Proof. The exact sequence \[1\longrightarrow S\longrightarrow U \xrightarrow{\det}\mathop{\mathrm{Res}}^1_{L/k}\mathbb G_m\longrightarrow1\] identifies each determinant fiber with an \(S\)-torsor. Harder’s global vanishing theorem and the local vanishing theorem of Bruhat–Tits give \(H^1(k,S)=1\) and \(H^1(k_v,S)=1\). These statements include characteristic two; a convenient uniform reference is (Conrad 2012, Theorem 5.1.1(i)), which recalls (Harder 1975, Satz A) and (Bruhat and Tits 1987, Theorem 4.7(ii)). They prove (i).

Strong approximation for simply connected almost simple groups over global function fields holds away from a place where the group is isotropic (Prasad 1977, Theorem A). There are infinitely many split places, by Chebotarev applied to a finite splitting field. To approximate at a given finite set of places, omit a split place outside that set. This proves weak approximation for \(S\). For \(U\), fix \(\theta\in L\setminus k\). The Cayley map \[ x\longmapsto (x+\theta)(x+\bar\theta)^{-1},\qquad x^*=x, \tag{2}\] is a birational map from the affine space of Hermitian elements of \(D\) to \(U\). Its inverse, on the open set where \(u-1\) is invertible, is \((u-1)^{-1}(\theta-u\bar\theta)\). The difference \(\theta-\bar\theta\) is nonzero also in characteristic two. Weak approximation for this affine space, and density of the Cayley open set in each local group, prove (ii).

Suppose that \(v\) does not split in \(L\). A unitary involution implies that the Brauer corestriction of \(D\) is zero. For a quadratic extension of nonarchimedean local fields, corestriction preserves the local Brauer invariant. Thus \(D\otimes_L L_v\) is a matrix algebra. The group \(S_{k_v}\) is consequently the special unitary group of an \(n\)-dimensional Hermitian form. The local classification of Hermitian forms over a separable quadratic extension gives anisotropic dimension at most two, in all characteristics. Since \(n\geq3\), this group is isotropic. These standard facts about unitary involutions and local Hermitian forms may be found in (Knus et al. 1998; Reiner 2003); for the characteristic-two classification see also (Elomary and Tignol 2001, Theorem 3.3 and Corollary 3.4).

At a split place, the two simple components of \(D\otimes_k k_v\) are interchanged by the involution. Choosing one component \(B_v\) identifies \(U(k_v)\) with \(B_v^\times\) and \(S(k_v)\) with \(\mathop{\mathrm{SL}}_1(B_v)\). If \(B_v\simeq M_r(\mathcal D_v)\), this last group has rank \(r-1\). It is anisotropic exactly when \(r=1\), proving (iii). ◻

Proof. This is the finite-index reduction in Margulis’s normal subgroup theorem (Margulis 1979, Theorem 2.4.6 and Corollary 2.4.9)(Margulis 1991). We recall the passage from an arithmetic lattice to all rational points. Let \(N\triangleleft S(k)\) contain a noncentral element \(z\). Choose a finite set of places \(T\) containing \(A\), the places where \(z\) is not integral, and two further places at which \(S\) is split. Fix compact open subgroups \(K_v\subset S(k_v)\) for \(v\notin T\), using an integral model outside finitely many places, and arrange that \(z\in K_v\) there. Put \[J_T=\prod_{v\notin T}'S(k_v),\qquad K^T=\prod_{v\notin T}K_v,\qquad \Gamma=S(k)\cap K^T.\] Here \(S(k)\cap K^T\) denotes the inverse image under the diagonal map to \(J_T\), and the prime denotes the restricted product with respect to the \(K_v\).

Reduction theory makes \(\Gamma\) an arithmetic lattice in \(\prod_{v\in T}S(k_v)\); compact factors may be projected away. Strong approximation, omitting any one noncompact factor, gives irreducibility. The two added split places have total rank at least four, since \(n\geq3\). The arithmetic normal subgroup theorem therefore applies. The subgroup \(\Gamma\) is Zariski dense, so a finite normal subgroup of \(\Gamma\) is central. Since \(z\in N\cap\Gamma\) is noncentral, the theorem gives \[ [\Gamma:N\cap\Gamma]<\infty. \tag{3}\] The lattice and strong-approximation assertions used here are valid over function fields (Harder 1969; Prasad 1977; Margulis 1991).

Let \(C\) be the closure of \(N\) in \(J_T\). Strong approximation gives density of \(S(k)\) in \(J_T\), hence normality of \(C\). It also gives density of \(\Gamma\) in \(K^T\). By (3), the closure of \(N\cap\Gamma\) has finite index in \(K^T\) and is therefore open. Thus \(C\) is open in \(J_T\). For every \(v\notin T\), the group \(C\cap S(k_v)\) is an open normal subgroup of \(S(k_v)\). It is noncentral, and \(S\) is isotropic at \(v\) because \(A\subset T\). Local simplicity and perfection imply \(S(k_v)\subset C\). Finite-support products are dense in the restricted product, so \(C=J_T\).

Every coset of \(N\) in \(S(k)\) is now dense in \(J_T\) and hence meets the open subgroup \(K^T\). Thus every such coset meets \(\Gamma\), and \[[S(k):N]\leq[\Gamma:N\cap\Gamma]<\infty,\] as required. ◻

Proof. The map \(g\mapsto g^e\) is dominant: over an algebraic closure it is surjective on each maximal torus, and conjugates of a maximal torus are dense in \(S\). This dominance does not require separability of the power map. Weak approximation implies that \(S(k)\) is Zariski dense, so the set of rational \(e\)th powers is Zariski dense as well. The characteristic subgroup \(R\) is therefore noncentral, and 3 gives its finite index.

Weak approximation gives density of \(\delta_A(S(k))\) in \(H_A\). A finite coset decomposition of \(S(k)\) by \(R\) then gives a finite coset covering of \(H_A\) by the closed subgroup \(P_0\). A closed subgroup of finite index is open. Normality follows from density and normality of \(R\), and the finiteness of \(V\) follows from that of \(S(k)/R\). Finally, \(U(k)\) normalizes \(S(k)\) and preserves its power subgroup. Taking closures at \(A\) shows that it also preserves \(P_0\) and \(V_0\). In fact weak approximation for \(U\) shows that \(P_0\) is normalized by \(\prod_{v\in A}U(k_v)\).

For the last assertion, let \(N\triangleleft S(k)\) be noncentral. By 3, the quotient \(S(k)/N\) is finite. Choose \(e\) divisible by its exponent; then \(R\subset N\). If \(V=1\), we have \(R=\delta_A^{-1}(P_0)\), and density induces an isomorphism \(S(k)/R\simeq H_A/P_0\). The inverse image in \(H_A\) of the subgroup corresponding to \(N/R\) is open and normal and has rational inverse image \(N\). This proves the claimed reduction. ◻

Proof. Strong approximation gives density of \(S(k)\) in \(S(\mathbb A^{v_0})\). Since \(R\) is a normal subgroup of finite index, its closure \(C\) is a closed normal subgroup of finite index, hence an open subgroup. At every isotropic place \(v\ne v_0\), the intersection \(C\cap S(k_v)\) is open, normal, and noncentral. The local result recalled above gives \(S(k_v)\subset C\). Taking closures of finite-support products, \(C\) contains the entire restricted product of these isotropic factors.

The remaining factors form the finite compact product \(H_A\). It follows that \(C=P\times\prod_{v\notin A\cup\{v_0\}}'S(k_v)\) for a closed subgroup \(P\subset H_A\). Since \(R\subset C\), we have \(P_0\subset P\). Conversely, the inverse image of \(P_0\) is closed and contains \(R\), so it contains \(C\) and gives \(P\subset P_0\). This proves (5). Translation gives the assertion for \(\gamma R\), and left translation by \(x_*\) gives the determinant-slice version. For finite local conditions that include \(v_0\), apply the same argument with a different isotropic place outside the specified finite set omitted instead. Finally, a nonempty Zariski open subset of a smooth geometrically irreducible variety over a local field meets every nonempty analytic open subset. Impose the Zariski conditions at one place, adding a place of approximation if necessary, and shrink its local open set. The same density argument then enforces them on the rational point. ◻

Proof. Strong approximation for \(H\) away from an auxiliary split place gives weak approximation by the same argument as in 2. In particular, \(H(F)\) is Zariski dense. The kernel of \(\psi\) has finite index, so connectedness of \(H\) makes its Zariski closure all of \(H\) as well. Thus this kernel is noncentral. By the assumed theorem, \[\ker\psi=\delta_{A_H}^{-1}(W)\] for an open normal subgroup \(W\) of \(\prod_{w\in A_H}H(F_w)\). This product is compact, so its quotient by \(W\) is finite. Weak approximation gives density of \(H(F)\) in the product, and hence a canonical isomorphism \[H(F)/\ker\psi\ \simeq\ \left(\prod_{w\in A_H}H(F_w)\right)/W.\] Compose the quotient map with this isomorphism and the induced injection into \(Q\). This constructs \(\widehat\psi\); density proves uniqueness and equality of images. An identity neighborhood contained in \(W\) gives the local smallness assertion. For a restriction of scalars, rational points and local points are the corresponding groups over \(F\) and products over the places above a given place. For a finite product, the images of the individual factors commute. Their continuous extensions therefore combine to give the stated factorwise conclusion. ◻

Proof. The restriction \(\pi|_Z\) is finite. Indeed, projection from \([d]\) extends to a morphism on the projective closure of \(Z\), since its center is disjoint from that closure. A fiber over an affine point can acquire no point at infinity: its projective fiber line meets the hyperplane at infinity only at \([d]\). Thus \(\pi|_Z\) is proper. It has finite fibers, because a positive-dimensional closed subset of a fiber line would be the whole line and would have \([d]\) in its projective closure. A proper morphism with finite fibers is finite. Its image \(Y\subset E/kd\) is closed, and \[\dim Y\leq \dim Z\leq \dim E-2<\dim(E/kd),\] so \(Y\) is proper.

After excluding finitely many places, the decomposition \(E=kd\oplus s(E/kd)\) is integral, \(d\) is a basis of its first summand, and \(\pi|_Z\) extends to a finite morphism of the models. A finite set of module generators for its coordinate algebra gives a uniform bound \(D\) for the cardinality of every geometric fiber. Include these exceptional places and all places with \(q_v\leq D\) in \(\Sigma_0\).

Linear projection is open over each completion. Additive strong approximation away from \(v_0\), equivalently the usual consequence of Riemann–Roch for a function field, therefore gives \[q\in (E/kd)(k)\setminus Y(k)\] that is integral outside \(\Sigma\) and belongs to \(\pi(\mathcal W_v)\) at every prescribed place. To ensure \(q\notin Y\), shrink one projected opening to avoid \(Y\); a proper algebraic subset has empty interior over an infinite local field. If there are no prescribed openings, impose such an additional opening inside the integral points at one auxiliary place. This does not change the integrality requirement. Put \(y=s(q)\).

The reduction of \(q\) belongs to the reduction of \(Y\) at only finitely many places outside \(\Sigma\). For example, choose a polynomial vanishing on \(Y\) but not at \(q\), clear its finitely many denominators, and use the fact that its nonzero value at \(q\) has only finitely many zeros. At each of these places the forbidden values of the residue of \(l\) number at most \(D<q_v\). There is therefore a nonempty congruence condition on \(l\in\mathcal O_v\) for which the reduction of \(y+ld\) avoids \(Z\).

At a prescribed place, \(q\in\pi(\mathcal W_v)\) says precisely that the set of \(l\in k_v\) with \(y+ld\in\mathcal W_v\) is nonempty and open. Additive strong approximation away from \(v_0\) now supplies \(l\in k\) satisfying these finitely many conditions and integral at every remaining place outside \(\Sigma\). Shrink the openings for \(l\) to be relatively compact. There are infinitely many such \(l\), since a nonempty open subset of the adeles away from \(v_0\) meets the dense diagonal copy of \(k\) in infinitely many points. If their absolute values at \(v_0\) were bounded, these points would lie in a compact subset of the full adele ring. Its intersection with the discrete subgroup \(k\) is finite. Hence \(|l|_{v_0}\) is unbounded, as required. ◻

Proof. Work first over an algebraic closure. Put \(H=d\lambda(1)\), a noncentral diagonal trace-zero matrix. The span of its conjugates contains every root vector \(E_{ij}\). Indeed conjugation by \(1+tE_{ij}\) subtracts from \(H\) a nonzero multiple of \(tE_{ij}\) whenever its \(i\)-th and \(j\)-th diagonal entries differ, and permutation conjugations supply every pair of positions. The span is conjugation-invariant. Conjugating \(E_{ij}\) by \(1+tE_{ji}\) and subtracting root-vector terms therefore supplies \(E_{ii}-E_{jj}\). These root vectors and diagonal differences span \(\mathfrak{sl}_n\) in every characteristic. In particular this argument does not require \(\mathfrak{sl}_n\) to be a simple Lie algebra.

The differential of \(\eta\) at \(1\) contains the line spanned by \(H\): vary just the absolute coordinate indexed by the identity of \(\mathop{\mathrm{Gal}}(M/k)\). Finitely many conjugates of that line span \(\mathop{\mathrm{Lie}}(S)\), and multiplication by \(e_0\) does not change the span. Thus suitable conjugators give a surjective differential for the product map at the identity. Surjectivity is an open condition on the conjugators. Since \(S(k)\) is Zariski dense, this nonempty open contains a rational tuple. Both source and target are smooth, so the differential criterion gives the asserted smoothness.

Let \(\Phi_0\) denote the product with exponent \(e_0\) in place of \(e\). Each chosen list has a proper closed nonsmooth locus in its own parameter torus. The full map \(\Phi_0\) is smooth wherever either list is smooth, since multiplication by the remaining factors is a translation when their parameters are fixed. The simultaneous nonsmooth locus has codimension at least two, because the two lists use disjoint coordinates. On its complement, smooth pullback preserves the lower bound of two for the codimension of \(Y\). Hence \(\Phi_0^{-1}(Y)\) has that bound everywhere.

Finally, \[\Phi=\Phi_0\circ[p^a],\] where \([p^a]\) raises every parameter to its \(p^a\)-th power. After splitting the parameter torus, this is coordinatewise Frobenius. It is a finite universal homeomorphism, and therefore preserves the dimensions and codimensions of inverse images of closed subsets. This proves the assertion for \(\Phi^{-1}(Y)\). Passing from an open subset of affine space to its closure does not change its dimension. ◻

Proof. The surjective differential for the exponent-\(e_0\) map remains surjective on reduction outside finitely many places. Its reduction is dominant, as is the exponent-\(e\) map obtained by composing with the surjective power map on the parameter torus. We also exclude places where the reduction of \(Y\) is not proper.

Choose affine coordinates on the underlying parameter space and equations for \(Y\). Pulling these equations back through the product map, and clearing the fixed torus denominators, gives polynomials of bounded degree. Their degrees are independent of the translating point \(x\). For every residue point \(\bar x\in X(\kappa_v)\), dominance implies that at least one such polynomial is nonzero. Multiply it by the product of the norm polynomials defining the torus boundary. The result is a nonzero polynomial, of degree bounded by a fixed integer \(C\), whose nonvanishing gives unit parameters and avoidance of \(Y\).

A nonzero polynomial of degree at most \(C\) in \(N\) variables over a field with \(q\) elements has at most \(Cq^{N-1}\) zeros; this follows by induction on \(N\). It therefore has a nonvanishing value when \(q>C\). Exclude the finitely many places with \(q_v\leq C\), and lift the resulting parameters to local units. ◻

Proof. We shall first choose the group-valued product and then apply Lemma 7 to its affine parameters. There are two reasons to enlarge the initial finite set of places: the initial rational point may have denominators, and the eventual projection may introduce further model exceptions. The former will be handled by extra factors; the latter by the fixed basic lists.

Choosing the basic lists and the initial point. Choose the conjugators of the two basic lists in the open subset supplied by (iii). Nonempty Zariski opens can be intersected here, since the product of copies of \(S\) is geometrically irreducible. Put into \(B\) the model exceptions for these fixed lists and the finite set from Lemma 9. At each newly added place, choose an opening contained in \(\mathcal U_v\) using (ii). These places lie outside \(A\), so the additional conditions are compatible with the closure of \(\mathcal C\).

Choose \(x\in\mathcal C\) meeting these local conditions and the nonempty Zariski open condition on \(x\) in (iii). This follows from Proposition 5: use an omitted isotropic place different from the finitely many places at which approximation is required. The Zariski condition can be included by shrinking the opening at \(v_0\), because a nonempty analytic open subset of a smooth geometrically irreducible variety is Zariski dense.

Let \(B_1\) be the finite set of places outside the enlarged \(B\) at which \(x\) is not integral. At each \(v\in B_1\) choose a nonempty local open subset of \(\mathcal U_v\).

Moving the denominator places. For \(v\in B_1\), the group \(S\) is isotropic over \(k_v\). The image \[\eta\bigl((M\otimes_k k_v)^\times\bigr)^e \subset S(k_v)\] is noncentral. Indeed its geometric image contains the noncentral cocharacter image, and local points of its source torus are Zariski dense. Local Kneser–Tits and Tits simplicity, as recalled in Section 2, imply that its normal closure is all of \(S(k_v)\). To be explicit, simplicity modulo the center first gives surjectivity onto that quotient; the remaining quotient is abelian and vanishes because \(S(k_v)\) is perfect. Every element of this normal closure is a finite product of conjugates of the displayed values; inverses are obtained by inverting the parameters.

Consequently finitely many factors of the form in (8) can move \(x_v\) into the chosen opening at every \(v\in B_1\). Use the largest number needed at these finitely many places, inserting identity factors where necessary. Approximate the local conjugators by rational conjugators, using strong approximation for \(S\) away from \(v_0\) (Prasad 1977); the hypothesis is satisfied because \(S\) is simply connected and \(S(k_{v_0})\) is noncompact. Choose the rational conjugators integral outside \(B\cup B_1\). The nonempty Zariski open condition on the appended conjugators in (iii) can be imposed at the same time: first shrink their local openings into that Zariski open and then apply strong approximation.

With these conjugators fixed, continuity gives nonempty open conditions on the parameter tuple at each place of \(B\cup B_1\). At \(B\) take all parameters close to \(1\). At \(B_1\) take the basic parameters close to \(1\) and the appended parameters close to the values effecting the local moves. Let \(r\) be the resulting total number of factors and retain the notation \(\Phi:\mathcal T_r\to X\).

The subset to be avoided. In \(E_r\) form the union \(Z\) of the following closed subsets: the closure of \(\Phi^{-1}(Y)\); every intersection of two distinct absolute boundary hyperplanes; and, on each boundary hyperplane, the closure of the locus where its pole tests fail away from the other hyperplanes. Include all Galois conjugates, so this union is defined over \(k\). Lemma 8 gives codimension at least two for the first part. The pairwise intersections have codimension two, and the remaining parts do also have codimension at least two, since each test succeeds on a nonempty open subset of its hyperplane.

Outside finitely many model exceptions, integral parameters whose reduction avoids \(Z\) have one of two forms. If every absolute coordinate is a unit, \(\Phi\) is integral and its reduction avoids \(Y\), so (i) applies. Otherwise exactly one coordinate has positive valuation, and its pole tests hold, so (iii) applies. In either case \(\Phi\) belongs to \(\mathcal U_v\).

Choosing the line and treating the remaining exceptions. Choose \(d=(d_1,\ldots,d_r)\in E_r(k)\) whose point at infinity misses the projective closure of \(Z\), and with each \(d_j\) sufficiently close to \(1\) at \(v_0\) that \(\Phi(d)\) satisfies the prescribed opening there. These requirements are compatible: the directions excluded at infinity form a proper algebraic subset, whereas the condition at \(v_0\) is a nonempty analytic open. Weak approximation on affine space supplies such a rational direction. Choose a rational linear section of projection along \(d\).

The equations defining \(Z\), the projection, and its fiber bound now give a fixed finite set of additional exceptional places as in Lemma 7. Let \(\Sigma\) contain those places, the exceptions in (iii), all exceptions to the preceding reduction argument, and \(B\cup B_1\). At a place of \(\Sigma\setminus(B\cup B_1)\), the initial point and every conjugator are integral. Lemma 9 therefore gives unit parameters on one basic list whose product with \(x\) reduces outside \(Y\); set all other parameters to \(1\). A sufficiently small local opening around this tuple maps into \(\mathcal U_v\) by (i). This argument is valid even when the projection or the pole equations have unsuitable reduction at that place. The small residue fields for which the basic-list argument could fail were already included in \(B\) before \(x\) was chosen.

Thus there are parameter openings at every place of \(\Sigma\), with the required local consequence, and no further choices of rational conjugators or initial points are needed. Apply Lemma 7, omitting the opening at \(v_0\). It gives \(y\in E_r(k)\) and parameters \(a=y+ld\) with \(|l|_{v_0}\) arbitrarily large, meeting all prescribed openings away from \(v_0\) and reducing outside \(Z\) at every place outside \(\Sigma\).

Recovering the opening at \(v_0\) and the exact coset. For each component of the parameter tuple, \[\eta(y_j+ld_j) =\eta(d_j+l^{-1}y_j) \longrightarrow\eta(d_j) \qquad\text{as }|l|_{v_0}\longrightarrow\infty,\] by (7). Hence \(\Phi(a)\) eventually satisfies the opening at \(v_0\). For such \(l\) all components \(a_j\) are nonzero at \(v_0\), so they belong to \(M^\times\) and \(a\in\mathcal T_r(k)\). At the other places of \(\Sigma\) the chosen openings give the required conclusion, and outside \(\Sigma\) avoidance of \(Z\) gives it. Finally (9) gives \[x'=\Phi(a)\in xR=\mathcal C.\] The conditions selected at the places added to \(B\) were contained in their original allowed sets \(\mathcal U_v\), so the conclusion holds for the original finite set as well. ◻

Proof. In the second assertion set \(a=1\) and \(b=\det h\). In both assertions we shall construct a regular semisimple first element \(x\) whose centralizer has local elements of determinant \(b\) at every place. The norm-torus theorem of Section 4 will then give a global second element, together with its approximations at \(A\).

We begin with the local conditions at \(A\). In the first assertion, perturb \(x_v\) within the determinant-\(a_v\) fiber of its given field torus until it is regular semisimple. The field is separable, so its norm is a smooth map. After extension to an algebraic closure, a norm fiber is a translate of \[\{(z_1,\ldots,z_n)\in\mathbb G_m^n:z_1\cdots z_n=1\}.\] No equality \(z_i=z_j\), with \(i\ne j\), holds identically on this fiber. Its regular elements therefore form a dense open subset, also dense in the local topology. Near a regular semisimple element the conjugation map from the group times its torus is a submersion. Consequently the centralizers of nearby elements are obtained by small conjugations. The same conjugations transport \(t_v\), preserving its determinant and the required approximation.

For the second assertion, first perturb \(h_v\) to a regular semisimple element in its determinant fiber. Such a perturbation is possible because the fiber is a smooth translate of \(S\) and its regular semisimple open is nonempty. In the resulting separable field torus choose a regular norm-one element close to \(1\) as the first element. The preceding conjugation argument again gives a local opening for \(x\). These openings near \(1\) are compatible with the prescribed coset \(\gamma R\): its closure at \(A\) is \(P_0\), and \(P_0\) is open.

Choose a finite set \(B\) containing \(A\), an isotropic place to be omitted in strong approximation, all exceptions for integral models, all places ramified in \(L/k\), and the places where \(a\) or \(b\) is not a unit. At the additional places we need local tori whose determinant images contain both \(a_v\) and \(b_v\). At a nonsplit place of \(L/k\), the algebra splits and a diagonal Hermitian torus has determinant image equal to the local norm-one group. At a split place outside \(A\), write a component of the algebra as \(M_l(\Delta)\). If \(\Delta\) is nontrivial, then \(l\geq2\). Use an unramified maximal subfield of \(\Delta\) in one diagonal block and a totally ramified separable maximal subfield in a second block. The first norm supplies all units, and the second norm supplies an element of valuation one; their combined image is all of \(k_v^\times\). Such separable totally ramified fields exist in every characteristic, for instance by separable Eisenstein polynomials, and embed in \(\Delta\) by the local invariant criterion. Complete the remaining blocks to a maximal étale subalgebra. If \(\Delta=k_v\), use the ordinary diagonal torus. In every case choose the first element regular with determinant \(a_v\) and take a sufficiently small opening around it. These constructions also give nonempty local openings at any further exceptional places introduced by the approximation theorem.

We add finitely many auxiliary places, split in \(L\) and in \(D\), at which \(a\) and \(b\) are units. Prescribe unramified regular centralizers having every possible permutation cycle type in \(S_n\), one type at each place. Unramified norms supply both unit determinants. We may use places with sufficiently large residue fields that regular elements with those determinants exist. These conditions will force the Galois group required by Proposition 12.

We now verify the hypotheses of Proposition 10 on the determinant-\(a\) slice \(X\). This slice is a rational translate of \(S\), since the determinant on \(U(k)\) is onto. For the first assertion, weak approximation supplies a rational point meeting the finitely many openings; we then apply the proposition to its coset modulo \(R\). For the second assertion, we apply it to \(\gamma R\). The closure statement of Proposition 5 permits the openings near \(1\) at \(A\) and places no restriction on the other local openings.

Define \(Y\subset X\) over \(k\) to be the closure of the geometric locus whose reduced characteristic polynomial has no root of multiplicity one. The root-multiplicity condition is preserved by the descent defining \(X\). We shall require integral reductions outside \(B\) to avoid the spread-out model of \(Y\). It has geometric codimension at least two in \(X\). Indeed, on a split matrix component there are at most \(\lfloor n/2\rfloor\) distinct eigenvalues on this locus. The fixed determinant reduces the eigenvalue parameter dimension by one, even if the resulting torus equation is inseparable. A conjugacy class with fixed eigenvalues has dimension at most \(n^2-n\). There are finitely many Jordan types, so \[\dim Y\leq n^2-n+\lfloor n/2\rfloor-1 \leq (n^2-1)-2.\] This codimension bound and the description by root multiplicities persist outside finitely many model exceptions.

Suppose that an integral regular semisimple element reduces outside \(Y\), and write \(K=k_v\) and \(L_K=L\otimes_k K\). A simple absolute residue root belongs to an irreducible residue factor of multiplicity one. Hensel lifting supplies an unramified field factor of the étale centralizer over \(L_K\). Take its full orbit under the involution and denote the resulting algebra by \(E_j\). All its factors are unramified over \(K\), since \(L_K/K\) is unramified or split. If the field factor is stable under the involution, its fixed field \(F_j\) is unramified over \(K\). If two factors are exchanged, their fixed algebra is the diagonal field under that exchange, and is again unramified. In both cases quadratic descent identifies the algebra with involution as \[E_j=F_j\otimes_K L_K,\] with the identity on \(F_j\) and quadratic conjugation on \(L_K\).

Its unitary torus contributes local determinant norms as follows. After an unramified extension splitting these algebras, it has one free multiplicative coordinate \(z_\tau\) for each embedding \(\tau:F_j\hookrightarrow\overline K\). The determinant is \(\prod_\tau z_\tau^{\epsilon_\tau}\) with \(\epsilon_\tau\in\{1,-1\}\): each field embedding occurs once in the reduced characteristic representation, and the sign records which member of its involution pair lies in the chosen \(L_K\)-component. Passing to the other member inverts the coordinate. The dual map from the rank-one character lattice of \(L_K^1\) therefore has primitive image. Thus the determinant is a surjection of unramified tori with a torus as kernel, in every characteristic. Lang’s theorem makes the map onto on residue-field points, and smooth lifting makes it onto integral points. It supplies the unit \(b_v\).

For a single pole, the approximation theorem places us at a place split in the fixed Galois field and gives \[x=g_L\mathop{\mathrm{diag}}(t^{e d_1},\ldots,t^{e d_n})g_R, \qquad d_1<\cdots<d_n,\] with \(\mathop{\mathrm{ord}}_v(t)>0\) and integral invertible side factors. Require the principal minor on the first \(j\) indices of \(g_Rg_L\) to be a unit for each \(1\leq j\leq n\). These are nonempty algebraic open conditions on each pole hyperplane. Indeed, the cyclic product \(g_Rg_L\) contains one entire undamaged basic parameter list. Keeping all other parameters fixed, that list varies \(g_Rg_L\) densely in the fixed determinant slice. The principal-minor conditions are nonempty on this slice, as witnessed by a diagonal matrix of the given determinant. Consequently, once the two basic lists have been chosen, every initial \(x\) and every appended conjugator tuple admit these pole tests; no further generic restriction on those choices is needed. Identity values of appended parameters preserve the original tests. If \(c_j\) is the \(j\)th elementary characteristic coefficient, its principal-minor expansion gives \[ \mathop{\mathrm{ord}}_v(c_j)=e(d_1+\cdots+d_j)\mathop{\mathrm{ord}}_v(t). \tag{17}\] The selected minor is the unique term of least valuation. Thus all Newton segments have length one and distinct slopes. The characteristic polynomial splits into distinct linear factors over \(k_v\), so its centralizer supplies every determinant.

For the allowed sets in Proposition 10, include all elements outside the regular semisimple open. On that open use the determinant-norm condition just proved. The integral and pole conditions hold, and the local tori constructed above provide the required nonempty openings. Impose regularity at one place of \(B\). The proposition now gives a global regular semisimple \(x\), with the prescribed determinant, local conditions, and, in the second assertion, its exact coset modulo \(R\).

Since \(D\) is division, \(E=L(x)\) is a maximal separable field in \(D\). It is stable under \(*\) because \(x^*=x^{-1}\). If \(F=E^{*=1}\) is its fixed field, then \(E=LF\) and \([F:k]=n\). Let \(\Gamma\) be the Galois group of the joint normal closure of \(F\) and \(L\); its natural action embeds it in \(S_n\times C_2\). The prescribed Frobenius classes at places split in \(L\) imply that the kernel of \(\Gamma\to C_2\) meets every conjugacy class of \(S_n\). Such a subgroup is all of \(S_n\). For, if it were proper, the transitive action on its cosets would have a derangement: the average number of fixed cosets is one, while the identity fixes more than one. A derangement cannot be conjugate to an element of the subgroup. Since \(\Gamma\) also surjects onto \(C_2\), we obtain \(\Gamma=S_n\times C_2\).

The unitary centralizer of \(x\) is \[T_E=\mathop{\mathrm{Res}}_{F/k}\bigl(\mathop{\mathrm{Res}}^1_{E/F}\mathbb G_m\bigr), \qquad \det|_{T_E}=N_{E/L}.\] The fiber \(N_{E/L}^{-1}(b)\) has points at every completion, by the local openings, integral conditions, and pole conditions. It is a torsor under the norm kernel of Proposition 12, whose Galois hypothesis we have just checked. That theorem gives a rational point \(t\) with the desired approximations at \(A\). It lies in the centralizer of \(x\) and has determinant \(b\). ◻

Proof. Choose an unramified maximal subfield \(E_K/K\) and a prime element \(\Pi\) of \(\mathcal D\) inducing a generator \(\sigma\) on \(E_K\). We normalize so that \(\mathop{\mathrm{ord}}_K(\mathop{\mathrm{Nrd}}\Pi)=1\); \(\Pi^n\) is a central uniformizer. Let \(\mathfrak P=(\Pi)\) be the maximal ideal of the maximal order of \(\mathcal D\), and let \(\mathfrak p\) be the maximal ideal of \(K\). Every element of \(S_K\) is integral. Its reduction has norm one, and the norm-one Teichmüller representatives in \(E_K\) give a multiplicative section of the reduction map onto \(T_K\).

Put \(S_r=S_K\cap(1+\mathfrak P^r)\) for \(r\geq1\). The norm filtration is \[ \mathop{\mathrm{Nrd}}(1+\mathfrak P^r)=1+\mathfrak p^{\lceil r/n\rceil}. \tag{19}\] For the inclusion, the roots of the reduced characteristic polynomial of an element of \(\mathfrak P^r\) have valuation at least \(r/n\); their elementary symmetric coefficients have the corresponding valuation bounds. For surjectivity, use norms of principal units in the unramified field \(E_K\): its norm maps \(1+\mathfrak p_{E_K}^{\lceil r/n\rceil}\) onto the right-hand side. The latter assertion follows successively from surjectivity of the finite-field trace and completeness.

Leading-coefficient coordinates now give \[ S_r/S_{r+1}\simeq \begin{cases} (\mathbb F_{q^n},+),&n\nmid r,\\ \ker\mathop{\mathrm{Tr}}_{\mathbb F_{q^n}/\mathbb F_q},&n\mid r. \end{cases} \tag{20}\] If \(n\nmid r\), the norm images at levels \(r\) and \(r+1\) agree in (19), so any leading coefficient can be corrected at the next level to give norm one. If \(r=nm\), the first norm coefficient of \(1+a\Pi^r\) is the residue trace of \(a\) at level \(m\); correction at the next level is possible exactly when that trace is zero. The trace map of a finite-field extension is surjective even when the characteristic divides \(n\).

For \(n\nmid r\), choose a Teichmüller element \(t\in T_K\) not fixed by \(\sigma^r\). Such a \(t\) exists because \(|T_K|=(q^n-1)/(q-1)\) is larger than the multiplicative group of any proper subfield of \(\mathbb F_{q^n}\). The commutator with \(t\) multiplies the leading coefficient at level \(r\) by the nonzero scalar \(t/\sigma^r(t)-1\). It therefore fills the grade in (20).

For \(n\mid r\), use commutators between levels \(1\) and \(r-1\). Neither level is divisible by \(n\), since \(n\geq3\). Choose leading coefficient \(1\) at level \(1\) and arbitrary coefficient \(b\) at level \(r-1\). The leading coefficient of their commutator is \(\sigma(b)-b\). These fill \[(\sigma-1)\mathbb F_{q^n}=\ker\mathop{\mathrm{Tr}}_{\mathbb F_{q^n}/\mathbb F_q}.\] Successive approximation in the complete filtration puts all of \(S_1\) in \(\overline{[S_K,S_K]}\). The reverse inclusion holds because the residue quotient is abelian, proving the first assertion.

Finally, commutators of \(\Pi\) with units of \(E_K\) have residues \(\sigma(z)/z\), which fill \(T_K\) by finite-field Hilbert 90. Together with \(S_1\) they are dense in \(S_K\); all such commutators have reduced norm one. A unit of \(\mathcal D\) acts trivially on its commutative residue field, whereas \(\Pi\) acts as \(\sigma\). Writing an arbitrary element as a unit times a power of \(\Pi\) proves the assertion about conjugation. ◻

Proof. Keep their two determinant coordinates fixed. By Proposition 17(i), their local components are limits of commuting rational pairs with those exact determinants. The lifts under the homomorphism \(\ell\) of each rational pair commute. For commuting base elements, the commutator of lifts is independent of the choices of lifts. Continuous local sections of (23) therefore allow us to take the limit of these commutators, giving the identity. ◻

Proof. If the correlation were positive, every color would occur as \(\phi(u)=\phi(u')\) for the fixed translation in (43). Indeed a point of the positive intersection supplies the two-coordinate return pattern in each palette copy, and every such finite pattern occurs rationally. The rational points for different colors may differ, which is all that is needed.

Write \(W=\phi(w)\), \(Z=\phi(z)\), and, for a return point, put \(X'=\phi(u)=\phi(u')\) and \(Y'=\phi(v)\). With \(B=Z^{-1}X'\), Equation (42) yields \[W^{-1}Y'\in C_{\mathcal F}(B),\qquad Z^{-1}Y'WZ\in C_{\mathcal F}(BZ^2).\] As \(WZ=ZW\), direct multiplication gives \[(W^{-1}Y')^{-1}(ZW^{-1})(Z^{-1}Y'WZ)=ZW.\] The return colors make \(B\) range over all of \(\mathcal F\), contradicting Theorem 20. The exceptional affine map preserves membership and hence does not arise under the stated hypotheses. ◻

Proof. By dominated convergence and (48), \(\mathbb E_jF_x\to\nu_x(\{1\})\). Since \(|F_x|\leq\delta\) and \(F_x\) vanishes on \(E\), this gives the lower-density assertion.

The union of the atoms of the measures \(\nu_x\) is countable. Pass to a subsequence on which \[c(\chi)=\lim_j\mathbb E_j \bigl(\boldsymbol 1_E(y)\chi(v(y))\bigr)\] exists for every character in that union. In particular, \(c(1)=d\). For any finite list of distinct characters, their Gram matrices for \(\mathbb E_j\) converge to the identity by (48). Bessel’s inequality therefore gives \(\sum_\chi|c(\chi)|^2\leq1\). Consequently the set of nontrivial atoms for which \(|c(\chi)|\geq d\delta/2\) is finite; denote it by \(\Lambda\).

The nonatomic part of \(\nu_x\) contributes zero to the limiting average against \(\boldsymbol 1_E\). Indeed, the mean square of its Fourier transform, evaluated at \(v(y)-a_x\), tends to zero: expand the square, apply (48) to \(\chi\overline\eta\), and integrate over two copies of the nonatomic measure. The diagonal has product measure zero. Cauchy–Schwarz now proves the assertion about the weighted average. The atomic part may be averaged term by term because its weights are summable. Thus \(F_x|_E=0\) implies \[0=d\nu_x(\{1\})+ \sum_{\chi\ne1}c(\chi)\chi(-a_x)\nu_x(\{\chi\}).\] The absolute contribution from characters outside \(\Lambda\cup\{1\}\) is at most \(d\delta^2/2\). The trivial character contributes at least \(d\delta^2\), and \(|c(\chi)|\leq1\). The remaining atoms must therefore have total mass at least \(d\delta^2/2\), as required. ◻

Proof. It suffices to consider the zero set of a nonzero polynomial \(f\) of total degree \(D\). The elementary grid bound, obtained by induction from the one-variable root bound, says that its zero proportion on \(A_1\times\cdots\times A_r\) is at most \(D/\min_i|A_i|\). This proves the second assertion.

For the first assertion, choose any finite additive subgroup \(H\subset k\). An exhausting sequence \(U_j\subset k^r\) eventually contains \(H^r\), and each such \(U_j\) is a disjoint union of its \(H^r\)-cosets. On a coset \(u+H^r\), apply the grid bound to the nonzero polynomial \(f(u+X)\). The zero proportion is at most \(D/|H|\), uniformly in the coset. Finite additive subgroups of \(k\) have arbitrarily large cardinality, so this bound proves the claim. ◻

Proof. Write \(\chi(l,s,r)=\chi_1(l)\chi_2(s)\chi_3(r)\). If \(\chi_3=1\), orthogonality on \(H_j\) proves the assertion when \(\chi_2\ne1\); when \(\chi_2=1\), the average of a nontrivial \(\chi_1\) on \(I_j\setminus\{0\}\) is eventually \(-1/(|I_j|-1)\).

Suppose \(\chi_3\ne1\). Fix distinct \(h_1,\ldots,h_m\in k\), which belong to \(H_j\) for all sufficiently large \(j\). For fixed \(l\in I_j\setminus\{0\}\), set \[f_l(s)=\chi_2(s)\chi_3((s^2+P_\theta)/l).\] The average of \(f_l\) equals the average of \(m^{-1}\sum_a f_l(s+h_a)\). In the squared modulus of the latter expression, the correlation for \(a\ne b\) is a constant of modulus one times the average on \(H_j\) of \[s\longmapsto\chi_3\bigl(2(h_a-h_b)s/l\bigr).\] Choose \(z_{ab}\in k\) with \(\chi_3(2(h_a-h_b)z_{ab})\ne1\). Once all \(z_{ab}\) belong to \(I_j\), the point \(lz_{ab}\in H_j\) detects this character, uniformly for every allowed \(l\). All these correlations are then zero. Cauchy–Schwarz gives \(|\mathbb E_{s\in H_j}f_l(s)|^2\leq1/m\), uniformly in \(l\). Letting \(m\) grow proves the character assertion. Polynomial avoidance follows from the product-grid assertion of Lemma 25. ◻

Proof. Fix a palette \(c\), a position \(h\), and the set \[E=\{y\in\mathcal A:c_{yh}\notin\mathcal S\}.\] For \(x=(l_x,s_x)\notin E\), define the surjective linear map \(L_x:k^3\to k^2\) by \[ L_x(l,s,r)= \left(\frac{P_\theta-s_x^2}{l_x}l+2s_xs-l_xr, 2s-\frac{2s_x}{l_x}l\right). \tag{51}\] Its kernel is \(kv(x)\). Substituting the parameters of \(yx^{-1}\) into Equation 46 gives exactly \(L_xv(y)\). First suppose \(c=c(p)\) is a rational palette, and let \(w_x=u(xhp)\) be the unitary representative at position \(xh\). The position \(yh\) is obtained from it by the scalar affine map \(yx^{-1}\). Using the map \(\mathcal B_{w_x}:k^2\to J\) of Section 6, define \[\nu_x= \bigl(\chi\longmapsto\chi\circ\mathcal B_{w_x}\circ L_x\bigr)_* \sigma.\] This positive measure on \(\widehat{k^3}\) has mass \(\delta\) and trivial atom at least \(\delta^2\). Its Fourier transform satisfies \(\widehat\nu_x(v(y))=K(\mathcal B_{w_x}(L_xv(y)))\). Proposition 23 consequently gives \[ \widehat\nu_x(v(y))=0\quad(y\in E),\qquad \operatorname{supp}\nu_x\subseteq(kv(x))^\perp. \tag{52}\] The support assertion holds because every character in the image annihilates \(\ker L_x\).

These measures exist for arbitrary palettes as well. Approximate \(c\) by rational palettes on successively larger finite sets of positions. For each fixed \(x\notin E\), take a weak limit of the corresponding measures on the compact dual. The equations in (52) pass to the limit, as does support in the closed annihilator \((kv(x))^\perp\). The lower bound on the trivial atom persists because \(\{1\}\) is closed.

We claim that either \(E=\mathcal A\), or \(E\) has density zero. By Lemmas 24 and 26, a nonempty complement has lower density at least \(\delta\). If \(E\) had positive density along a subsequence, those lemmas would give a finite set of nontrivial characters such that, for every \(x\notin E\), at least one of them annihilates \(kv(x)\).

For any fixed nontrivial \(\chi\), this annihilation condition is a proper polynomial condition on \((l_x,s_x)\). To treat arbitrary additive characters, fix a primitive \(p\)-th root of unity and regard characters as \(\mathbb F_p\)-linear functionals. The space \(\mathop{\mathrm{Hom}}_{\mathbb F_p}(k,\mathbb F_p)\) is a \(k\)-vector space under \[(a\lambda)(t)=\lambda(at).\] If \(\chi\) has component functionals \(\lambda_1,\lambda_2,\lambda_3\), annihilation of \(kv(l,s)\) says \[l\lambda_1+s\lambda_2+\frac{s^2+P_\theta}{l}\lambda_3=0.\] This is a rational vector equation in the finite-dimensional span of the three functionals. It is not an identity: the vectors \(v(l,s)\) span \(k^3\), as follows on multiplying any putative linear relation among their coordinates by \(l\). Clearing denominators therefore places its solutions in a proper polynomial locus. A finite union of these loci has density zero, contradicting the positive lower density of \(\mathcal A\setminus E\). The claim follows.

It remains to turn this density dichotomy into actual invariance. Let \(q_0=|\mathcal F\setminus\mathcal S|/|\mathcal F|\). For every fixed \(h\), uniform marginals give \(\int\mathbb E_j\boldsymbol1_E\,d\mu=q_0\). The claim and dominated convergence show that \(\mu(E=\mathcal A)=q_0\). For each \(x\in\mathcal A\), the event \(E=\mathcal A\) is contained in \(\{c_{xh}\notin\mathcal S\}\), whose probability is also \(q_0\). Their difference is null. There are only countably many \(x\) and \(h\), so outside a single null set membership is constant on every one of the asserted families of positions. ◻

Proof. Put \(E_0=\{x:\phi(w_x)\notin\mathcal S\}\) and \(Q_0=H_0^2+TH_0\). The polynomial \(Q_0\) is nonconstant and additive. Equation 46, applied to scalar translations, gives \[ K\bigl(B(x)Q_0(y-x)\bigr)=0\qquad(x\notin E_0,\ y\in E_0), \tag{53}\] where \[ B(x)=\frac{w_x+w_x^*}{T} =\frac{T}{(Y+H_0(x))^2+T(Y+H_0(x))+P_\theta}\in J. \tag{54}\] In any fixed \(k\)-basis of \(J\), the coordinates of \(B(x)\) are rational functions of \(x\) tending to zero at infinity. This follows also directly by expanding the inverse denominator in \(F((x^{-1}))\). The denominator never vanishes for \(x\in k\), since its two factors are \(Y+H_0(x)+\theta\) and \(Y+H_0(x)+\bar\theta\), and \(\theta\notin F\).

Push \(\sigma\) forward by the character map \[\chi\longmapsto \bigl[y\longmapsto\chi(B(x)Q_0(y))\bigr]\] to obtain a measure \(\nu_x\) on \(\widehat k\). It has mass \(\delta\) and trivial atom at least \(\delta^2\). Equation (53) is precisely the vanishing required by Lemma 24, with \(v(y)=y\) and \(a_x=x\). Ordinary orthogonality on exhausting additive subgroups supplies (48). Thus, if \(k\setminus E_0\ne\varnothing\), it has positive lower density. If \(E_0\) also had density tending to some \(d>0\) along a subsequence, there would be a fixed finite set \(\Lambda\subset\widehat k\setminus\{1\}\) such that \[ \sigma\{\chi:\chi(B(x)Q_0(\cdot))\in\Lambda\} \geq d\delta^2/2\qquad(x\notin E_0). \tag{55}\]

We prove that, for every fixed \(\chi\in\widehat J\) and every fixed nontrivial \(\lambda\in\widehat k\), the equality \[ \chi(B(x)Q_0(\cdot))=\lambda \tag{56}\] holds on a set of density zero. This is the point where the additive polynomial and the use of arbitrary additive characters both matter. Regard the characters as \(\mathbb F_2\)-linear functionals, with the \(k\)-vector space structure used in the preceding subsection. Write \(Q_0(y)=\sum_i a_i y^{2^i}\), and choose \(M=2^r\geq\deg Q_0\). The extension \(k/k^M\) is finite. For a basis \(e_1,\ldots,e_N\) over \(k^M\), the representation \[x=\sum_{j=1}^N e_jx_j^M\] is unique and defines an additive group isomorphism \(k^N\to k\). Let \(b_\ell(x)\) be the coordinates of \(B(x)\), and \(\chi_\ell\) the component functionals of \(\chi\). For each \(i\), choose a basis \((f_{im})_m\) of \(k/k^{2^i}\). Expansion in that basis gives rational functions \(R_{\ell im}\in k(X_1,\ldots,X_N)\) such that \[ a_i b_\ell\left(\sum_j e_jX_j^M\right) =\sum_m f_{im}R_{\ell im}(X)^{2^i}. \tag{57}\] Indeed the left side belongs to \(k(X_1^{2^i},\ldots,X_N^{2^i})\), and this field has basis \((f_{im})_m\) over \(k^{2^i}(X_1^{2^i},\ldots,X_N^{2^i})\).

Define fixed functionals \(\psi_{\ell im}(y)=\chi_\ell(f_{im}y^{2^i})\). The left side of (56), in functional notation, is now \[ \sum_{\ell,i,m}R_{\ell im}(X)\psi_{\ell im}. \tag{58}\] It is a rational map into a finite-dimensional \(k\)-vector space. Moreover, it tends to zero when all \(X_j\) are multiplied by a common indeterminate tending to infinity. For completeness, the left side of (57) tends to zero under this scaling because each \(b_\ell\) vanishes at infinity. If some \(R_{\ell im}\) did not tend to zero, the terms of largest Laurent degree on the right could not cancel: the \(f_{im}\) are linearly independent over \(k^{2^i}(X_1^{2^i},\ldots,X_N^{2^i})\). Thus every one of the rational functions in (58) tends to zero.

It follows that this rational map cannot be identically equal to the fixed nonzero functional \(\lambda\). If \(\lambda\) is outside its finite-dimensional span the equality is impossible; otherwise, taking coordinates and clearing denominators puts its solutions in a proper polynomial locus. Any exceptional denominator locus is proper as well. Lemma 25, transported through the additive isomorphism \(k^N\to k\), proves the asserted density zero.

Average (55) over \(x\). Its right side, restricted to \(k\setminus E_0\), has positive lower average. Its left side has average tending to zero: for each fixed \(\chi\), the finite union of the sets (56), with \(\lambda\in\Lambda\), has density zero, and dominated convergence applies to the finite measure \(\sigma\). This contradiction proves the density dichotomy. ◻

Proof. Fix \(E=LF\) as above and let \(A_E=\phi((E/F)^1)\), a finite abelian subgroup of \(\mathcal F\). If \(A_E\cap\mathcal S\) is empty there is nothing to prove. Otherwise Theorem 20 says that \(A_E\setminus\mathcal S\) is a subgroup, say \(A_E^0\). Define the homomorphism, with additive target notation, \[\psi:E^\times\longrightarrow A_E/A_E^0, \qquad \psi(a)=\phi(a/\bar a)\bmod A_E^0,\] where bar denotes the involution on \(E\). Membership of \(\phi(q(Z))\) in \(\mathcal S\) is equivalent to \(\psi(Z+\theta)\ne0\).

For \(Y\in F\), consider the finite-valued function \(g(s)=\psi(Y+s+\theta)\) on \(k\). We show that its additive period group \[\Pi=\{t\in k:g(s+t)=g(s)\text{ for every }s\in k\}\] has finite index. For \(t\ne0\), agreement of these two values is equivalent to vanishing of \(\psi\) on \[\begin{align*} &(Y+s+t+\theta)(Y+s+\bar\theta)\\ &\hspace{1cm}=Y^2+(T+t)Y+P_\theta+s^2+(T+t)s+t\bar\theta. \end{align*}\] Here \(\psi(\bar a)=-\psi(a)\), so the product measures the difference of the two values of \(g\). Division by \(t\in k^\times\) does not change \(\psi\). With \[Z_t=\frac{Y^2+(T+t)Y+P_\theta}{t},\qquad H_t(s)=\frac{s^2+(T+t)s}{t},\] the agreement condition becomes \(\psi(Z_t+H_t(s)+\bar\theta)=0\). Replacing \(\bar\theta\) by \(\theta\) negates this value and preserves its zero set. The polynomial \(H_t\) is nonconstant and additive, so Lemma 28 says that either agreement holds for every \(s\), or its set has density zero. Thus every \(t\notin\Pi\) gives density-zero agreement.

Choose translations of \(g\) indexed by distinct cosets of \(\Pi\). Any two agree on a set of density zero, because the averages are eventually invariant under each fixed additive shift. If there were more such translates than values of \(g\), then at every point two would agree. A finite union of density-zero sets cannot cover \(k\). This proves that \([k:\Pi]\) is finite.

Finally apply Lemma 28 with \(H_0(s)=s\). The zero set of \(g\) is either all of \(k\) or has density zero. Since \(g\) is constant on cosets of \(\Pi\), any nonempty zero set contains a coset of density \([k:\Pi]^{-1}\). Hence \(g\) is either identically zero or nowhere zero. Membership is therefore constant on \(\{q(Y+s):s\in k\}\). Every finite rational point belongs to such a field chart, and \(\infty\) is fixed by translations. This proves the rational-point assertion. For fixed \(s\in k\) and \(h\in\mathcal H\), the equality of membership indicators \(\boldsymbol1_{\mathcal S}(c_{\tau_s h})= \boldsymbol1_{\mathcal S}(c_h)\) is a closed condition. It holds for every rational palette, hence for every palette in their closure. ◻

Proof. We first obtain all \(F\)-translations, then use a central rotation to generate \(\mathop{\mathrm{SL}}_2(F)\), and finally express a square rotation in Cayley coordinates. Choose \(n_0=q(x_0)\in(E/F)^1\cap\ker\phi\), with \(x_0\) generating \(F/k\), such that the polynomials \[ (s+x_i+\theta)(s+x_i+\bar\theta) \tag{59}\] have pairwise disjoint root sets as \(x_i\) runs over the conjugates of \(x_0\) under embeddings fixing \(L\). These are nonempty Zariski open conditions: over a splitting field the conjugate coordinates of \(\mathop{\mathrm{Res}}_{F/k}\mathbb A^1\) are independent, and all prohibited equalities are proper linear conditions. They can be met inside \(\ker\phi\) because powers annihilating \(\mathcal F\) are Zariski dense in the norm-one torus. Indeed its rational points are Zariski dense by the Cayley parametrization, and every positive power map is dominant, even when it is inseparable.

Since the color set is finite, there are \(t_0\in k\) and an infinite set \(\Sigma\subset k\) for which \(\phi(\tau_s(n_0))=\phi(\tau_{t_0}(n_0))\) for every \(s\in\Sigma\). Here translations on unitary elements are understood through the Cayley chart. Put \[g_s=L_{n_0}^{-1}\tau_s^{-1} L_{\tau_s(n_0)\tau_{t_0}(n_0)^{-1}} \tau_{t_0}L_{n_0}\in\mathcal G_c.\] Each \(g_s\) fixes the unitary representative \(1\), which is the point \(\infty\) in additive coordinates. Its fractional matrix has entries in \(E\) and is upper triangular. To compute its diagonal entries, the exact matrix of \(L_{q(z)}\), for \(z\in F\), is \[M_z=(z+\bar\theta)^{-1} \begin{pmatrix}z&-P_\theta\\1&z+T\end{pmatrix}.\] The factor \((z+\bar\theta)^{-1}\) is retained here because it need not be central in \(D\). Multiplying the matrices in the expression for \(g_s\) gives diagonal entries \[a_s=\frac{x_0+t_0+\bar\theta}{x_0+s+\bar\theta},\qquad d_s=\frac{x_0+s+\theta}{x_0+t_0+\theta}.\] Thus its affine multiplier on the field line \(F\cup\{\infty\}\) is \[ m_s=\frac{(x_0+t_0+\theta)(x_0+t_0+\bar\theta)} {(x_0+s+\theta)(x_0+s+\bar\theta)}\in F^\times. \tag{60}\] This is also the ratio of the logarithmic derivatives of \(q\) at \(x_0+s\) and \(x_0+t_0\).

The same multiplier controls conjugation of scalar translations on the whole chart \(J\). Indeed an upper triangular fractional matrix \(\left(\begin{smallmatrix}a&b\\0&d\end{smallmatrix}\right)\) acts by \(x\mapsto(ax+b)d^{-1}\). Conjugating \(x\mapsto x+r\), \(r\in k\), gives \(x\mapsto x+r ad^{-1}\). Here \(ad^{-1}=m_s\), so \(\mathcal G_c\) contains translations by \(km_s\).

The elements \(m_s\), \(s\in\Sigma\), span \(F\) over \(k\). Otherwise a nonzero \(k\)-linear functional on \(F\) would annihilate them. After passing to a normal closure, such a functional is a linear combination of the distinct embeddings of \(F\). It would give a linear relation among the rational functions \[s\longmapsto \frac{(x_i+t_0+\theta)(x_i+t_0+\bar\theta)} {(x_i+s+\theta)(x_i+s+\bar\theta)}\] at infinitely many \(s\), and therefore an identity. Each summand has a pole that occurs in no other summand by (59); its numerator is nonzero. Every coefficient must consequently vanish, a contradiction. Composing the translations already obtained now gives every \(F\)-translation on \(J\).

For a central rotation, take the particular scalar \(a=q(0)=\theta/\bar\theta\ne1\). Its color is trivial, so \(L_a\in\mathcal G_c\). In additive coordinates it acts by \(x\mapsto-P_\theta(x+T)^{-1}\). Consequently \(W=L_a\tau_{-T}\) is represented by \(\left(\begin{smallmatrix}0&-P_\theta\\1&0\end{smallmatrix}\right)\), and direct multiplication gives \[W\begin{pmatrix}1&b\\0&1\end{pmatrix}W^{-1} =\begin{pmatrix}1&0\\-b/P_\theta&1\end{pmatrix} \qquad(b\in F).\] Thus all lower as well as upper unipotent matrices over \(F\) act in \(\mathcal G_c\). Hence \(\mathcal G_c\) contains the fractional action on \(J\) of \(\mathop{\mathrm{SL}}_2(F)\), since upper and lower elementary matrices generate that group.

Finally set \[C=\begin{pmatrix}1&\theta\\1&\bar\theta\end{pmatrix}.\] For \(\xi\in(E/F)^1\), the matrix \[ C^{-1}\begin{pmatrix}\xi&0\\0&\xi^{-1}\end{pmatrix}C =\frac1{\bar\theta-\theta} \begin{pmatrix} \bar\theta\xi-\theta\xi^{-1}&P_\theta(\xi-\xi^{-1})\\ \xi^{-1}-\xi&\bar\theta\xi^{-1}-\theta\xi \end{pmatrix} \tag{61}\] has all entries in \(F\) and determinant one. It belongs to the \(\mathop{\mathrm{SL}}_2(F)\) just generated. In unitary coordinates its action is \(u\mapsto\xi u\xi\), as is seen by applying the diagonal matrix to the column \((x+\theta,x+\bar\theta)^{\mathsf t}\). Composing this action with the unitary conjugation \(u\mapsto\xi u\xi^{-1}\) gives \(u\mapsto\xi^2u\). This is the required left rotation. ◻

Proof. Every color is represented by an element \(\xi\in V_0\) lying in a separable field of the kind used in Lemma 30. Indeed, start with any representative and perturb it within its \(R\)-coset to a regular semisimple element, using Proposition 5 and the nonempty local regular semisimple open subset. In a division algebra its generated algebra over \(L\) is a field, stable under the involution, and regular semisimplicity makes that field separable. This also covers characteristics dividing the odd degree \(n\).

Lemma 30 and the left-rotation palette rule therefore make the membership indicator invariant under left multiplication by the square of every element of \(\mathcal F\). These squares generate \(\mathcal F\): their generated subgroup is normal, and if it were trivial then \(\mathcal F\) would have exponent two and hence be abelian. Since \(\phi\) is surjective, the positions obtained from any one position by left rotations realize every color. The indicator must be constant on \(\mathcal F\), contrary to the choice of the nonempty proper set \(\mathcal S\). ◻

Proof. The variety defined by Equation 62 is a torsor under \(S\): over a separable closure the Hermitian equation has a solution, and the prescribed determinant is compatible because \[\mathop{\mathrm{Nrd}}(sh)=s^{n+1}=N_{L/k}\bigl(s^{(n+1)/2}\bigr).\] The simply connected \(H^1\)-vanishing theorem over global function fields therefore gives a rational point (Harder 1975; Conrad 2012). This applies without any restriction on the characteristic or on its divisibility into \(n\).

The cyclic approximation theorem for global fields supplies a cyclic extension \(F/k\) of degree \(n\) with full local degree at each place below the Brauer support of \(D\) (Lorenz and Roquette 2003, Theorems 1–2 and Section 4). Prescribing a degree-\(n\) local field at one of these places ensures that the global cyclic Galois algebra is a field. The characteristic-primary part is covered by Artin–Schreier–Witt approximation, and the Wang condition is automatic in positive characteristic. Since \(n\) is odd, \(F\) and \(L\) are disjoint. The local invariant criterion shows that \(LF\) splits \(D\). Its degree over \(L\) is \(n\), so the embedding criterion gives an embedding \(\alpha:LF\hookrightarrow D\) (Reiner 2003).

Let bar on \(LF\) fix \(F\) and induce the nonidentity automorphism of \(L/k\). Skolem–Noether provides \(h\in D^\times\) with \[\alpha(\bar z)^*=h\alpha(z)h^{-1}\qquad(z\in LF).\] Taking adjoints shows that \(h^*\) is another intertwiner, whence \(c=h^{-1}h^*\in\alpha(LF)^\times\). On this field \(x^*=h\bar xh^{-1}\), and consequently \(\bar c=c^{-1}\). Hilbert 90 gives \(t\in\alpha(LF)^\times\) with \(t/\bar t=c\). Replacing \(h\) by \(ht\) makes it Hermitian, since \((ht)^*=h\bar t c=ht\), and preserves the intertwining identity. Choose \(a\) as in Equation 62 for this \(h\). Then the embedding \(z\mapsto a\alpha(z)a^{-1}\) is \(*\)-stable: \[(a\alpha(\bar z)a^{-1})^* =(a^*)^{-1}h\alpha(z)h^{-1}a^* =a\alpha(z)a^{-1}.\] Its image is the required field \(E\). ◻

Proof. The unitary multiplications leave Equation 66 unchanged. For the pair transformation put \(b=ed\) and \(z=b^*+e-a^*c\). Expanding in the indicated order gives \[z+z^*=(a-c)^*(a-c),\qquad zd+d^*z^*=(a-cd)^*(a-cd).\] The pair transformation is invertible, since \(d\ne1\), so \(z\ne0\). Set \(P=ce^{-1}-ab^{-1}\) and \(Q=b^{-*}ze^{-1}\), where \(b^{-*}=(b^*)^{-1}\). Direct multiplication gives \[P^*P=Q+Q^*,\qquad Q=(b^{-1})^*(zd)b^{-1}.\] The multiplicative identity that relates their characters is \[ 1-PQ^{-1}P^*=m_bm_zm_e^*, \tag{68}\] where the defining vectors for \(m_b,m_z,m_e\) are \(a,c-a,c\), respectively. One can check the identity without commuting any factors: use \(Q^{-1}=ez^{-1}b^*\) and \[\begin{align*} Pe&=-ab^{-1}z+m_b(c-a),\\ b^*P^*&=ze^{-*}c^*+(c-a)^*m_e^*. \end{align*}\] In multiplying these expressions, the terms not containing \(-m_b(c-a)z^{-1}(c-a)^*m_e^*\) sum to \(m_bm_e^*\); here one uses \(m_ba=-ab^{-1}b^*\) and \(c^*c=e+e^*\). By congruence invariance the left side of Equation 68 has character \(f(zd)\). Hence \(f(zd)=f(b)+f(z)-f(e)\), which is precisely the assertion. ◻

Proof. Put \(M=\mathbb ZU(k)\); it is a subring because \(U(k)\) is a group. Choose a place of \(k\) split in \(L\). Hilbert 90 and weak approximation give a central phase \(z\in L^1\) with a simple pole at one of the two places above it and a simple zero at the other. Then \(t_0=z+z^{-1}\in k\cap M\) has a simple pole at the chosen place. In particular \(k/ k_0\) is finite separable for \(k_0=\mathbb F_p(t_0)\), where \(p=\operatorname{char}(k)\). The group \(M\) is a module over \(R=\mathbb F_p[t_0]\).

Fix a finite prime \(\pi\) of \(R\), let \(K\) be the corresponding completion of \(k_0\), and let \(C\) be the additive closure of \(M\) in \(D_K=D\otimes_{k_0}K\). It is a closed module over the valuation ring \(\mathcal O_K\). Weak approximation for \(U\) puts the product of all its local groups above \(\pi\) inside \(C\). The largest \(K\)-vector subspace contained in \(C\) is \[W=\bigcap_{r\geq0}\pi^r C.\] Indeed the right side is closed, is an \(\mathcal O_K\)-module, and is stable under division by \(\pi\). Left and right multiplication by each local unitary group preserve \(W\).

Their \(K\)-linear span is the full algebra \(D_K\). To check this, view \(\mathop{\mathrm{Res}}_{k/k_0}U\) as a group over \(k_0\) and extend scalars from \(K\) to an algebraic closure. Separability of \(k/k_0\) and \(L/k\) gives independent pairs of matrix factors, on which a unitary element has the form \((g,g^{-t})\) with \(g\in\mathop{\mathrm{GL}}_n\). These pairs span both matrix factors: varying a scalar multiple of \(g\) separates the weights \(\lambda\) and \(\lambda^{-1}\), and invertible matrices span \(M_n\). Independent factors give the whole product. The local points of the smooth unitary groups are Zariski dense, so the same linear-span assertion holds over \(K\). It follows that \(W\) is a two-sided ideal of \(D_K\).

Every simple factor occurs in this ideal. At a nonsplit place of \(k\) the algebra over \(L\) splits and the unitary group is isotropic because \(n\geq3\). Take an unbounded sequence in that local group and the identity in every other component. For suitable integers \(r_\nu\to\infty\), multiplication by \(\pi^{r_\nu}\) gives a subsequence converging to a nonzero vector in that factor and to zero elsewhere. For every fixed \(r\), this sequence eventually belongs to \(\pi^rC\); closedness therefore puts its limit in \(W\). At a split place, the unitary group contains reciprocal central scalars. The same argument with \((\pi^{-r},\pi^r)\), and then with the factors reversed, isolates each of the two simple factors. Thus \(W=D_K\), and \(C=D_K\).

It remains to pass from all these local closures to an exact equality. The \(k_0\)-linear span of \(M\) is \(D\), since a proper finite-dimensional subspace would remain proper after completion. Choose a full \(R\)-lattice \(\Lambda\subset M\). For \(x=\pi^{-j}\lambda\), with \(j\geq0\) and \(\lambda\in\Lambda\), local density gives \(a\in M\) with \(a-x\in\Lambda\otimes_R\mathcal O_K\). Choose a polynomial \(q\in R\) which clears all denominators of \(a\) away from \(\pi\) and satisfies \(q\equiv1\pmod{\pi^j}\). The Chinese remainder theorem permits both conditions. Then \(qa-qx\) belongs to the local lattice at every finite prime of \(R\), hence to \(\Lambda\), and \((q-1)x\in\Lambda\). It follows that \(x\in M\). Partial fractions now give every element of \(k_0\Lambda=D\) inside \(M\), proving the lemma. ◻

Proof. We first construct a transformation triangular at a single block. The first-column slope of \(B_h(t)B_{hz}(t)\) is \(h(1+z)/(1+tz)\). At \(t=t_j\), its norm from \(E\) to \(F\) is \[ \ell_j(z)=\frac{(1+z)^2}{(1+t_jz)(z+\bar t_j)}. \tag{70}\] This is a separable rational map of degree two. In characteristic two, its derivative is nonzero because \(t_j\bar t_j\ne1\); in other characteristics separability is automatic. For a generic \(z\in E^1\), the other point of the fiber is \[z'=\frac{(t_j-\bar t_j)z+1+t_j\bar t_j-2\bar t_j} {(1+t_j\bar t_j-2t_j)z+\bar t_j-t_j}.\] It is distinct from \(z\) and lies in \(E^1\): conjugate reciprocation preserves the two roots and fixes \(z\), so also fixes \(z'\). Writing \(s_t(z)=(1+z)/(1+tz)\), choose \(h'=s_{t_j}(z)/s_{t_j}(z')\in E^1\) and set \[g_j(t)=\bigl(B_{h'}(t)B_{h'z'}(t)\bigr)^{-1}B_1(t)B_z(t).\] Its lower-left numerator is \[(1+tz')(1+z)-h'(1+z')(1+tz).\] This nonzero linear polynomial has its unique root at \(t_j\); it is nonzero as a polynomial because the two fractional slope functions have distinct poles. Thus this entry is nonzero at every other \(t_\ell\) and every \(\bar t_\ell\).

Matrices of the form \[\begin{pmatrix}\alpha(t)&t\overline{\beta}(t)\\ \beta(t)&\overline{\alpha}(t)\end{pmatrix},\] with bar acting on coefficients and fixing \(t\), are closed under multiplication and inversion. Therefore the upper-right entry of \(g_j(t)\) is nonzero at all \(t_\ell\), including \(t_j\). All diagonal entries can also be kept nonzero. Indeed at \(z=-1\) the fiber interchange in Equation 70 fixes \(-1\) and has derivative \(-1\) (equal to \(1\) in characteristic two). Hence \(h'\to-1\), both boost products tend to \((1-t)I\), and \(g_j(t)\to I\). The excluded conditions are consequently proper algebraic conditions on the rational norm-one curve, whose rational points are infinite. We obtain an upper triangular nondiagonal matrix at \(j\) and a matrix with all four entries nonzero at each other block. Using \(\bar t_j\) in the same construction gives the lower triangular analogue.

Fix \(i\). For \(j\ne i\), the subgroup generated by \(g_j\) and diagonal phases is triangular at \(j\) and projectively Zariski dense at \(i\). For the second assertion, its closure contains the diagonal torus and a matrix with four nonzero entries; a proper algebraic subgroup of \(\mathop{\mathrm{PGL}}_2\) containing that torus lies in a Borel subgroup or in the normalizer of the torus, and neither contains such a matrix. The second derived subgroup is trivial at \(j\) but remains projectively dense at \(i\), by algebraic perfectness of \(\mathop{\mathrm{PGL}}_2\).

Use the full normal kernels \[K_j=\ker\bigl(H\longrightarrow\mathop{\mathrm{GL}}_2(E) \text{ on block }j\bigr).\] Each \(K_j\) has projectively dense image at \(i\). Successive commutators of these normal kernels are supported on the intersection of their supports and still have projectively dense image at \(i\), since the commutator map is regular and its image generates the derived algebraic group \(\mathop{\mathrm{PGL}}_2\). More explicitly, successive groups \(N_{r+1}=[N_r,K_j]\), beginning with one \(K_j\), remain in every kernel already used because those kernels are normal. Consequently the image \(P_i\) of \(\bigcap_{j\ne i}K_j\) is projectively dense and is normal in the full image \(H_i\) of \(H\) at \(i\).

We next produce actual unipotents in \(P_i\). Commutators of the triangular matrices at \(i\) with diagonal phases give upper and lower elementary unipotents in \(H_i\) with nonzero coefficients. Conjugating by phases and adding coefficients shows that both coefficient groups contain scalar multiples of \(\mathbb ZE^1\). We claim that they contain a common nonzero ideal in a suitable ring of \(S\)-integers of \(E\). Choose a place of \(F\) split in \(E\) and a phase \(z\) with a simple pole above it, as in the proof of Lemma 35. Then \(t_0=z+z^{-1}\in F\) has a simple pole, so \(E/\mathbb F_p(t_0)\) is finite separable and \(\mathbb ZE^1\) is a module over \(R=\mathbb F_p[t_0]\). The \(k_0\)-linear span of \(E^1\) is \(E\), where now \(k_0=\mathbb F_p(t_0)\). Indeed the group \(\mathop{\mathrm{Res}}_{F/k_0}(\ker(N_{E/F}:\mathop{\mathrm{Res}}_{E/F}\mathbb G_m\to\mathbb G_m))\) has rational points \(E^1\). After extending scalars to an algebraic closure of \(k_0\), its coordinates are independent pairs \((x_\ell,x_\ell^{-1})\). A linear relation \(\sum_\ell(a_\ell x_\ell+b_\ell x_\ell^{-1})=0\) forces all coefficients to vanish, so these pairs span their ambient product. Hilbert 90 gives Zariski density of its rational points. Thus \(\mathbb ZE^1\) contains a full \(R\)-lattice. If \(\mathcal O\) is the integral closure of \(R\) in \(E\), any two scalar multiples of such a lattice contain a common nonzero ideal \(I\subset\mathcal O\): clear the finitely many denominators of generators of the finite \(R\)-module \(\mathcal O\) relative to both lattices. There are distinct places \(P,Q\) of \(E\) above the pole of \(t_0\). The nonzero divisor \(\deg(Q)P-\deg(P)Q\) has degree zero. Finiteness of the degree-zero divisor class group over the finite constant field gives a positive multiple equal to the divisor of a nonconstant function. That function is an infinite-order unit of \(\mathcal O\) (Rosen 2002, chap. 14).

Choose \(g\in P_i\) carrying \(\infty\) to a finite point \(r\) of \(\mathbb P^1(E)\). Fix \(0\ne s\in I\) and choose an infinite-order unit \(u\in\mathcal O^\times\) sufficiently congruent to \(1\) that \[u=1+st\quad\text{for some }t\in I, \qquad (1-u^2)r\in I.\] A power of an infinite-order unit satisfies the finitely many required congruences. With \(U(x)=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right)\) and \(L(x)=\left(\begin{smallmatrix}1&0\\x&1\end{smallmatrix}\right)\), the identity \[U(s)L(t)U(-s/u)L(-tu)=\mathop{\mathrm{diag}}(u,u^{-1})\] shows that \(H_i\) contains the affine transformation \(h:x\mapsto u^2x+(1-u^2)r\). It fixes \(r\) and \(\infty\). Normality puts \(q=h^{-1}g^{-1}hg\) in \(P_i\). This element fixes \(\infty\) and has affine slope \(u^{-4}\): the derivatives of \(h\) at \(\infty\) and \(r\) are \(u^{-2}\) and \(u^2\), respectively. Hence, for \(0\ne x\in I\), \[qU(x)q^{-1}U(-x)=U((u^{-4}-1)x)\] is a nonidentity upper unipotent in \(P_i\). By definition of \(P_i\) it lifts to a transformation supported only on block \(i\). Interchanging the two axes proves the lower assertion. ◻

Proof. If \(c\ne0\), one of its \(E\)-block coordinates is nonzero. The pure upper shear from Lemma 36 changes \(a\) by some fixed nonzero \(y\in D\), leaving \(c\) unchanged. Conjugating by left multiplication of the first coordinate by \(v\in U(k)\) replaces this increment by \(vy\). Products and inverses therefore add every element of \((\mathbb ZU(k))y=D\), by Lemma 35. The lower shears similarly permit arbitrary changes of \(c\) when \(a\ne0\). These operations connect every nonzero pair to one with both entries nonzero, and connect any two such pairs. Thus \(F_d\) is constant.

Take \(e=(d-\bar d)^{-1}\), \(c=0\) and \(a=1\) in Equation 66. The elements defining \(f(e)\) and \(f(ed)\) are \(1\) and \(\bar d/d\), respectively. This evaluates the constant. Any \(b,bd\in\mathcal D\) yield a pair \((a,c)\) through their defining equations; it is nonzero since the Sylvester operator has zero kernel. This proves Equation 71. ◻

Proof of Theorem 32. Choose \(u\in U(k)\) with \(\mathop{\mathrm{Nrd}}(u)\ne1\); a central phase suffices, since the norm-one torus of \(L/k\) has infinitely many rational points. Division makes \(1-u\) invertible. For \(b=(1-u)^{-1}\), \[ b+b^*=1,\qquad b^*=-ub,\qquad f^\#(b)=\chi(u). \tag{75}\] If \(u'\in U(k)\) has the same reduced norm as \(u\), put \(b'=(1-u')^{-1}\). Oddness gives \(\overline{\mathop{\mathrm{Nrd}}(b)}=-\mathop{\mathrm{Nrd}}(u)\mathop{\mathrm{Nrd}}(b)\), and the same identity for \(b'\), so \[q=\mathop{\mathrm{Nrd}}(b')/\mathop{\mathrm{Nrd}}(b)\in k^\times.\]

For \(u'\) sufficiently close to \(u\) at \(A\), choose \(c\in D^\times\) with \(\mathop{\mathrm{Nrd}}(c)=q^{(n-1)/2}\) and \(c\) close to \(1\) at \(A_D\). Existence without approximation follows from \(H^1(L,\mathop{\mathrm{SL}}_1(D))=1\). Locally the reduced norm is smooth, including when \(p\mid n\): its differential is the nonzero reduced trace. It therefore has solutions near \(1\) for norms near \(1\). Weak approximation for \(\mathop{\mathrm{SL}}_1(D)\) adjusts the global solution to these local solutions. Set \(b''=q^{-1}c^*b'c\). Then \[\mathop{\mathrm{Nrd}}(b'')=q^{-n}q^{n-1}\mathop{\mathrm{Nrd}}(b')=\mathop{\mathrm{Nrd}}(b), \qquad f^\#(b'')=f^\#(b').\] Thus \(r=b^{-1}b''\in\mathop{\mathrm{SL}}_1(D)(L)\) is close to \(1\) at \(A_D\) and belongs to \(J\). Equation 74 gives \(\chi(u')=\chi(u)\). Taking \(u'=us\) proves continuity of \(\chi|_{S(k)}\) for the topology induced by \(S_A\).

For clarity, this continuous finite-valued homomorphism extends to \(S_A\). Weak approximation makes \(S(k)\) dense. If \(K\) is its kernel, then \(\overline K\cap S(k)=K\) by continuity. The finite cosets of \(K\) cover \(S(k)\); their closures therefore cover \(S_A\). Thus \(\overline K\) is an open normal subgroup, and \(S(k)/K\simeq S_A/\overline K\), giving the extension.

At \(v\in A\) write \(U(k_v)=D_v^\times\) and \(S(k_v)=\mathop{\mathrm{SL}}_1(D_v)\). Approximate any two elements of \(D_v^\times\) by rational unitary elements, prescribing the identity at the other places of \(A\). Their rational commutators have character zero, so continuity kills every local commutator. By Lemma 18, these commutators generate a dense subgroup of \(\mathop{\mathrm{SL}}_1(D_v)\). The extension is therefore zero on every factor of \(S_A\), proving \(\chi(S(k))=0\). If \(A\) is empty, the preceding continuity assertion already gives the result. ◻

Proof. First let \(H\) be any local form of \(\mathop{\mathrm{SL}}_2\). Choose \(g_0\in H(K)\) such that \(g_0^e\) is regular semisimple. Such elements exist: rational points are Zariski dense, and the power map is dominant, including when the characteristic divides \(e\). For a regular semisimple element \(a\), the differential at \(1\) of \(x\mapsto axa^{-1}x^{-1}\) has image \((\operatorname{Ad}(a)-1)\mathop{\mathrm{Lie}}(H)\). Over an algebraic closure this image contains both root lines. Their conjugates span \(\mathfrak{sl}_2\). This remains true in characteristic two: conjugating the upper root vector by a lower unipotent gives \[E_{12}+tI+t^2E_{21},\] so the scalar direction is in the span as well. Zariski density permits a finite set of rational conjugates of \(g_0^e\) with these spanning properties. The product of the corresponding commutator maps therefore has surjective differential at the identity. Its image contains an identity neighborhood by the local submersion theorem, and every value belongs to the subgroup generated by \(e\)-th powers. That subgroup is thus open.

For the division form, let \(\nu_B\) be the division valuation on \(B\). For \(g\ne1\), \[\nu_B(g-1)=\tfrac12\mathop{\mathrm{ord}}_{K'}\mathop{\mathrm{Nrd}}_B(g-1) =\tfrac12\sum_\lambda\mathop{\mathrm{ord}}_{K'}(\lambda-1).\] Consequently eigenvalues sufficiently close to \(1\) put \(g\) in any specified identity neighborhood of \(\mathop{\mathrm{SL}}_1(B)\), in particular in its open power subgroup.

There are only finitely many possibilities for \(K'\) up to topological field isomorphism when \([K':K]\leq r\): these fields are Laurent series fields over finite fields, and their residue degrees are bounded by \(r\). Over each there is exactly one quaternion division algebra. Topological field isomorphisms preserve the normalized valuation and the subgroup generated by powers. Taking a maximum of the finitely many thresholds, and using the bound on the ramification index to convert \(\mathop{\mathrm{ord}}_{K'}\) to \(\mathop{\mathrm{ord}}_K\), proves the first assertion. Finally, sufficiently small ambient matrices have all their reduced eigenvalues close to \(1\). A homomorphism of exponent dividing \(e\) kills the generated power subgroup, which proves the last assertion. ◻

Proof. In odd characteristic choose \(S_0=k(s)\) with \(s^2=r\) and \(r\in N_{L/k}(L^\times)\). In characteristic two choose \(s^2+s=r\), with \(r\in k\). We specify finitely many local conditions on this quadratic algebra. At the exceptional nonsplit places of \(L/k\), require it to split. The exceptional set is chosen so that at every remaining nonsplit place, \(L/k\) is unramified and the involution is represented, up to scalar, by a unimodular Hermitian form. At a split place of \(L/k\) where the index of \(D\) does not divide \(m\), require \(S_0\) to be a field. Require the same at one further place split in \(L\). Weak approximation supplies these conditions. In odd characteristic one chooses an element of \(L^\times\) first and takes its norm; at split places there is no norm restriction, and at the specified nonsplit places a square value is available. The extra split place ensures that \(S_0\) is a field and is distinct from \(L\).

We check local embeddability with the stated multiplicities. Where \(L\) splits, this is the usual embedding criterion for central simple algebras. If \(S_0\) splits, the local index divides \(m\), so there are two blocks of degree \(m\). If \(S_0\) is a field, the index after quadratic extension divides \(m\): multiplying a local invariant by two removes precisely the possible extra factor of two in the index of an algebra of degree \(2m\). The involution pairs the two \(L\)-components and supplies the embedding on the second component. Where \(L\) is a field and \(S_0\) splits, use an orthogonal decomposition into two \(m\)-dimensional spaces.

It remains to consider a nonexceptional place where both are fields. The local Hermitian form has norm determinant. If \(S_0\) and \(L\) agree locally, the form is hyperbolic, since \(L\) is unramified; let \(S_0\) act through its two embeddings on dual totally isotropic spaces. If the two quadratic fields differ, use \(m\) copies of the Hermitian plane obtained by tracing a line over their composite. In odd characteristic use half the trace; in the basis \(1,s\) its determinant is \(r\), a norm from \(L\). In characteristic two use the trace; its matrix is \[\begin{pmatrix}0&1\\1&1\end{pmatrix},\] with determinant \(1\). Local Hermitian classification identifies these forms with the required one. This classification applies also in characteristic two; see Section 2.

To pass from local embeddings to a global embedding, let \(X\) be their variety, with the multiplicities fixed as above. It is a homogeneous space under \(S\), with smooth connected reductive geometric stabilizer \[\mathrm{S}(\mathop{\mathrm{GL}}_m\times\mathop{\mathrm{GL}}_m).\] Over a separable closure, an embedding is just a decomposition into complementary subspaces of dimension \(m\), which gives this description. The toric quotient of the stabilizer has a rank-one character lattice with its induced Galois action, either trivial or quadratic sign. Its \(\mathop{\mathrm{Sha}}^1\) is zero. In the trivial case this follows from the absence of nonzero continuous homomorphisms from a profinite group to \(\mathbb Z\). In the sign case a class factors through the quadratic quotient, and a nonzero class is detected at an inert Frobenius place by Chebotarev. Global duality therefore gives \(\mathop{\mathrm{Sha}}^2(k,T)=0\) for this toric quotient \(T\) (Demarche and Harari 2020, Theorem 5.2).

The local embeddings give adelic points on \(X\); integral points at almost all places follow by spreading out and Lang’s theorem for the connected stabilizer. For a simply connected semisimple acting group, the obstruction in the proof of (Demarche and Harari 2022, Theorem 2.5) lies in \(\mathop{\mathrm{Sha}}^2(k,T)\), and its vanishing gives a rational point. Applying that theorem proves the lemma. ◻

Proof. The centralizer \(C_D(F)\) is a quaternion algebra over \(LF\), understood componentwise. Its canonical quaternion conjugation commutes with \(*\). Their composite is a semilinear algebra automorphism of order two over \(LF/F\), and descent gives a quaternion algebra \(Q/F\). On \(LS_0F\), canonical conjugation exchanges the two \(s\)-labels. Composing with \(*\) therefore fixes \(J_0F\), proving the stated inclusion.

On the two-dimensional block associated with an eigenvalue of \(h\), both \(P\) and \(P'\) have rank one. The rank assertion for \(P'\) follows from the two terms defining \(h\) and the invertibility of \(h\) and \(1-h\). With \(P\) the first-coordinate projection, write \[P'=\begin{pmatrix}h&x\\y&1-h\end{pmatrix},\qquad z=\begin{pmatrix}h&-x\\y&h\end{pmatrix}.\] The equation \(P'^2=P'\) gives \(xy=h(1-h)\). Thus \(z\) has quaternion trace \(2h\) and norm \(h\), and \(z+z^*=2h\) shows that its canonical conjugate equals \(z^*\). Hence \(z\in Q\). The element \(z-h\) is invertible, exchanges the \(J_0F\)-lines, and satisfies \[(z-h)^2=-h(1-h).\] It realizes \(Q\) as the cyclic quaternion algebra for the separable quadratic extension \(J_0F/F\) and this scalar. The usual norm criterion for a cyclic algebra proves the splitting assertion. This argument also applies in characteristic two: the quadratic extension is separable and \(z-h\) implements its nontrivial automorphism.

The first norm equation says \(aa^*=h\), so \(w\) has quaternion norm one and belongs to \(S\). Also \[q q^*=a(p^*)^{-1}p^{-1}a^*=a h^{-1}a^*=1, \qquad wq=z(p^*)^{-1}=u.\] Therefore \(q\in\mathop{\mathrm{U}}(C_s)\cap S\). Finally, \[\mathop{\mathrm{Nrd}}_{C_s}(q)=N_{J_0F/J_0}(a)d_0^{-1},\] so the second norm equation puts \(q\) in \(\mathop{\mathrm{SU}}(C_s)\). ◻

Proof. Write \(\kappa\) for canonical quaternion conjugation on \(C_s\). It commutes with \(*\), since reduced trace commutes with \(*\). Consequently \(*\circ\kappa\) is a semilinear algebra automorphism of order two relative to \(E/S_0\). Its fixed algebra \(B_0\) is a quaternion algebra over \(S_0\), and Galois descent gives \[B_0\otimes_{S_0}E=C_s.\] The restriction of \(*\) to \(B_0\) is canonical conjugation. This descent is valid in characteristic two as well, because \(E/S_0\) is separable; see (Knus et al. 1998).

Fix \(B_s\) at this point, containing all places below ramification of \(B_0\) and sufficiently large for the following integral properties outside \(B_s\). The fields \(L\) and \(S_0\) are unramified, the fixed algebras and embeddings have unramified integral models, and the involution at a nonsplit place of \(L/k\) is given, up to scalar, by a unimodular Hermitian form. When \(S_0\) splits, its two orthogonal Hermitian blocks are unimodular as well. These properties follow by spreading the fixed algebras, involution and separable embedding; the local classification of unitary involutions gives the stated Hermitian description (Knus et al. 1998).

We first construct an abstract quadratic extension that embeds in \(B_0\). Suppose the characteristic is odd. Choose \(i_0\in L^\times\) with \(\mathop{\mathrm{Tr}}_{L/k}(i_0)=0\). For a scalar \(c\in k^\times\), consider the quadratic algebra over \(S_0\) with generator \(y\) and equation \[y^2=i_0^2c.\] An embedding of this algebra in \(B_0\) has \(y^*=-y\). The element \(f=y/i_0\in C_s\) therefore satisfies \[f^*=f,\qquad f^2=c,\qquad\mathop{\mathrm{Trd}}_{C_s}(f)=0.\] In characteristic two, write \(L=k(\iota)\) with \(\iota^2+\iota=b_L\in k\). Instead use the quadratic algebra over \(S_0\) with equation \[y^2+y=c+b_L,\qquad c\in k.\] Its generator has reduced trace one, so \(y^*=y+1\) in \(B_0\). Then \(f=y+\iota\) satisfies \[f^*=f,\qquad f^2+f=c,\qquad\mathop{\mathrm{Trd}}_{C_s}(f)=1.\] These formulas specify how a quadratic embedding in \(B_0\) produces a Hermitian quadratic embedding in \(C_s\) in either characteristic.

A quadratic separable field over \(S_0\) embeds in \(B_0\) precisely when it remains a field at every ramified place of \(B_0\). Indeed a quadratic local field extension kills the invariant \(1/2\) of a quaternion division algebra, whereas the split algebra does not. The global Brauer invariant theorem then gives the splitting, and hence the embedding, criterion (Reiner 2003; Milne 2020). The resulting local conditions on \(c\) occur above \(B_s\) and can always be met. In odd characteristic, the kernel of \[k_v^\times/k_v^{\times2} \longrightarrow S_{0,w}^\times/S_{0,w}^{\times2}\] has at most two elements when \(S_{0,w}/k_v\) is quadratic, and one when \(S_{0,w}=k_v\). The square-class group of an odd-characteristic local field has four elements. We can therefore choose \(c\) so that \(i_0^2c\) is nonsquare at each required place. In characteristic two the same argument applies to the restriction map on Artin–Schreier classes \[k_v/\{a^2+a:a\in k_v\} \longrightarrow S_{0,w}/\{a^2+a:a\in S_{0,w}\}.\] Its kernel has at most two elements, whereas its source is infinite. Thus \(c+b_L\) can be required to have nonzero image. If \(S_0\) has two places over \(v\), both completions equal \(k_v\) and the exclusions at those places are identical.

At every \(v\in T\), require that \(c\) define a split quadratic algebra over \(k_v\): take \(c\) to be a nonzero square in odd characteristic and an Artin–Schreier value in characteristic two. At one further place \(v_*\), split completely in \(E\) and outside \(B_s\cup T\), require the quadratic algebra defined by \(c\) to be a field. Chebotarev supplies \(v_*\), and weak approximation supplies \(c\) satisfying all these local conditions. The conditions are open in the local fields; for Artin–Schreier classes this follows also from the surjectivity of \(a\mapsto a^2+a\) on the maximal ideal. The field \(F'\) defined by \(c\) is disjoint from \(E\): otherwise it would be one of the three quadratic subfields of \(E\), all of which split at \(v_*\). The associated algebra for \(y\) over \(S_0\) is therefore a field, and the embedding criterion gives an embedding in \(B_0\). The preceding formulas give \(F'\subset C_s\); moreover \(EF'\) is a field of degree two over \(E\), hence a maximal subfield of \(C_s\).

We next choose this embedding so that the two binary groups split at the places in \(T\), using the integral properties secured when \(B_s\) was fixed.

Fix \(v\in T\) and put \(K=k_v\). When \(L\) splits at \(v\), the algebra on each \(L\)-component is \(M_4(K)\). Put \(A_0=S_0\otimes_kK\). The four-dimensional module is free of rank two over \(A_0\), and its \(s\)-centralizer is \(\operatorname{End}_{A_0}(A_0^2)\). For the two roots \(\lambda_1,\lambda_2\in K\) of the chosen polynomial of \(f\), take \(f=\mathop{\mathrm{diag}}(\lambda_1,\lambda_2)\) in this centralizer. Its two \(K\)-eigenspaces are copies of \(A_0\), of dimension two over \(K\), whether \(A_0\) is a field or split. Both special groups are \(\mathop{\mathrm{SL}}_2\), and the involution specifies the paired \(L\)-component. After splitting \(A_0\), each eigenvalue occurs once in each of the two \(s\)-blocks, as required.

Suppose instead that \(L\otimes_kK\) is the unramified quadratic field over \(K\). If \(S_0\) splits, decompose each of its two Hermitian blocks into orthogonal lines of unit norm values. Such decompositions follow by diagonalizing the reduction and lifting an orthogonal basis, in characteristic two as well. Assign to each \(f\)-eigenvalue one line from each \(s\)-block. Each resulting binary Hermitian space has unit determinant. All units of \(K\) are norms from the unramified quadratic extension, so its negative determinant is a norm and the space is hyperbolic. Its special unitary group is therefore split.

If \(S_0\) is nonsplit, its completion equals the unique unramified quadratic extension \(L\otimes_kK\). The two \(s\)-eigenspaces over that extension are dual totally isotropic two-spaces. Choose paired bases \(e_1,e_2\) and \(e'_1,e'_2\) with \(\langle e_i,e'_j\rangle=\delta_{ij}\), and assign one \(f\)-eigenvalue to each hyperbolic plane \(\langle e_i,e'_i\rangle\). This again makes both binary groups split.

All the local elements \(f\) just constructed are Hermitian and have each eigenvalue once in each \(C_s\)-block. They therefore give local embeddings in \(B_0\) by the same descent formulas. For clarity, in characteristic two \(y=f+\iota\) has reduced trace one and satisfies \[y^*=y+1=\kappa(y),\] so it is fixed by \(*\circ\kappa\). Thus it really lies in \(B_0\). In odd characteristic the corresponding identity is \((i_0f)^*=-i_0f=\kappa(i_0f)\).

Finally, local embeddings of the fixed quadratic algebra for \(y\) are conjugate by \(B_0^\times\), by Skolem–Noether. Weak approximation in the open affine variety \(B_0^\times\) over \(S_0\) lets a global conjugator approximate all the prescribed local conjugators. The splitting property persists under sufficiently close approximation: the eigenspaces vary continuously and the determinant norm classes of their Hermitian forms are locally constant. The resulting global embedding has all the asserted properties. ◻

Proof. First fix a field \(F_1\) given by Lemma 42, with no extra local prescriptions, and fix its embedding in \(C_s\). Write \(G_i=\mathop{\mathrm{SU}}(C_D(F_i),*)\) when a field \(F_i\) has been specified. The field \(F_1\) will remain fixed when the neighborhoods \(W_v\) are chosen; a second field \(F_2\) will be chosen after \(q\).

For an element \(q\) in the proposition put \[r_0=\mathop{\mathrm{Nrd}}_{C_s}(q)\in E^\times.\] Let \(\alpha\) and \(\beta\) be the automorphisms of \(E/k\) fixing \(S_0\) and \(L\), respectively. Unitarity gives \(\alpha(r_0)=r_0^{-1}\), and ambient determinant one gives \[N_{E/L}(r_0)=1,\qquad\beta(r_0)=r_0^{-1}.\] Hence \(\alpha\beta\) fixes \(r_0\), so \(r_0\in J_0^1\). Write a bar for the nontrivial automorphism of \(J_0/k\). Choose once and for all \(c_*\in J_0\) with \(c_*+\bar c_*=1\), and set \[ d=c_*+r_0\bar c_*. \tag{84}\] This formula is valid in both characteristics. It satisfies \(r_0\bar d=d\) and gives \(d=1\) when \(r_0=1\). We will require \(q\) close enough to \(1\) at \(B\) that \(d\) is invertible, in which case \(r_0=d/\bar d\).

We describe the neighborhoods \(W_v\) before making any further choices. The local norm map for the fixed separable quadratic algebra \(J_0F_1/J_0\) is a submersion at \(1\): its differential is the surjective trace map, also in characteristic two. For \(d\) sufficiently close to \(1\), it therefore has solutions \[ N_{J_0F_1/J_0}(x_{1,v})=d,\qquad x_{1,v}\longrightarrow1 \quad\text{as }d\longrightarrow1, \tag{85}\] in the product of the local algebras above \(v\). Because the field and embedding are fixed, the elements \(b_{1,v}=x_{1,v}/x_{1,v}^*\) tend to \(1\) in the ambient algebra \(D\otimes_k k_v\).

By the binary case of MP and Proposition 6, the quotient maps on \(G_1(F_1)\) and on \(\mathop{\mathrm{SU}}(C_s,*)(S_0)\) extend continuously to their respective products of rank-zero local groups. Choose \(W_v\) so small that the solutions in Equation (85) make \(b_{1,v}\) belong to the needed identity neighborhoods at every rank-zero place of \(G_1\) over \(v\). At the same time require \(q\) and \(b_{1,v}\) close enough to \(1\) that \[(b_{1,v}b_{2,v})^{-1}q\] belongs to the needed identity neighborhoods for \(\mathop{\mathrm{SU}}(C_s,*)\) whenever \(b_{2,v}\) is sufficiently close to \(1\) and the displayed element has determinant one in \(C_s\). These requirements concern finitely many fixed groups and continuous multiplication maps, so they hold after shrinking \(W_v\). They also ensure that \(d\) is invertible. For a later field \(F_2\), its local target at every \(v\in B\) will be \(x_{2,v}=1\). Once \(F_2\) is fixed, approximation to that target can be made arbitrarily close both in its binary group and in \(D\); no choice of \(F_2\) enters the present neighborhoods.

Now let \(q\) satisfy these conditions, and hence fix \(r_0\) and \(d\). Say that a place \(v\notin B\) is covered by \(F_i\) if

1. \(d\) is a norm from \(J_0F_i/J_0\) at every place of \(J_0\) above \(v\); and

2. \(G_i\) is isotropic at every place of \(F_i\) above \(v\).

All but finitely many places are covered by the fixed field \(F_1\). Indeed, away from the ramification and the divisors of \(d\), norms from unramified quadratic algebras are surjective on units. Also \(G_1\) is an unramified binary group at all but finitely many places, and is split there: a quaternion algebra over a local field with unramified integral model has trivial Brauer class, since the residue field is finite.

Let \(T\) be the finite set of places outside \(B\) not covered by \(F_1\). Apply Lemma 42 to choose \(F_2\) split at every place of \(T\), with both binary groups split there. The algebra \(J_0F_2/J_0\) is then locally split at every place over \(T\), and its norm is surjective. Thus every place outside \(B\) is covered by at least one of \(F_1,F_2\).

We seek invertible elements \(x_i\in J_0F_i\) satisfying \[ N_{J_0F_1/J_0}(x_1)N_{J_0F_2/J_0}(x_2)=d. \tag{86}\] At the places above \(B\), take the local solutions \(x_1=x_{1,v}\) already specified and \(x_2=1\). At any other \(k\)-place choose a field covering that entire place, let its variable carry the norm \(d\) at every \(J_0\)-place above it, and set the other variable equal to \(1\). This gives local solutions everywhere. It also arranges the following useful condition: if \(G_i\) has a rank-zero place above \(v\notin B\), then \(F_i\) cannot cover \(v\), so the other field covers it and \(x_i=1\) on all components above \(v\).

We justify both the Hasse principle and the required weak approximation for Equation (86), including in characteristic two. Put \(K=J_0\), \(E_i=J_0F_i\) and \(y_2=x_2^{-1}\). On the locus where the variables are invertible the equation becomes \[ N_{E_1/K}(x_1)-dN_{E_2/K}(y_2)=0. \tag{87}\] Both norm forms are nonsingular binary quadratic forms because \(E_i/K\) is separable; their direct sum in this display is a nonsingular four-variable quadratic form. This remains true in characteristic two, where its polar form is the direct sum of two nondegenerate alternating forms. Its projective zero locus \(X_d\subset\mathbb P^3_K\) is consequently a smooth quadric surface, becoming \(\mathbb P^1\times\mathbb P^1\) over a separable closure.

Here is a local-global argument for this surface that works uniformly in the characteristic. Its two rulings determine a quadratic étale algebra \(K'/K\). The automorphisms of \(\mathbb P^1\times\mathbb P^1\) act separately on the two factors or exchange them. Descent therefore expresses \(X_d\) as the restriction of scalars, from \(K'\) to \(K\), of a Severi–Brauer conic over \(K'\); when \(K'=K\times K\) this means a product of two such conics. A local point of \(X_d\) supplies points on the conic at every completion of \(K'\). The Brauer local-global theorem over the global field \(K'\), componentwise in the split case, makes the conic split. Hence \(X_d(K)\) is nonempty; see (Milne 2020) for the Brauer theorem.

A smooth quadric surface with a rational point is rational, by projection from that point. Its affine cone minus the vertex is rational as well: over a rational affine open of the surface the tautological line bundle is trivial, so the punctured cone is birational to \(\mathbb A^2\times\mathbb G_m\). Weak approximation follows on its smooth locus by using such a rational chart and density of a nonempty Zariski open in each local smooth variety. Remove also the two norm-zero loci. The local solutions above belong to the resulting open set; its rational points still approximate any finite collection of those local solutions. Inverting \(y_2\) returns the same Hasse principle and weak approximation for Equation (86).

There are only finitely many rank-zero places of \(G_1\) and \(G_2\). Apply this weak approximation at their underlying \(k\)-places and at \(B\). At a rank-zero place outside \(B\), the variable belonging to that group has local target \(1\), as arranged above. Approximate it closely enough that \(x_i/x_i^*\) lies in an identity neighborhood killed by the continuous extension of the finite quotient on \(G_i\). At \(B\), approximate the specified \(x_{1,v}\) and \(1\) closely enough to retain all the conditions used in choosing \(W_v\), and also the analogous kernel conditions for \(G_2\).

For the resulting global solution set \[b_i=x_i/x_i^*\quad(i=1,2).\] Equation (83) places \(b_i\) in \((\mathop{\mathrm{Res}}_{F_i/k}G_i\cap\mathop{\mathrm{U}}(C_s,*))(k)\), and the local conditions just imposed show that both have trivial image in \(S(k)/R\). Their determinants satisfy \[\mathop{\mathrm{Nrd}}_{C_s}(b_1)\mathop{\mathrm{Nrd}}_{C_s}(b_2) =\frac{N_{E_1/K}(x_1)N_{E_2/K}(x_2)} {\overline{N_{E_1/K}(x_1)N_{E_2/K}(x_2)}} =\frac d{\bar d}=r_0.\] Thus \(q'=(b_1b_2)^{-1}q\) belongs to \(\mathop{\mathrm{SU}}(C_s,*)(S_0)\). At all its rank-zero places it lies in the prescribed kernel neighborhoods, since those places lie above \(B\). The binary case of MP therefore makes \(q'\) trivial in the quotient as well. The identity \(q=b_1b_2q'\) proves the proposition. ◻

Proof. Choose \(X\in\mathop{\mathrm{Lie}}(S)(K)\) such that the geometric matrix \(X_{+-}X_{-+}\) is invertible and has separable characteristic polynomial. This is a nonempty Zariski open condition, as is seen in split coordinates, and the \(K\)-points of this vector space are Zariski dense. Smoothness supplies an analytic curve \(u(\varepsilon)\) through \(1\) with derivative \(X\). Formula (78) gives \[(h-1)/\varepsilon^2 \longrightarrow X_{+-}X_{-+}\] on one block, with the same characteristic polynomial on the other. For small nonzero \(\varepsilon\), therefore, \(u(\varepsilon)\in\Omega\).

Identify the étale algebras generated by \((h-1)/\varepsilon^2\) with the étale algebra generated by its limit. These identifications vary continuously: the characteristic polynomials have simple roots, and simple-root lifting, followed by the basis of powers, identifies the embeddings in the ambient algebra. The same holds after adjoining \(J_0\). Under these identifications, \(h\to1\) and \(d_0\to1\).

Consider the simultaneous norm map from \((J_0F)^\times\) to the torus of pairs \((h,d_0)\) satisfying (80). This map is smooth and surjective as a morphism of tori. Indeed over a splitting field, write the coordinates of \(a\) as \((a_i,b_i)_{1\leq i\leq m}\); then the map records \[h_i=a_i b_i,\qquad d_+=\prod_i a_i,\qquad d_-=\prod_i b_i,\qquad \prod_i h_i=d_+d_-.\] Its kernel is the torus \(b_i=a_i^{-1}\), \(\prod_i a_i=1\), of dimension \(m-1\). There is no inseparability in this assertion, even when the characteristic divides \(m\). The local submersion theorem supplies solutions \(a\to1\). Formula (82) then gives \(w,q\to1\). If only the first norm is specified, the same argument uses the ordinary quadratic norm map. Around any sufficiently small fixed nonzero \(\varepsilon\), these properties persist on an open set, by the same continuous étale identifications and norm submersion. ◻

Proof. The set is nonempty and excludes the identity. If \(W\in\mathcal S\) and \(Z\) commutes with \(W\), then \(Z\) lies in the same conjugate of \(C\) as \(W\). Consequently \(Z\notin\mathcal S\) forces \(Z=1\). More generally, if an abelian subgroup \(A\) meets \(\mathcal S\), conjugating one member of \(A\cap\mathcal S\) into \(C\) puts all of \(A\) in \(C\); hence \(A\setminus\mathcal S=\{1\}\). This proves the complement assertion.

Conjugate \(W\) into \(C\). Testing Equation (92) with every conjugate of a fixed nonidentity element of \(C\) shows that every conjugate \(v\) of \(W\) belongs to \(Cv^{-1}C\). If \(v=a v^{-1}b\), \(a,b\in C\), then \[(v b^{-1})^2=a b^{-1}\in C.\] If this square is nonidentity, \(v b^{-1}\) centralizes a nonidentity element of \(C\), so \(v\in C\). Otherwise \(v\in I(\mathcal F)C\). Since \(I(\mathcal F)\) is conjugacy-invariant, \(I(\mathcal F)C=CI(\mathcal F)\). Thus the whole conjugacy class of \(W\) lies in \(CI(\mathcal F)\), and \[\frac{|\mathcal F|}{c}\le c|I(\mathcal F)|.\] The Frobenius–Schur formula and Cauchy–Schwarz give \[|I(\mathcal F)| =\sum_{\chi\in\operatorname{Irr}(\mathcal F)}\nu_2(\chi)\chi(1) \le\sum_\chi\chi(1) \le\sqrt{k(\mathcal F)|\mathcal F|}.\] These inequalities contradict Equation (93). ◻

Proof. Put \[\overline H=H/Z(H),\qquad \overline T=T/Z(H),\qquad c=\frac{q^p+1}{q+1},\qquad d=(p,q+1).\] The actual centralizer of \(w\) is \(T\). If \(g\) centralizes its projective image, then \(gwg^{-1}=\lambda w\) for a scalar \(\lambda\) of norm one. Taking determinants gives \(\lambda^p=1\), so \[ |C_{\overline H}(\overline w)|\le dc. \tag{94}\]

Suppose the displayed condition in the lemma holds. Every conjugate \(v\) of \(w\) then lies in \(Tv^{-1}T\). The calculation used in Lemma 45 gives an element \(u=v b^{-1}\) whose square lies in \(T\). If \(u^2\) is nonscalar, it has field degree \(p\), because \(p\) is prime; its centralizer is \(T\), and hence \(u\in T\). If \(u^2\) is scalar, \(\overline u\) has square one. Consequently the entire projective conjugacy class of \(\overline w\) lies in \(\overline T I(\overline H)\).

The determinant quotient of \(\overline H\) has order \(d\), and \(\overline T\) maps onto it. To see this, write \(T\) as the norm-one subgroup of \(\mathbb F_{q^{2p}}^\times\) over \(\mathbb F_{q^p}\). The determinant of a field multiplication is its norm to \(\mathbb F_{q^2}\). On this torus, that norm takes a generator to a generator of the scalar norm-one group of order \(q+1\); its exponent is congruent to \(c\) modulo \(q^p+1\). Modding out by scalar matrices mods the determinant out by \(p\)th powers, giving the quotient of order \(d\).

All members of the conjugacy class have the same determinant in this quotient. For each fixed \(i\in I(\overline H)\), at most \(c/d\) elements \(t\in\overline T\) have \(ti\) with that determinant. Combining this with Equation (94) gives \[\frac{|\overline H|}{dc} \le \frac c d |I(\overline H)|, \qquad\text{and therefore}\qquad |\overline H|\le c^2|I(\overline H)|.\] We show that the reverse strict inequality always holds.

Every projective involution lifts to an involution in \(H\). Indeed, if \(g^2=\alpha I\), then \(\alpha^p=\det(g)^2\) in the scalar cyclic group of order \(q+1\). As \(p\) is odd, \(\alpha\) is a square in that group, and scalar rescaling gives an involution.

For odd \(q\), an involution is determined up to conjugacy by the dimensions \(a,p-a\) of its two nondegenerate eigenspaces. Projectively the unordered partitions are indexed by \(1\le a<p/2\), and their centralizers have order at least \[ \frac{|\mathop{\mathrm{U}}_a(q)|\,|\mathop{\mathrm{U}}_{p-a}(q)|}{q+1}. \tag{95}\] For even \(q\), the Jordan type is \(2^a1^b\), where \(b=p-2a\). Its full unitary centralizer has order \[ q^{a^2+2ab}|\mathop{\mathrm{U}}_a(q)|\,|\mathop{\mathrm{U}}_b(q)|. \tag{96}\] We recall the structure behind this formula, including in characteristic two; see also (Wall 1963, sec. 2.6, Case (A)(ii),(iv), pp. 34–36). In the ambient general linear group the connected matrix centralizer has reductive quotient \(\mathop{\mathrm{GL}}_a\times\mathop{\mathrm{GL}}_b\) and unipotent radical of dimension \(a^2+2ab\). The Hermitian Frobenius gives the unitary forms on both multiplicity spaces. Equivalently, for \(g=1+N\) of order two, \(N=N^*\); the induced forms on \(\ker N/\mathop{\mathrm{im}}N\) and, using \(N\), on the quotient by \(\ker N\) are the forms on these two spaces. Nondegenerate Hermitian forms of a given dimension over a finite field are isometric. Connectedness and Lang’s theorem give one full-unitary class for each occurring type and the stated order. Dividing by \(q+1\) gives a lower bound for its projective centralizer.

The unitary order formula implies \[ |\mathop{\mathrm{U}}_j(q)|=q^{j^2}\prod_{r=1}^j(1-(-q)^{-r})\ge q^{j^2}. \tag{97}\] For completeness, pair consecutive factors: their product is \[(1+q^{-(2r-1)})(1-q^{-2r}) =1+q^{-2r}(q-1-q^{-(2r-1)})>1.\] An unpaired final factor also exceeds one. Thus every nonidentity projective involution class has centralizer order at least \[M=\frac{q^{(p^2+1)/2}}{q+1}.\] There are at most \((p-1)/2\) such classes, and including the identity gives \[\frac{|I(\overline H)|}{|\overline H|}\le\frac{p+1}{2M}.\] When \(p\ge5\), we have \(c<q^{p-1}\) and \[\frac{M}{q^{2p-2}} =\frac{q^{(p^2-4p+5)/2}}{q+1} \ge\frac{2^p}{3}>\frac{p+1}{2}.\] The first inequality uses \((p^2-4p+5)/2\ge p\) and monotonicity in \(q\ge2\). This proves \(c^2|I(\overline H)|<|\overline H|\).

It remains to check \(p=3\). Now \[c=q^2-q+1,\qquad h=|\overline H|=q^3(q^3+1)(q^2-1).\] There is one nonidentity involution class. In odd characteristic its centralizer lower bound is \(M=q(q-1)(q+1)^2\), and \(h/M=q^2c\). In even characteristic it is \(M=q^3(q+1)\), and \(h/M=(q^2-1)c\). In both cases \(|I(\overline H)|\le1+q^2c\), whereas \[h-c^2(1+q^2c) =c\bigl((3q-4)q^4+(q-2)q^2+q-1\bigr)>0 \quad(q\ge2).\] This includes \(\mathop{\mathrm{U}}_3(2)\) and finishes the proof. ◻

Completion of the proof of Theorem 20. The classification consists of the alternating groups, the classical and exceptional groups of Lie type, and the sporadic groups. All families have been covered above. The low-rank orthogonal groups have their usual linear, unitary, or symplectic realizations; in particular plus type \(l=3\) is \(\mathop{\mathrm{PSL}}_4\), and rank two symplectic type was treated separately. The exceptional small derived simple groups are \(G_2(2)'\simeq\mathop{\mathrm{PSU}}_3(3)\), \({}^2G_2(3)'\simeq\mathop{\mathrm{PSL}}_2(8)\), and the Tits group, already included in Table 1. Every chosen \(\mathcal S\) is nonempty, proper, and conjugacy-invariant, and for it Equation (90) has been excluded for every commuting pair with \(W\in\mathcal S\), \(Z\notin\mathcal S\). The complement assertion was established with each construction: it is \(A[2]\) for the square sets, \(\{1\}\) for the self-centralizing torus sets, and the kernel of a restriction or projection homomorphism in the remaining cases. Finally, if \(s\in\mathcal S\) but \(s^{-1}\notin\mathcal S\), the complement subgroup in \(\langle s\rangle\) would contain \(s^{-1}\) and hence \(s\), a contradiction. This proves invariance under inversion and completes Theorem 20. ◻