initial affine position, the scalar parameters in Equation 6.14 become linear functions of these three coordinates. We average over l\in I_j\setminus\{0\} and s\in H_j independently and uniformly, where I_j,H_j are increasing finite additive subgroups exhausting k, and

Such sequences exist by taking finite-dimensional vector spaces over the prime field and enlarging H_j as necessary.

7.3 Characteristic two

Assume now that \operatorname{char}k=2, so T\ne0. We shall prove scalar-translation invariance directly on rational points. A unitary element w belongs to the field E=L[w]\subset D, which is stable under * because w^*=w^{-1}. With F=E^{\langle *\rangle}, we have E=LF. The degree [E:L] divides the odd degree of D, so E/L, and hence F/k, is separable. The Cayley chart identifies (E/F)^1 with F\cup\{\infty\}.

We first prove a density statement on each of these fields. Throughout this subsection, densities on k are taken over an arbitrary fixed increasing sequence of finite additive subgroups exhausting k.

The density statement alone allows exceptional points. The abelian structure of the colors on a field removes them.

7.4 Generating rotations

In either characteristic, choose a palette c satisfying (7.1). Define

This is a subgroup. In addition to all \tau_s, it contains left rotations L_a with \phi(a)=1, and all unitary conjugations: these assertions follow from the palette rotation identities and the conjugacy invariance of \mathcal S. We now explain how these transformations generate squares of arbitrary separable field rotations. The construction adapts [23, Section 5.6], with the field-spanning step made explicit in positive characteristic.

8 Excluding finite abelian characters

Throughout this section, D is a central division algebra of odd degree n\geq3 over the quadratic extension L/k, with unitary involution *. We write U=\U(D,*) and S=\SU(D,*). The remaining alternative in Proposition 5.1 is excluded by the following result.

The difference-function argument below adapts the corresponding argument of [23, Section 4]. Its two arithmetic ingredients have particularly useful function-field forms: every nonsingular Hermitian element admits a scalar normalization, and the additive span of U(k) is all of D. The latter is an equality of abstract additive groups. Local density alone would not suffice for the exact transformations used here. Fix a homomorphism \chi:U(k)\to B, with B finite and written additively. We construct a function on D whose differences are controlled by \chi, and prove that exact transformations of pairs make those differences constant. Scalar normalization then gives a function on all of D^\times, invariant under a noncentral normal subgroup of the inner group \SL_1(D)(L). The known inner-type theorem turns this invariance into continuity of \chi|_{S(k)} at A; local unitary commutators then force its vanishing.

8.1 Hermitian normalization and a cyclic maximal field

We first arrange a commutative field inside D on which both conjugation and the involution can be calculated explicitly.

Write \bar z=z^* on E, and put E^1=\{z\in E^\times:z\bar z=1\}. This group has infinitely many points, by the Hilbert 90 parametrization. We shall use parameters d\in E^\times with the following properties:

The last condition implies t_j\bar t_j\ne1 for every j. It ensures separability of a degree-two parameter map even in characteristic two.

8.2 A difference function and its exact identities

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

Multiplication gives m_bm_b^*=1. If a'{}^*a'=a^*a\ne0, then a'=va for v\in U(k), so the resulting elements m_b are conjugate. When a=0, m_b=1. Thus f is well-defined, including on skew elements. Using ax in the defining equation also gives

Fix d satisfying Equation 8.2. For (a,c)\in D^2\setminus\{(0,0)\} there is a unique e\in D with

Indeed, solve the linear equation

After splitting D, its linear part is a Sylvester operator whose eigenvalues are the differences of the disjoint reduced spectra of d and d^*. It is therefore invertible. Taking adjoints shows that c^*c-e^* is a second solution of the same linear equation, proving both identities in Equation 8.5. The solution is nonzero, since e=0 would force a=c=0. We may consequently define

To make this invariance useful, we must show that the transformations act transitively on nonzero pairs. We first prove the additive assertion that will turn one nonzero elementary shear into arbitrary addition.

8.3 Separating the blocks by actual shears

Skolem–Noether gives elements w_j\in D^\times and a decomposition

Thus D\oplus D is the direct sum of n two-dimensional E-blocks. Let H be the subgroup of the invariance group of F_d generated by \diag(\lambda,\mu), with \lambda,\mu\in E^1, acting simultaneously on every block, and by the simultaneous matrices

The phases act independently on the two D coordinates; no independent choice between blocks is assumed. These transformations are available: changing the sign of the second output in Equation 8.6 gives B_{-1}, and diagonal phase conjugation gives every B_h. Their determinants 1-t_j are nonzero.

8.4 From the difference identity to continuity

Lemma 8.2 extends f to every nonzero element of D. For b+b^*\ne0, choose s\in k^\times with sb\in\mathcal D and put f^\#(b)=f(sb); for skew b put f^\#(b)=0. This does not depend on the choice of s. If both sb and tb are permitted, reduced norms of their defining Hermitian equations show that (t/s)^n is a norm from L. The quotient k^\times/N_{L/k}(L^\times) has exponent two, so oddness of n implies that t/s itself is a norm. Central congruence in Equation 8.4 then gives the same value. Consequently

Let m\geq1 annihilate B, and let T=\ker(N_{E/L}:\Res_{E/L}\bbG_m\to\bbG_m). Choose d\in T(L)^m satisfying Equation 8.2. Such choices are Zariski dense in a nonempty open subset of \Res_{L/k}T. Over a separable closure of k, the two embeddings of L give two independent lists (x_1,\ldots,x_n) and (y_1,\ldots,y_n), each with product one. These determinant-one eigenvalue lists can avoid all stated equalities, and n\geq3 also allows their coordinatewise products to be unequal. Hilbert 90 for the cyclic extension E/L gives Zariski density of T(L), and the power map is dominant even when p\mid m.

For every such d and every b\in D^\times,

Indeed normalize b and bd separately into \mathcal D. Their ratio replaces d by a nonzero k-multiple, preserving Equation 8.2 and the ratio \bar d/d. If d=y^m, then \bar d/d=(\bar y/y)^m has zero character, so Proposition 8.6 applies. Congruence invariance extends Equation 8.12 to every D^\times-conjugate of d.

Let J be the subgroup of \SL_1(D)(L) generated by all these conjugates. It is noncentral and normal. The established inner-type Margulis–Platonov theorem over L implies that J contains every rational point sufficiently close to 1 at the places

[28, Sections 3.4–3.5][32]. These are precisely the two places of L over each place of A. Since f^\# is defined on all of D^\times, products of the invariant steps give

We now use this identity to obtain the continuity of \chi on S(k).

9 Induction on the degree

We now prove that the power defect vanishes in every degree. The induction is simultaneous over all global function fields: passing to a finite separable extension changes the ground field, but every group to which we apply the induction hypothesis has smaller absolute degree. The factorization used in the even step is adapted from [23, Section 7]. We give its algebra and approximation conditions in detail, since the local groups occurring in the factorization vary with the element being factored.

Throughout the proof, e is fixed and

denotes the finite quotient map of Proposition 2.3. Its target has exponent dividing e. We say that an element is killed if its image under \pi is trivial. The inner type A case, including degree two, is already known. Isotropic groups also satisfy the assertion by the reductions in Section 2. These cases are available whenever they occur in the induction.

9.1 Identity neighborhoods for smaller groups

Assume the induction hypothesis for a fixed smaller-degree special unitary group embedded in S. Proposition 2.5 then extends the restriction of \pi to its product of anisotropic local groups. An element sufficiently close to 1 at those places is consequently killed. We need a version of this observation for binary groups which have not yet been chosen.

In our applications a binary group over a finite extension F/k acts, after passage to an algebraic closure, by ordinary two-dimensional blocks. The degrees [F:k] are bounded by n. We may therefore use Lemma 9.2 at a place of k before constructing F or its binary group. At split binary factors there is no condition: the inductive Margulis–Platonov assertion makes every finite quotient of the rational group depend only on its anisotropic factors.

9.2 Odd degree

Suppose first that D is division and n is odd. If V_0/R were nontrivial, Proposition 5.1 would give either a finite abelian character of U(k) nontrivial on S(k), or a nonabelian finite simple quotient as in that proposition. The first possibility is excluded by Theorem 8.1, and the second by Proposition 7.8. Thus V_0=R in odd division degree, without any induction hypothesis.

Now write D=M_l(\Delta), where \Delta is a unitary division algebra of degree d, and suppose that n=ld is odd and l>1. Realize S as the special unitary group of an l-dimensional Hermitian space over \Delta. Both d and l are odd, and l\geq3. If d=1, Hermitian forms of fixed dimension and determinant over L/k are isometric: after choosing a compatible determinant for an isometry, its existence is an instance of Harder’s vanishing for a special-unitary torsor. There is a form of the prescribed determinant containing a hyperbolic plane. This would make S isotropic, so the anisotropic case has d>1.

Every subspace of this anisotropic Hermitian space is nondegenerate. Given a special unitary transformation and a nonzero vector, choose a two-dimensional subspace containing that vector and its image. Witt extension gives an isometry of the plane carrying one to the other. Its determinant can be corrected on the perpendicular line inside the plane: the determinant map on that line’s unitary group is surjective, since each fiber is a torsor under a simply connected special unitary group and has a rational point. Extending this special plane isometry by the identity, and dividing the original transformation by it, leaves a transformation fixing the chosen vector. Repeat in its orthogonal complement, ending in dimension two. Hence S(k) is generated by special plane groups.

Each plane group has degree 2d<n. It has no anisotropic place. At a place nonsplit in L, it is the group of an ordinary Hermitian form of dimension 2d\geq6, and is isotropic. At a split place it is an inner form represented by a matrix algebra with matrix size at least two. The induction hypothesis therefore makes every plane group trivial in the finite quotient S(k)/R. This proves the assertion in all odd degrees.

9.3 A fixed quadratic subfield in even degree

Let n=2m with m\geq2. We will factor a suitably chosen element of each coset of V_0/R as a product of an element in a binary group and an element in a fixed unitary centralizer of degree m. The first step is to construct that centralizer.

Fix s as in Lemma 9.3, and put

Then C_s is a central simple algebra of degree m over LS_0, and * restricts to a unitary involution with fixed ground field S_0. The unitary groups of C_s are defined over S_0. Whenever they are embedded in a group over k, restriction of scalars from S_0 is understood; their rational elements are thus the ordinary unitary elements of C_s.

9.4 The algebraic factorization

The fixed field gives two blocks. Comparing them with their images under u\in S produces a quaternion algebra over a field that depends on u. The factorization below reduces the global problem to one or two norm equations in that field.

Over a splitting field, let P be either spectral idempotent of s. It is fixed by the extended involution. Define

The expressions p,z,h are unchanged when P is replaced by 1-P, so they descend to k. Direct multiplication gives

In particular h commutes with both P and P'.

Write a split matrix component of u in blocks as

The two blocks of h have the same characteristic polynomial, denoted by \Psi. For example, where the relevant blocks are invertible,

the corresponding other product is conjugate to this one. The identity extends to all u by density. Exchanging the labels preserves \Psi, and h^*=h; hence \Psi\in k[T]. Similarly, complementary-minor identities and \det u=1 show that

Indeed label exchange sends this determinant to \Nrd_{C_s}(p), as does the involution on the center over S_0. Their composite fixes d_0, and its fixed field is J_0.

Let \Omega\subset S be the open set on which \Psi is separable and h(1-h) is invertible. For u\in\Omega, put F=k(h). This is an étale algebra of degree m, and its regular characteristic polynomial is \Psi. Moreover, F and LS_0 generate their full tensor product inside D: in either s-block the element h has m distinct eigenvalues, so its minimal polynomial over LS_0 still has degree m. The same notation applies over a completion, component by component. We will later choose u so that F and LS_0F are fields globally. Taking determinants of h=pp^* gives the compatibility

For m\geq3 we will solve both equations in (9.6) using Proposition 4.1. When m=2 we solve only the first and then correct the determinant of q. Figure 1 records the fixed fields and the varying norm equations. The following construction makes the degree-four correction possible with neighborhoods fixed before the element q is chosen.

9.5 Auxiliary fields for the degree-four correction

Suppose in this subsection that D has degree four over L. Retain the Hermitian quadratic field S_0=k(s) already constructed, put E=LS_0 and C_s=C_D(s), and let J_0 be the third quadratic subfield of the biquadratic extension E/k. Thus C_s is a quaternion algebra over E, with a unitary involution over S_0. When its unitary groups are viewed inside the ambient groups over k, restriction of scalars from S_0 is understood. We will correct the determinant of an element of \U(C_s,*)\cap S by elements in binary special unitary groups. The auxiliary fields in the following lemma let us prescribe where those binary groups are split. This construction is the degree-four determinant correction of [23, Section 7], with separable quadratic descent used also in characteristic two.

For each field F' in Lemma 9.5, put G_{F'}=\SU(C_D(F'),*), a group over F'. Its restriction of scalars embeds in S, since over a separable closure it acts by special transformations on the two binary F'-eigenspaces. We will use the inclusion

Here (J_0F'/F')^1 denotes the elements of norm one. Indeed EF' is a maximal subfield of C_D(F') as well as of C_s. Its norm to LF' computes the determinant in the binary algebra; the nontrivial automorphism of EF'/LF' restricts to the nontrivial automorphism of J_0F'/F'. On this last field that automorphism is *. Thus norm one gives both the unitary and special conditions. The determinant in C_s is the norm from EF' to E, whose restriction to J_0F' is the norm displayed in Equation (9.8).

9.6 Correcting the determinant in degree four

We now isolate the local conditions on the factor q that suffice to remove its determinant. The neighborhoods in the next proposition are fixed before q is chosen. This will allow the preceding factorization to use them as ordinary local approximation conditions.

9.7 Local factorizations near the identity

We return to even degree n=2m. For m=2, Proposition 9.6 supplies identity neighborhoods in \U(C_s)\cap S that can be fixed before q is chosen; for m\geq3, the induction hypothesis supplies kernel neighborhoods in the fixed group \SU(C_s). It remains to choose a representative of each coset in V_0/R and solve the norm equations so that both factors w and q are killed. We first construct local open sets near 1 on which the norm equations are soluble and their solutions make both factors small in the ambient algebra.

At auxiliary places split in LS_0 and in D, these neighborhoods can also prescribe a Frobenius cycle type for \Psi. In split coordinates take

Its determinant is one and h_+=I+\varepsilon^2X. Choose X integral with invertible separable residue characteristic polynomial of the specified factorization type. At sufficiently large residue fields all cycle types are available. Taking \varepsilon small gives the norm solutions and smallness of Lemma 9.7 while retaining that type for the étale field generated by h.

9.8 The approximation conditions for the even step

Assume the induction hypothesis in degrees smaller than 2m. We will choose a representative u of an arbitrary coset in V_0/R for which Lemma 9.4 can be applied globally. The choice must control the anisotropic places of the variable quaternion algebra Q, as well as the fixed group \SU(C_s).

Choose a finite set B containing A, the places below the anisotropic places of \SU(C_s), all exceptions to integral models of the fixed data, and an isotropic place v_0 of S. In degree four include B_s and apply Proposition 9.6 at this point. Denote this set by B_{\mathrm{corr}}; its field F_1 and neighborhoods W_v, v\in B_{\mathrm{corr}}, are now fixed. Subsequent enlargements of B retain those neighborhoods on B_{\mathrm{corr}} and impose no additional determinant-correction condition. At every added place we may still choose near-identity openings using Lemma 9.7.

Enlarge B by sufficiently large auxiliary places split in LS_0 and in D at which we impose, using (9.13), every permutation cycle type of degree m. These conditions force the splitting field of \Psi over LS_0 to have group S_m. To recall the group-theoretic argument, the transitive S_m-action on the cosets of a proper subgroup has a derangement, as follows by averaging the numbers of fixed points. An element conjugate into that subgroup fixes a coset, so the derangement’s conjugacy class misses the subgroup. A subgroup containing every cycle type therefore cannot be proper.

At each place of B, use Lemma 9.7. Require u to lie sufficiently near 1 that its A-coordinate belongs to P_0; then these openings are compatible with every coset in V_0/R by Proposition 2.4. Require w to be small enough for Lemma 9.2. When m\geq3, require q to be small enough to be killed in the fixed smaller group \SU(C_s). When m=2, require q_v\in W_v for v\in B_{\mathrm{corr}}; at the other places of B we may require any sufficiently small identity neighborhood.

Outside B we allow either u\notin\Omega, or a point u\in\Omega for which the required norm equations are soluble and Q is split at every local factor of F. We also allow a nonempty open set near 1 from Lemma 9.7, where the norm equations are soluble and w is small enough for Lemma 9.2; this second open set will only be used at the denominator places B_1 introduced in Proposition 3.4. Once such a place is known, we choose an indicated local solution a there. No extra condition on q is needed outside B, because all anisotropic places of the fixed group lie above B.

We verify the two remaining hypotheses of the approximation proposition: a codimension-two exclusion at integral reductions and nonempty algebraic tests at a single pole. This will ensure that the only nonsplit quaternion factors occur above B\cup B_1, where their elements are already small.

9.8.1 Integral reduction

Over a splitting field consider the four divisors

in the split matrix group. Exclude their pairwise intersections. If m\geq3, impose also the following conditions. Off all four divisors, \Psi must have at least one simple absolute root. On one divisor alone, require 0 to be a simple root for a diagonal divisor, and 1 to be a simple root for an off-diagonal divisor. The asserted root already occurs by the complementary-minor identities; for example

For m=2 there is no simple-root requirement.

These exclusions are invariant under descent. Exchanging s-labels interchanges the two diagonal divisors and the two off-diagonal divisors. The unitary diagram action does the same, because it acts by inverse transpose and complementary minors. More explicitly, the automorphisms of LS_0/k fixing L and S_0 each interchange both pairs, while their product fixes each divisor. The permutation character on either pair is therefore the quadratic character of J_0/k. The excluded closures define a k-subvariety Y.

We claim that Y has geometric codimension at least two. Work first in \GL_{2m}. The four determinant hypersurfaces are distinct and irreducible, so their pairwise intersections have codimension at least two. On each one, the simple-root requirement holds generically: use independent two-dimensional transformations on m pairs, with one line in each block, and arrange exactly one of the resulting values to be 0 or 1. Off the divisors the matrix B'D'^{-1}C'A'^{-1} is a submersive parameter. For m\geq3, matrices with no simple characteristic root have codimension at least two. Indeed their distinct eigenvalues have at most \lfloor m/2\rfloor parameters, and each Jordan orbit has dimension at most m^2-m. The resulting dimension is at most m^2-2. Finally all these tests are invariant under scalar multiplication of u, which carries the codimension statement to \SL_{2m}. Enlarge B to make these statements valid on the integral models.

Let an integral point reduce outside Y, and assume it belongs to \Omega. If J_0 is nonsplit at this place, its Frobenius interchanges both pairs of divisors. A rational residue point lying on one divisor would therefore lie on its mate, which is excluded. Thus it lies on none, and h and 1-h are units in every local factor of F. The quadratic extension defined by J_0 is unramified, so -h(1-h) and h are norms of units. This proves that Q is split and gives a unit solution of the first equation in (9.6).

For m\geq3 the second equation can be imposed as well. Given a unit solution of the first, its determinant discrepancy is a norm-one unit in J_0. A simple residue root of \Psi gives an unramified factor of F over the local ground field. Write the discrepancy as y/\bar y by Hilbert 90, taking y a unit; its valuation can be removed by a ground-field uniformizer. The norm on units from that unramified factor, after extending to J_0, is surjective. Lift y through that norm and divide the lift by its conjugate. This changes the second norm by the required discrepancy without changing the first.

If J_0 splits, the quaternion algebra is split and the first equation is immediate. For the second equation, identify J_0 with two copies of the local field. Choose the first component of a with prescribed norm equal to the first component of d_0, and determine the other component from h; compatibility (9.5) gives its correct norm. Off the divisors, the prescribed norm is a unit and the simple residue root again supplies an unramified factor. On one divisor, the simple residue root 0 or 1 lifts to a degree-one factor of F, whose norm has no restriction on valuation. This proves all required local assertions for integral points reducing outside Y.

9.8.2 A single pole

Take the fixed Galois field M in Proposition 3.4 large enough to split all fixed data. A single pole then occurs only at a place split completely in M. In degree four, J_0 is consequently split, so there is no additional pole condition. For m\geq3, it is enough to force \Psi to split, which makes both norm equations soluble.

Order the cocharacter exponents as d_1<\cdots<d_{2m}. At a single pole the split matrix expression is

with integral invertible side factors and \ord(t)>0. For the b-th elementary coefficient of \Psi, 1\leq b\leq m, require

These equalities follow from finitely many nonvanishing minor tests. Indeed write the coefficient as the exterior trace for P u P u^{-1}P. The uniquely lowest-weight term uses the first b indices in the positive exterior power and the last b in the inverse. Its coefficient is a product of opposite minors in the two rank-m projections

Requiring these minors to be nonzero on reduction gives (9.14). Such minors are generically nonzero on the variety of projections: one may use complementary subspaces in general position, or independent two-dimensional projections on the paired coordinates. The successive differences of the displayed valuations are strictly ordered. Every segment of the Newton polygon therefore has length one, and \Psi splits over the local field.

We must also check the order of the generic choices in Proposition 3.4. On the hyperplane belonging to a factor with conjugator k_j, put all other parameters equal to 1. Then, in the corresponding fixed bases, g_R=k_j^{-1} and g_L=xk_j, where x is the initial rational point. For the two basic lists first choose the conjugators so that the right-hand minor tests hold. Next choose x, subject also to the prescribed local openings, so that all left-hand tests hold. These are finitely many nonempty Zariski open conditions. For every appended conjugator, x has already been fixed; both conditions are still nonempty open conditions on that conjugator, since conjugation sweeps out the full variety of rank-m projections. Impose the tests also on all Galois-conjugate hyperplanes. They form one finite algebraic family independent of the place, and inserting identity parameters in appended factors preserves all earlier tests. This is exactly the pole hypothesis of the approximation proposition.

9.8.3 Choosing the representative and solving the norms

All hypotheses of Proposition 3.4 are now verified. Apply it to a prescribed coset in V_0/R and the local openings at B. At the finite set B_1 of denominator places of its initial point, use the near-identity open sets already described, with local solutions a making w sufficiently small. At any further model or projection exceptions introduced in the proof of that proposition, use the integral-reduction open sets supplied by its original basic list. These later places therefore introduce no additional nonsplit quaternion factors requiring smallness. We obtain u in the prescribed coset, with

with all required norm equations locally soluble, and with Q split outside B\cup B_1.

The Galois condition implies that F=k(h) is a field. If \widetilde F is its splitting field over k, then \Gal(\widetilde F/k)\subset S_m already contains \Gal(\widetilde F LS_0/LS_0)=S_m. Thus this group is S_m and \widetilde F\cap LS_0=k. In particular the pair F,J_0 has the joint group required by Proposition 4.1.

For m\geq3, apply that theorem to (9.6), using (9.5). It gives a global a approximating the selected local solutions at B\cup B_1. The binary factor w is killed: its quaternion algebra is split away from these places, and at every anisotropic factor above them its eigenvalues satisfy Lemma 9.2. The factor q\in\SU(C_s) is killed by the induction hypothesis and its smallness at the anisotropic places of this fixed degree-m group. Hence u=wq is killed.

For m=2, the first equation of (9.6) is a cyclic norm equation over F. The Hasse norm theorem gives a global solution [22]. Its solution variety is a torsor under the norm-one torus of a separable quadratic extension; after choosing a solution this torus is rational, so it has weak approximation. We may therefore approximate all selected local solutions at B\cup B_1. The binary factor w is killed by the same argument, and q\in\U(C_s)\cap S lies in the neighborhoods fixed in Proposition 9.6. That proposition kills q as well. This completes the even step.

10 Proof of the main theorem

A The finite simple-group obstruction

We prove Theorem 6.1, following the finite-group argument of [23, Section 6]. In addition to the double-coset obstruction proved there, we verify for each constructed set that its complement in an abelian subgroup meeting it is a subgroup. In characteristic two, this property supplies the finite abelian quotient used in Proposition 7.6. The resulting functions have additive period groups of finite index, which rules out the exceptional points left by the density argument.

The proof uses the classification of finite simple groups [1, Classification Theorem, p. 737]. Its common mechanism is to locate a part of a commuting pair on which centralizers are small. For alternating and linear groups, this part consists of two points or a line and a hyperplane. For the remaining classical groups it is a large irreducible subspace. Most exceptional and sporadic groups instead have an abelian subgroup whose nonidentity elements have exactly that subgroup as centralizer. We first establish the two elementary tests behind these constructions.

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. For every choice we must prove both that Equation (A.1) fails for some B whenever W\in\mathcal S and Z\notin\mathcal S commute, and that the complement condition in Theorem 6.1(ii) holds.

A.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 every abelian subgroup A, its complement is the subgroup A[2]=\{a\in A:a^2=1\}. Thus this choice automatically satisfies Theorem 6.1(ii). 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 (A.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.

A.2 Alternating, linear, and sporadic groups

For alternating and projective special linear groups we use Equation (A.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 (A.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 [5, 46]. 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 A.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 [5, 46]; the online database labels this permutation representation 12a (the other degree-12 action interchanges the classes 8A and 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 (A.2).

A.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.

A.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 (A.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. More explicitly, the Hermitian form h(x,y)=\Tr_{\bbF_{q^{2p}}/\bbF_{q^2}}(\kappa xy^*) in the quadratic case satisfies h(x,x)=Q(x): on \bbF_{q^p} the trace to \bbF_{q^2} equals the trace to \bbF_q, since p is odd. Thus H preserves the original form on E in both cases. 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 (A.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 A.2.

To verify the complement assertion, let A be abelian and choose W\in A\cap\mathcal S. Its marked block E is now fixed. The preceding centralizer argument gives a well-defined homomorphism

Indeed every lift restricts on E to a field multiplication, and changing the lift multiplies it by an ambient central scalar lying in \bbF_{q^2}^{\times}. The complement of E is too small to carry a marked block. For a restriction z\in T, its square has proper degree over \bbF_q precisely when z^2\in\bbF_{q^2}: the other maximal proper subfield is \bbF_{q^p}, whose norm-one elements are \pm1. Because p is odd, z^2\in\bbF_{q^2} is equivalent to z\in\bbF_{q^2}. Full degree automatically gives self-duality by the norm-one equation. Thus A\setminus\mathcal S=\ker\rho, as required.

A.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. Writing w for any one of them, 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 (A.1) restricts to the relation prohibited by Lemma A.2.

For the complement assertion, fix W\in A\cap\mathcal S in an abelian subgroup A. The projective-centralizer argument just proved gives a restriction homomorphism as in Equation (A.10), now with T the torus on the p-dimensional Hermitian block. A restriction z has square of degree less than p precisely when z^2, equivalently z, is in \bbF_{q^2}. Since the complementary subspace has dimension less than p, this characterizes nonmembership in \mathcal S. Once again A\setminus\mathcal S=\ker\rho.

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 (A.3). If an abelian A meets this \mathcal S, fix the irreducible block using any member of the intersection. Every other member restricts to a norm-one field scalar z, and the calculation above gives

The last condition says precisely that its projective image has square one. Thus A\setminus\mathcal S=A[2]. The nonsimple group \Sp_4(2) is not needed; its simple derived group is A_6.

A.6 Small tori in exceptional groups

We have proved the obstruction for all classical simple groups. For most exceptional families we apply Lemma A.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 [42] and [19, Propositions 25.1–25.3, pp. 219–220]. Semisimple centralizers in a simply connected semisimple group are connected [41, Remark 2.10]. The tori below and their self-centralizing property in the simple quotient occur in [39, Corollary 8.2 and Lemma 8.7]. We verify that property here by a root-lattice estimate before applying Lemma A.1. Their orders outside type G_2 are also recorded in [12, 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 [40, 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 A.1 are now verified.

For the numerical inequality, we use

where l is the absolute rank. This follows, with a smaller constant, from [10, 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, pp. 208, 211] 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 A.1 completes these families.

A.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. Geometrically M is (E_{6,\mathrm{sc}}\times\bbG_m)/\mu_3; the central one-dimensional torus has Frobenius action \epsilon q on its character lattice. The kernel K_M=\ker\pi is therefore 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 a prime-order power of each such element to be regular not only in E_6 but in E_7. For W\in M(q) with nonidentity projection in C_0, choose a prime r dividing the order of \pi(W) and a power t of W of order r. Because K_M has order prime to r, \pi(t)\ne1; the quotient calculation above also puts t in D_M(q). 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 (A.11) bounds the resulting weight-lattice index by 8\sqrt3<19, smaller than the primitive prime. Thus no root vanishes on t, and its algebraic centralizer in E_7 is precisely the maximal torus containing it.

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 (A.1) to \mathcal F_0 therefore puts the nonidentity projection w of W in

The counting proof of Lemma A.1, already verified for this E_6 group, excludes this. This proves the obstruction in E_7.

We must also identify the complement inside an abelian subgroup A meeting \mathcal S. Fixing a member of the intersection conjugates A into the image \overline T of the torus just used. The map \pi:T\to C_0 descends to \overline T, since the ambient center acts trivially on adjoint E_6. Every element with nonidentity projection belongs to \mathcal S by definition. Conversely, every element of \mathcal S has order divisible by a prime dividing |C_0|: its nonidentity projection has such order, and passage to projective E_7 removes at most a factor two. No element of the image of K_M has such a prime divisor. Therefore even conjugacy into a different torus cannot put an element of \ker\pi into \mathcal S, and

This completes both assertions in type E_7.

For the simple Suzuki and rank-one Ree groups, use Equation (A.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 [12, 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 [10, Table 1, p. 3049]. Thus

and Lemma A.1 applies.