2 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV For any m which is not squarefree and any l, there exists a division algebra D of index m over K = Q

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1 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV 1. Introduction An irreducible algebraic variety X dened over a eld K is called K- rational (resp. stably K-rational) if X (resp. X IA m K for some m 2 IN) is birationally isomorphic over K to an ane space. The present paper is devoted to the rationality problem for group varieties. This problem has a long history and is especially interesting for semisimple group varieties. Let G be a semisimple algebraic group dened over a eld K. It is well known that the algebraic variety G is unirational over K, i.e. there exists a dominant rational map : IA m K! G over K or equivalently, the eld K(G) of K-rational functions on G is contained in a purely transcendental extension of K (see [DG], Exp. XIV ). Rationality problem. 1) Is the variety G rational over K? In other words, is K(G) a purely transcendental extension of K? 2) If no, is the variety G then stably rational over K? If G is K-split (i.e. there exists a maximal torus T splitting over K), then G is K-rational. This follows immediately from the Bruhat decomposition: G is birationally isomorphic over K to the product variety U T U where U is the unipotent radical of a K-dened Borel subgroup B of G containing T. In particular, over an algebraically closed eld K, the variety G is always K-rational. Other examples of rational group varieties are provided by groups of small ranks. It is known that for any eld K all algebraic K-tori of dimension at most two are rational over K (see [V] ). This fact combined with the Chevalley-Grothendieck theorem and the K-rationality of the variety of maximal tori of G (see [Ch], [DG], Exp. XIV ) implies that G is K-rational if rank G 2. In particular, varieties of all K-groups of types A 1 ; A 2 ; B 2 = C 2 ; G 2 are rational over K. However in general (i.e. for arbitrary G and K), this problem seems to be dicult. For a long time, the following question was open: Is the variety G rational over K if G is simply connected? It turned out that this is not the case. First examples of non-rational varieties of simply connected groups were constructed by the second-named author ([P1]): Supported in part by an NSERC grant. 1

2 2 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV For any m which is not squarefree and any l, there exists a division algebra D of index m over K = Q(x j 1 ; : : : ; x n ), or Q j p (x 1 ; : : : ; x n ); n 2, such that the variety G = SL l (D) is not stably rational over K. Here j Q and j Q p are, respectively, the elds of rational and p-adic numbers and x 1 ; : : : ; x n are independent variables. In fact, even more was proven: G(K) = SL l (D) does not have the weak approximation property. The following results were inspired by a question posed by P. Deligne. Let f be a non-degenerate quadratic form of rank n over a eld K of characteristic 6= 2. It is known that the variety SO(f) is K-rational (see Lemma 5 in x4). In 1978, P. Deligne asked the following question: IR)? Is the spinor variety Spin(f) rational over K (in particular, over K = We answered (see [P2], [PC1], [C1] ) this question positively in the following cases. 1) If a quadratic form f is K-isotropic, then Spin(f) is K-rational. In particular, if K is either an algebraic extension of j Q p or a totally imaginary number eld or a global function eld, then Spin(f) is K-rational. 2) Spin(f) is rational over K = IR or j Q. 3) Spin(f) is rational over any K if f is either a sum of squares or a Pster form. For an algebraic number eld K, which is not totally imaginary, the question of the rationality of Spin(f) is still open. Since all groups of rank 2 are K-rational, the minimal n for which, a priori, there is a possibility of Spin(f) being non-rational over K, is n = 6. The next result was relatively surprising ( [P3] ): Let n = 4m + 2; n 6, and let x 1 ; : : : ; x n?2 be algebraically independent variables over some eld L. If f(y 1 ; : : : ; y n ) = y1 2 + x 1y x n?2yn?1 2 + x 1 x 2 x n?2 yn 2, then the spinor variety Spin(f) is not stably rational over K = L(x 1 ; : : : ; x n?2 ). After the discussion above, it becomes clear that it is possible to deal with the question of the rationality of an arbitrary almost simple algebraic K- group only for "good" elds. In the present paper we consider, for instance, the rationality problem for arbitrary almost simple algebraic K-groups over arithmetic elds. Main Theorem. Let G be an almost simple algebraic group dened over K, where K is either an algebraic extension of Qp j or a totally imaginary number eld or a global function eld. If G is not of type A n ; n 3, then G is a rational variety over K. This statement cannot be extended to groups of type A n ; n 3. In fact, now due to A. Merkurjev and M. Rost (see [M1] ) we know that the variety G = SL 1 (D), where D is a division algebra of degree 4m over any eld K, is not rational over K. In particular, this implies that there exists a j Q p -form G of type D 3 such that G is not rational even over j Q p (recall that D 3 = A 3 ), in contrast to the case D n ; n 4. If ind(d) is prime we have the following conjecture which is still open:

3 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 3 Conjecture (Platonov [P1] ). Let D be a division algebra of prime index over an arbitrary eld K. Then the variety SL 1 (D) is rational over K. If ind(d) is not squarefree, then there exists the following conjecture which is also still open: Conjecture (Suslin [Su] ). If ind (D) of a simple algebra D is not squarefree, then the generic element D of SK 1 (D K(X) ), where X = SL 1 (D), D K(X) = D K K(X), is not trivial; in particular, the variety SL 1 (D) is not K-rational. These conjectures and results both of the present paper (see xx 3-10) and Merkurjev's paper [M2] on the rationality of adjoint classical groups suggest that the answer to the question whether the variety of a given almost simple algebraic K-group G is rational over K very likely depends only on some discrete parameters such as for example the K-rank, the Tits K-index of G, indices of Tits algebras and so on. Our main theorem yields immediately Corollary. Let K be as in the main theorem and G be a semisimple algebraic group over K which is either simply connected or adjoint without almost simple components of types A n ; n 3. Then G is K-rational. An intermediate semisimple group G (i.e. one which is neither simply connected nor adjoint) is not necessarily K-rational. There exists even a quasi-split semisimple K-group G which is not K-rational. To produce such an example consider 1 ; 2 ; 3 2 K which are independent modulo (K ) 2 and any positive number n. Let f l = nx i=1 x 2 i? nx j=1 yj 2 + u 2 l? lvl 2 ; l = 1; 2; 3; be (2n + 2)-dimensional quadratic forms over K. Then we have Proposition. (SO(f 1 ) SO(f 2 ) SO(f 3 ))= < (?1;?1;?1) > is not a K-rational group. Another example of a non-rational semisimple group was constructed by Ph. Gille. In fact, the rst example of a non-rational semisimple algebraic group which is neither simply connected nor adjoint is due to J-P. Serre. In [S] (P ) he constructed a group G which does not have the weak approximation property; in particular, its variety is not rational (recall that every smooth rational variety has the weak approximation property). The strategies used to prove the non-rationality of a given variety are very dierent from those used to prove its rationality. A proof of the non-rationality is usually connected with some new invariants (for instance, SK 1 (D)). On the other hand, a proof of the rationality of an algebraic variety involves a direct parametrization of its K-points depending on that particular variety. So it is natural that the proof of our main theorem depends on the distinct types of G. In fact, we prove that the variety of any almost simple group with a given Tits' index (see [T]), which is admissible over the local and global elds under consideration, is rational over an arbitrary eld K. In particular, we prove that if the anisotropic semisimple kernel of

4 4 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV an almost simple group G dened over any eld K, char(k) 6= 2, has the property that any almost simple K-component of has rank at most 2, then the variety of G is K-rational. Besides, we also consider some additional Tits indices which do not occur over local and global elds. It seems unlikely to obtain an essential generalization of these results. A sketch of the proof of our main theorem for p-adic (i.e. nite extensions of Q j p ) elds was published in [PC2]. Acknowledgements. This joint work began in 1995 while the rstnamed author visited the University of Bielefeld as a guest of SFB 343 \Diskrete Strukturen in der Mathematik" and the second-named author visited the Max-Planck-Institute of Mathematics. The nal version of this paper was written during our stay at the University of Bielefeld in 1996 supported by the Alexander von Humboldt Foundation. The authors gratefully acknowledge the support of all these institutions and the hospitality of the Max-Planck-Institute of Mathematics and the University of Bielefeld. 2. Some properties of algebraic groups over arbitrary elds Let G be an almost simple algebraic group dened over an arbitrary eld K. If S is a maximal K-split torus in G, then the generalized Bruhat decomposition (see [BT] ) implies that G U Z G (S) U over K, where we use to denote birationally isomorphic varieties over a basic eld and U is the unipotent radical of a minimal parabolic K-subgroup P of G. Since U is K-rational, it suces to show the K-rationality of the variety Z G (S). But, in turn, the bundle Z G (S)! Z G (S)=S is locally trivial, since S is the K-split torus, and so Z G (S) (Z G (S)=S) S. Thus, if we prove that for a given group G the quotient Z G (S)=S is K-rational, then G will also be so. It is known that Z G (S) is a reductive group. So Z G (S) = S 0, where S 0 is the central torus of Z G (S) and = [Z G (S); Z G (S)] is the commutator subgroup called the semisimple K-anisotropic kernel of G. There are two possibilities for S 0 : (i) S 0 = S (this is true for any inner type group); (ii) S 0 6= S, and so G is an outer type group. Usually we will reduce our consideration to the case when for any extension F of K one has H 1 (F; S 0 ) = 1, and hence by virtue of Lemma 1 below, we will get Z G (S) (Z G (S)=S 0 ) S 0 (= \ S 0 ) S 0 ; therefore, the variety G is K-rational if = \ S 0 and S 0 are. For our considerations it is essential to know the intersection \ S 0 in an explicit form. Let T be a maximal torus of G dened over K and containing S. Let = R(T; G) be the root system of G relative to T, and = f 1 ; : : : ; n g be a system of simple roots. If fx ; 2 ; H 1 ; : : : ; H n g is a Chevalley basis of the Lie algebra, then G is generated by the corresponding root subgroups G = < x (t) j t 2 K >, and the torus T is generated by T = T \ G = < h (t) >, where h (t) = w (t)w (1)?1 and w (t) =

5 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 5 x (t)x? (?t?1 )x (t) (see [St] ). Furthermore, if G is simply connected then T = < h 1 (t 1 ) > < h n (t n ) > and for all roots ; 2 one has (1) h (t)x (u)h (t)?1 = x (t <;> u); where < ; > = 2(; )=(; ) (see [St] ). In this notation our torus S is the identity component of the subgroup dened by the following system of equations: (2) 8 < : (t) = ()(t) for all 2 and all 2? = Gal(K=K) i (t) = 1 for all non-distinguished vertices i of the Tits K-index where t 2 T and denotes Q the natural action of? on (see [T] ). If G n is simply connected and t = j=1 h j (t j ) then it follows from (1) that the second equation of the system (2) is equivalent to (3) ny j=1 t 2( i; j )=( j ; j ) j = 1 Recall also that for groups G of inner type, we have () = for all 2 and all 2?. If X then the subgroup generated by G for all 2 X will be denoted by G X. We also need to consider more general bundles, for which we use the following Lemma 1. Let G 1 be a connected algebraic group dened over an arbitrary eld K. If G 2 is a connected subgroup of G 1 dened over K such that for any extension E=K the kernel of the natural morphism H 1 (E; G 2 )! H 1 (E; G 1 ) is trivial, then the variety G 1 is birationally isomorphic to (G 1 =G 2 )G 2 over K. We will assume from now on that chark 6= 2 (unless otherwise stated). 3. Groups of type A n 3.1. Isotropic simply connected groups of inner type A n Let G be an isotropic simply connected K-group of type 1 A n. Denote its K-rank by r. Then G is isomorphic over K (see [T] ) to the special linear group SL r+1 (D), where D is a central division algebra of degree d over K and (r + 1)d = n + 1. Proposition 1. The variety G is birationally isomorphic over K to the product SL 1 (D) IA t K, where t = (n + 1)2? d 2. Remark 1. From what has been said in x2 and the direct matrix realization of G it is easy to give a simple proof consisting of a few lines only. However we prefer to present a more complicated proof since it can be applied to groups of other types, too. Proof. The Tits K-index of G is as follows: r 1 : : : ir d : : : ir rd : : : r n

6 6 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV So the system (2) can be written in the form 8 >< >: 1 (t) = 1; : : : ; d?1 (t) = 1 d+1 (t) = 1; : : : ; 2d?1 (t) = 1 n?d+1 (t) = 1; : : : ; n (t) = 1 Let t = Q n i=1 h i (t i ). Then taking into account (3) one can rewrite the latter in the form (4) 8 < : t 2 1 = 1; t2 2 =t 1t 3 = 1; : : : ; t 2 d?1 =t d?2t d = 1 t 2 n?d+1 =t n?dt n?d+2 = 1; : : : : : : ; t 2 n =t n?1 = 1 The semisimple K-anisotropic kernel of G coincides with the product of the simply connected groups H 1 = G X1 ; : : : ; H r+1 = G Xr+1, where X 1 = f 1 ; : : : ; d?1 g; : : : ; X r+1 = f n?d+1 ; : : : ; n g; each of them being isomorphic over K to SL 1 (D). Considering the connected component of the subgroup dened by (4) and adding equations t d = 1, t 2d = 1, : : :, t rd = 1 one can easily obtain that the intersection \ S 0 ( = \ S) is generated by all the products of the form a 1 a 2 a r+1, where (5) a 1 = h 1 ( 1 )h 2 (1 2) h d?1 (1 d?1 ) a r = h n?2d+1 ( r )h n?2d+2 (r 2 ) h n?d?1 (r d?1 ) a r+1 = h n?d+1 ( 1?1 r?1 )h n?d+2 ( 1?2 r?2 ) h n (?(d?1) 1 r?(d?1) ) and d i = 1 for all i = 1; : : : ; r. Let Z 1 ; : : : ; Z r+1 be the centers of the simply connected groups H 1 ; : : : ; H r+1 dened above. It is clear that Z i can be identied with d, where d is the Gal(K=K)-module d-th roots of unity. Consider the following exact sequence 1?! ( d ) r '?! H 1 H r+1?! = \ S?! 1; where '( 1 ; : : : ; r ) = (a 1 ; : : : ; a r+1 ) and the elements a 1 ; : : : ; a r+1 are given by (5). Our goal is to prove that the variety = = \ S is birationally isomorphic over K to the product (1 1 H r+1 )? = (1 1 H r+1 ) : By Lemma 1, we need only to check that for any extension E=K one has Ker : H 1 (E; (1 1 H r+1 ))?! H 1 (E; ) = 1: Let 2 Ker. Since the restriction j 11Hr+1 is injective there exists a unique 0 2 H 1 (E; 1 1 H r+1 ) such that = ( 0 ), where is the natural bijection : H 1 (E; 1 1 H r+1 )?! H 1 (E; (1 1 H r+1 )) : By construction, ( 0 ) is trivial in H 1 (E; ), hence

7 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 7 (6) 0 2 Im? H 1 (E; ( d ) r )?! H 1 (E; H 1 H r+1 ) : Identify H 1 (E; d ), H 1 (E; H i ) with E = (E ) d, E =Nrd ((D K E) ), respectively. Then 0 = (1; : : : ; 1; a Nrd(D E )) for some a 2 E and condition (6) implies that there exist b 1 ; : : : ; b r 2 E such that b 1 Nrd(D E ) = Nrd(D E ) b r Nrd(D E ) = Nrd(D E )? b?1 1 b?1 r Nrd(DE ) = a Nrd(D E ) It follows immediately that a 2 Nrd(D E ), i.e. 0 = 1 and hence = 1. Thus, we proved that (1 1 H r+1 )? = (1 1 H r+1 ) : Now to complete the proof it remains to observe that the variety = (1 1 H r+1 ) (H 1 H r+1 )=(1 1 H r+1 )' (( d ) r ) (H 1 =Z 1 ) (H r =Z r ) PSL 1 (D) PSL 1 (D) is clearly rational over K, yielding the required result. Corollary 1. For any central subgroup G the variety of the group G= is rational over K if d 3. Proof. Keep the notation above. It suces to prove the K-rationality of the variety X = = \< S; >. If \< S; > = \ S then X = and hence X is rational over K since SL 1 (D) has rank at most 2. Otherwise the same arguments as above show that X is birationally isomorphic over K to the variety (H 1 =Z 1 ) (H r+1 =Z r+1 ) PSL 1 (D) PSL 1 (D) and hence is K-rational. Remark 2. If d = 4, then G is not K-rational. This was proved in [P4] for special elds, and in [M1] for arbitrary elds Simply connected groups of outer type A n Let L=K be a separable quadratic eld extension and D be a central division algebra of nite dimension over L with an involution of the second kind on D, trivial on K. Consider a right vector space V over D of nite dimension and a non-degenerate hermitian form h on V. Then the unitary group U(V; h) of isometries of V with respect to h coincides with U(B; ), where B = End D (V ) and is the adjoint involution of h (see [Sch] ), and the special unitary group SU(V; h) equals SU(B; ) and has type 2 A n. It follows from [CM] that SU(B; ) is stably birationally isomorphic over K to SU(D; ) for some involution of D of the second type, and this fact will be crucial when we consider groups of type E 6. So for the sake of expository completeness we include here the proof of this statement given in [CM] and inspired by the proof of Proposition 1 in [PC1].

8 8 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV Let w 2 V be an anisotropic vector and let W = w? be the orthogonal complement to w. A non-trivial isometry 2 U(V; h), which is the identity on W, is called a reection on V. The image of? id equals wd. Hence the space W is determined by as the orthogonal complement to (v)? v for any v 2 V, such that (v) 6= v (see [Sch], p.268 ). Lemma 2. Let v and v 0 be anisotropic vectors in V, such that h(v; v) = h(v 0 ; v 0 ) and v 0? v is anisotropic. Then there is a unique reection 2 U(V; h), such that (v) = v 0. Proof. Denote v 0? v by w and w? by W. Since h(v; v) = h(v 0 ; v 0 ) = h(v; v) + h(v; w) + h(w; v) + h(w; w); it follows that v =2 W. Hence the linear map, dened by the conditions (v) = v 0 and j W = id j W, is clearly a reection. Conversely, let be a reection, such that (v) = v 0. Since w = (v)?v is non-trivial, must be identical on W and hence is uniquely determined. Let v 2 V be an anisotropic vector. Denote by V 0 the orthogonal complement of v in V and by h 0 the restriction of h to V 0. Then h 0 is a non-degenerate hermitian form on V 0 and U(V 0 ; h 0 ) can be identied with the subgroup in U(V; h) consisting of all ', such that '(v) = v. Denote the quotient U(V; h)=u(v 0 ; h 0 ) by X and the subvariety of reections by I U(V; h). Lemma 3. The restriction of the natural surjection : U(V; h)! X to I is a birational isomorphism. Proof. Denote by I 0 the open subvariety in I consisting of all reections, such that (v) 6= v, i.e. (v)? v is anisotropic, and by U 0 the open subvariety in U(V; h) consisting of all ', such that '(v)? v is anisotropic. By Lemma 2, U 0 = I 0 U(V 0 ; h 0 ) ' I 0 U(V 0 ; h 0 ); i.e. the restriction of to I 0 yields an isomorphism between I 0 and the image of U 0 in X. Consider the following commutative diagram with exact rows? y? y 1! SU(V 0 ; h 0 )?! SU(V; h) 1! U(V 0 ; h 0 )?! U(V; h)?! X! 1?! X! 1: By Lemma 3, the map splits rationally. We want to show that is also rationally split. Let E be the function eld of X. Since splits, there is ' 2 U(V E ; h E ) such that E (') is the generic point in X(E). Denote by T the norm 1 torus of the extension L=K. More precisely, T = ker R L=K (G m;l ) N L=K?! G m;k : The reduced norm gives the homomorphism Nrd : U(B; )! T:

9 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 9 Consider also the group homomorphism = B : R L=K (GL 1;B )! T; dened by the formula (b) = Nrd(b)=? Nrd(b). Proposition 2. ([M3], Prop. 6.1 ) For any eld extension E=K the images of Nrd E and E coincide. By Proposition 2, the image of the reduced norm homomorphism Nrd : U(V E ; h E )! (L K E) equals (1? ")Nrd(B E ) = (1? ")Nrd(D E ) (here " is the generator of the Galois group of L K E=E) and hence does not depend on the hermitian form h. In particular, the reduced norm homomorphism Nrd : U(V 0 E ; h0 E)! (L K E) has the same image. Therefore, we can modify ' by an element from U(VE 0 ; h0 E ) such that Nrd(') = 1, i.e. ' 2 SU(V E; h E ) and hence splits rationally. In particular, SU(V; h) is birationally isomorphic to SU(V 0 ; h 0 )X. Lemma 4. The variety SU(V; h) is birationally isomorphic over K to the product SU(V 0 ; h 0 ) IA s K for some s 2 IN. Proof. By Lemma 3, it suces to show that I is a rational variety. Denote by P D (V ) the subvariety of 1-dimensional D-subspaces in V. This is a rational projective homogeneous variety. Consider the dominant morphism : I! P D (V ), taking a reection to the image of? id. The ber over a given anisotropic 1-dimensional D-subspace V 1 V is isomorphic to the unitary group U(V 1 ; h 1 ), where h 1 is the restriction of h on V 1. Since the variety of the unitary group is rational (see Lemma 5 below), so are the generic ber of and hence also I. Proposition 3. There is an involution of the second kind on D such that SU(V; h) is birationally isomorphic over K to SU(D; ) IA r K for some r 2 IN. Proof. By Lemma 4 and induction, it suces to consider the case dim(v ) = 1. But in this case SU(V; h) = SU(D; ) for a certain involution on D of the second kind. Corollary 2. Let B be a central simple L-algebra with an involution of the second kind, trivial on K. If ind(b) 3, then for any central subgroup SU(B; ) the variety of the group G = SU(B; )= is rational over K. Proof. By the Wedderburn theorem, B ' End D (V ), where D is a central division algebra over L and V is a vector space over D. Then the involution is isomorphic to the adjoint involution with respect to a certain nondegenerate hermitian form h on V. Repeating verbatim the same arguments as in Corollary 1 and using Proposition 3, we obtain that the variety G is birationally isomorphic over K to the product of either IA t K SU(D; ) or IA t K PSU(D; ). In both cases the second factor is a semisimple group of rank at most 2, and hence is rational.

10 10 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV Remark 3. Since SU(B; ) L ' SL 1 (B), it follows from [M1] that the group SU(B; ) is not rational if ind(b) is divisible by 4. Corollary 3. Let (B; ) and (B 0 ; 0 ) be central simple L-algebras with involutions of the second kind. If B and B 0 are Brauer equivalent algebras, then the groups SU(B; ) and SU(B 0 ; 0 ) are stably birationally isomorphic. Proof. By Proposition 3, it suces to show that groups SU(D; ) and SU(D; 0 ), where and 0 are involutions of the second kind on a division algebra D, are stably birationally isomorphic. We have 0 = Int(d) for some -symmetric invertible element d 2 D. Consider the 2-dimensional D-space V = D D and the hermitian diagonal form h = h1; di on V. By Lemma 4, the groups SU(D; ) and SU(D; 0 ), being subgroups in SU(V; h), are stably birationally isomorphic to this group. Remark 4. Corollary 3 shows that the stable birational type of the group variety SU(B; ) depends only on the Brauer class of B and does not depend on the involution. 4. Isotropic groups of type B n Proposition 4. Let G be a K-isotropic group of type B n. Then the variety G is rational over K. Proof. There are two possibilities for G to be either G ' SO 2n+1 (q), where q is a K-isotropic quadratic form of dimension 2n + 1, or G ' Spin(q). In the rst case one can apply Lemma 5. Let L be either the eld K or a quadratic extension of K and B any central simple algebra over L with an involution of either the rst or the second kind. Then the connected component of the unitary group U(B; ) is rational over K. Proof. The map ' : U?! (U) o ; X?! (1? X)=(1 + X); where U is the corresponding Lie algebra, consisting of skew elements in B with respect to, is clearly a birational K-isomorphism. In the simply connected case we rst consider the Tits K-index of G : ir ir ir ir r r r r 1 2 : : : r : : : n?1 > n It follows that the system (1) is equivalent to r+1 (t) = 1; : : : ; n?1 (t) = 1; n (t) = 1: Putting t = Q n i=1 h i (t i ) and taking into account (3), from the above we obtain an equivalent system 8 < : t 2 r+1 =t rt r+2 = 1; : : : ; t 2 n?2 =t n?3t n?1 = 1; t 2 n?1 =t n?2t 2 n = 1; t 2 n =t n?1 = 1

11 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 11 Again, considering the connected component of the subgroup dened by the latter system and adding equations t 1 = : : : = t r = 1, we nally get that \ S 0 = \ S = < h n (?1) > is the center of G. This means that = \ S = G fr+1 ;::: ; n g= < h n (?1) > ' SO(q an ); where q an is the anisotropic part of q, and it remains to apply Lemma Isotropic groups of type C n Any almost simple simply connected K-group G of type C n can be realized (see [T] ) as a special unitary group SU 2n=d (D; h), where h is a non-degenerate hermitian form on the 2n=d-dimensional right vector space V = D 2n=d, D being a central division algebra over K, provided with a proper involution of the rst kind. Hence it follows from Lemma 5 that the variety G is always rational over K. Let be the centre of G. To study the variety of the adjoint group G = G= we will rst describe the maximal K-split torus S in G. The Tits K-index of the group under consideration is as follows: r r r ir r r ir r r ir r r r 1 2 : : : d : : : 2d : : : rd : : : n?1 < n Hence the semisimple anisotropic kernel is clearly the direct product of groups G X1 ; : : : ; G Xr?1, where X 1 = f 1 ; : : : ; d?1 g; : : : ; X r?1 = f (r?1)d+1 ; : : : ; rd?1 g, each of them being isomorphic to SL 1 (D), and G Xr, where X r = f rd+1 ; : : : ; n g. Furthermore, S is described by the system 8 >< >: 1 (t) = 1; : : : ; d?1 (t) = 1 (r?1)d+1 (t) = 1; : : : ; rd?1 (t) = 1 rd+1 (t) = 1; : : : ; n (t) = 1 Adding equations t d = t 2d = = t rd = 1, we conclude that \ S coincides with the product of the centers of the groups G X1 ; : : : ; G Xr?1 : Consider the canonical map Since '(S) is a K-split torus we have ' : C G (S)?! C G (S) = C G (S)= C G (S) ' '(S) C G (S)='(S) '(S) (C G (S)=< S; > ) ' '(S) =[ \ < S; >] But is generated by the element Q [(n+1)=2] i=1 h 2i?1 (?1), where the square bracket means the integer part of a number, and hence the intersection \ < S; > is exactly the center of. Therefore, Thus, we proved = \ < S; > ' G X1 G Xr?1 G Xr IA r(d2?1) PSU(D; h an ) ' '

12 12 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV Proposition 5. Let G be as above. Then G PSU(D; h an ) IA t K, where t = dim G? dim PSU(D; h an ). Corollary 4. G is rational over K, if n? rd Isotropic groups of type 1 D n Any almost simple simply connected group G of type 1 D n can be realized as a spinor group Spin(D; h) associated to a non-degenerate hermitian form h of discriminant one over a central division algebra D of degree d with an involution of the rst kind and of orthogonal type. There are two dierent cases to be considered: the Witt index r of the hermitian form h is either maximal possible, i.e. r = n=d, or not. In the rst case the Tits K-index of G is of the form (7) i r 1 r 2 : : : r r d r : : : r (r?1)d r r : : : r r rd?1 and in the second case (8) i i H H r ri rd r 1 r 2 : : : r r d r : : : r r r : : : r r rd n? The case rd = n and G is simply connected i H H The assumption rd = n implies (see (7)) that the semisimple anisotropic kernel of G is isomorphic over K to the direct product of r copies of the group SL 1 (D). Denote the simple components of by H 1 = G f1 ;::: ; d?1 g; : : : ; H r = G f(r?1)d+1 ;::: ; rd?1 g and their centers by Z 1 ; : : : ; Z r. It is clear that Z i can be identied with d for all i = 1; : : : ; r. As in the previous sections a straightforward computation shows that \ S is generated by the following elements: (; 1; : : : ; 1; ); (1; ; : : : ; 1; ); ; (1; : : : ; 1; ; ); (1; : : : ; 1; ) where (resp. ) is a primitive root of unity of degree d (resp. d=2). Put H r = H r = d=2 ' SL 1 (D)= d=2 and consider the exact sequence 1?! r?1 d '?! H 1 H r?1 H r?! = = \ S?!1; where '( 1 ; : : : ; r?1 ) = ( 1 ; : : : ; r?1 ; 1 : : : r?1 ) and i denotes the image of i under the canonical map H r?!h r. As in x3 (see Proposition 1) one can easily prove that for all extensions E=K Ker H 1 (E; (1 1 H r ))?! H 1 (E; ) = 1: It follows immediately that is birationally isomorphic over K to the direct product of the variety H r and the variety = (1 1 H r ) ' H 1 =Z 1 H r?1 =Z r?1 : But H i =Z i ' PSL 1 (D) is rational over K and therefore, we have r r n

13 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 13 Proposition 6. Let G be as above. Then the variety G is birationally isomorphic over K to the product SL 1 (D)= d=2 IA t K, where t = dim G? (d2? 1). Corollary 5. Let G be as above and d 4. Then G is K-rational. Proof. If d = 1, then G is split and there is nothing to prove. If d = 2, then clearly d=2 = 1 and SL 1 (D)= d=2 = SL 1 (D) is rational over K. Finally, if d = 4, then the group SL 1 (D)= 2 is of type D 3 = A 3 and so one can apply Lemma The case rd = n and G is not simply connected Consider the corresponding exact sequence 1?!?! G e?! G?! 1, where G e is the simply connected covering. Let S e be a maximal K-split torus in e G, S its image in G. If d = 1, then G is split and hence K-rational. So we may assume in the sequel that d 2. The center Z of e G is isomorphic to Z=2ZZ=2Z. Therefore, there are four possibilities for to be considered: (i) = < h 1 (?1)h 3 (?1) h rd?1 (?1) > (ii) = < h 1 (?1)h 3 (?1) h rd (?1) > (iii) = < h rd?1 (?1)h rd (?1) > (iv) = < h 1 (?1)h 3 (?1) h rd?1 (?1); h rd?1 (?1)h rd (?1) > From the system (2) one can easily obtain that in case (i) one has e S; therefore, we have C G (S) ' S (C G (S)=S) ' S (C e G ( e S)= e S); which implies that G is birationally K-isomorphic to the product of an ane space and the variety SL 1 (D)= d=2 (see Proposition 6); in particular, the variety G is K-rational, if d 4. In cases (ii), (iii) one can make sure that the non-trivial element of can be written as the product of an element belonging to the torus S e and a generator of the center of H r = Gf(r?1)d+1 e ;::: ; rd?1 g. It follows that C G (S) ' S (C G (S)=S) ' S C eg ( e S)=( e S; ) S (H 1 =Z 1 H r =Z r ) ' S PSL 1 (D) PSL 1 (D); hence in these cases G is K-rational for any d. Finally, in case (iv) arguing as above we obtain that C G (S) ' S (H 1 =Z 1 H r =Z r ) is also a K-rational variety. Thus, summarizing we have Proposition 7. Let G be as above. If d 4, then G is rational over K The case n? rd > 0 and G is simply connected As above, put = H 1 H r+1, where H 1 = G f1 ;::: ; d?1 g; : : : ; H r = G f(r?1)d?1 ;::: ; rd?1 g; H r+1 = G frd+1 ;::: ; ng : Identify the centers Z 1 ; : : : ; Z r of H 1 ; : : : ; H r with d. As in the previous sections it is easy to show that the intersection \S consists of all elements '

14 14 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV of the form ( 1 ; : : : ; r ; d=2 1 r d=2 ), where i d = 1 for all i = 1; : : : ; r, and d=2 1 r d=2 is considered as an element of the group Z r+1 =< 1 > = < h n?1 (?1)h n (?1) > : Put H i = H i = d=2 ' SL 1 (D)= d=2, i = 1; : : : ; r, and consider the exact sequence 1?! r 2 '?! H 1 H r H r+1?! = = \ S?! 1; where '( 1 ; : : : ; r ) = ( 1 ; : : : ; r ; 1 r ) and as above 1 ; : : : ; r, 1 r are considered as central elements of the groups H 1 ; : : : ; H r ; H r+1. Lemma 6. For any extension E=K one has Ker H 1 (E; (H 1 1 1))?! H 1 (E; ) = 1 Proof. Pick any 0 in the kernel of the morphism above. The restriction j H1 11 is clearly injective and so is the image of the unique element 0 2 H 1 (E; H 1 1 1). By assumption, ( 0 ) is trivial in H 1 (E; ), where : H 1 (E; H 1 1 1)?! H 1 (E; ) is the canonical map. This means that (9) 0 2 Im H 1 (E; r 2)?! H 1 (E; H 1 H r H r+1 ) Identify H 1 (E; r 2 ) with? E =(E ) 2 r. Then in view of (9) we conclude that there are b 1 ; : : : ; b r 2 E such that (10) ( 0 ; 1; : : : ; 1) = (' 1 (b 1 (E ) 2 ); : : : ; ' r (b r (E ) 2 ); ' r+1 (b 1 b r (E ) 2 )); where ' i : H 1 (E; 2 )! H 1 (E; H i ) is the canonical map. First, we are going to show that b 2 ; : : : ; b r are squares in E Nrd(D K E). To this end, consider the commutative diagram x?? x?? 1?! d=2?! H i ' SL 1 (D)?! H i?! 1 1?! d=2?! d?! (1)?! 1 It induces the commutative diagram with exact rows modulo (11) H 1 (E; d=2 ) 6?! H 1 (E; SL 1 (D)) 2?! H 1 (E; H i ) x??1 x??4 H 1 (E; d=2 ) 5?! H 1 (E; d ) 3?! H 1 (E; (1)) As above, identify H 1 (E; d ), H 1 (E; d=2 ), H 1 (E; (1)) respectively with E =(E ) d, E =(E ) d=2 and E =(E ) 2. Then one can easily make sure that? 3 x(e ) d? = x(e ) 2 and 5 x(e ) d=2 = x 2 (E ) d. So in view of (10) we have b i (E ) d? = 4 bi (E ) 2 = 1: ( 4 3 )

15 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 15 On the other hand, it follows from diagram (11) that ( 4 3 ) b i (E ) d = ( 2 1 ) b i (E ) d ;? therefore, 1 bi (E ) d 2 Ker 2 = Im 6, forcing b i ; i = 2; : : : ; r, to be squares in E modulo Nrd(D K E), as claimed. From (10) it also follows that b 1 b r (E ) 2 lies in the kernel of the natural map H 1 (E; 2 )?! H 1 (E; H r+1 ) consisting of spinor norms. However, any spinor norm can be written as the product of a square and an element, which is a reduced norm in D K E (see [Z] ). So without loss of generality we may assume that b 1 (E ) 2 = Nrd (b)(e ) 2 for some b 2 D K E. Then, by construction and from diagram (11), we have 0 = 4? Nrd (b)(e ) 2 = ( 4 3 )? Nrd (b)(e ) d = obtaining the desired result. ( 2 1 )? Nrd(b) (E ) d = 2 (1) = 1; Proposition 8. Let G be a simply connected group of type 1 D n with the Tits index (8). Then G IA t K SL 1(D)= d=2 for some t 2 IN. Proof. Lemma 6 combined with Lemma 1 implies that the variety is birationally isomorphic over K to the product of H 1 ' SL 1 (D)= d=2 and a K-rational variety X = = (H 1 1 1) for X ' H 1 H r H r+1 = (H1 1 1)'( r 2 ) ' is clearly rational over K. H 2 =(1) H r =(1) H r+1 =(1) ' PSL 1 (D) PSL 1 (D) SU(D; h an ) Corollary 6. Let G be as above. Then the variety G is K-rational, if d The case n? rd > 0 and G is not simply connected This case can be handled just as the case rd = n. Namely, consider the corresponding exact sequence 1?!?! e G?! G?! 1, where e G is simply connected covering. Let e S be a maximal K-split torus in e G, S its image in G. There are four possibilities for to be considered: (i) = < h n?1 (?1)h n (?1) >; (ii) = < h 1 (?1)h 3 (?1) h n?1 (?1) > (n is even), (iii) = < h 1 (?1)h 3 (?1) h n (?1) > (n is even); (iv) coincides with the center of e G: In the rst case G ' SU(D; h) and by Lemma 5, the variety G is K- rational. In cases (ii), (iii) the non-trivial element of can be written as product of an element belonging to e S and the element = h rd+1 (?1)h rd+3 (?1) h n?1 (?1)

16 16 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV belonging to the center of H r+1 = G frd+1 ;::: ; ng. So keeping the notation of the section 6.3 we have C G (S) ' S [C G (S)=S] ' S hc eg ( e S)= < e S; > i S (H 1 H r H r+1 )= < '( r 2 ); > : Again, arguing as in 6.3 one can nally show that (H 1 H r H r+1 )= < '( r 2 ); > ' H 1 PSL 1 (D) PSL 1 (D) H r+1 =Z r+1 ; where Z r+1 denotes the center of H r+1. arguments show that ' Finally, in case (iv) analogous C G (S) ' S PSL 1 (D) PSL 1 (D) H r+1 =Z r+1 : Summarizing we have Proposition 9. 1) In case (i) G is rational over K. 2) In cases (ii), (iii) we have G IA t K PSU(D; h an) SL 1 (D)= d=2 3) In case (iv) we have G IA t K PSU(D; h an) Corollary 7. G is rational over K if d 4 and n? rd 3. Proof. It is enough to observe that all K-anisotropic adjoint groups of type 1 D l are K-rational if l 3. Indeed, the case l 2 is trivial. If l = 3 then the group under consideration is K-isomorphic to PSL 1 (T ) for some central simple algebra T and hence is rational. 7. Isotropic groups of type 2 D n As in the previous section any simply connected group e G of the type under consideration is the spinor group Spin(D; h) associated to a non-degenerate hermitian form h of discriminant 6= 1 over a division algebra D of degree d with an involution of the rst kind and of orthogonal type. Let r be its K-rank, any central subgroup and G = e G=. Let us also label simple roots as in x 6. Then there are three possibilities to be considered: coincides either with 1 or with the center of e G or = < hn?1 (?1)h n (?1) >. Denote the quotient e G= by G The case n? rd > 1 The Tits K-index is as follows: r r : : : r ri r : : : r ri r : : : r r r 1 2 d rd n?1 n Proposition 10. 1) e G IA t K SL 1 (D)= d=2 for some t 2 IN; in particular, eg is K-rational, if d 4. 2) If = < h n?1 (?1)h n (?1) >, then G is rational over K. 3) If coincides with the centre of G, then G IA t K PSU(D; h an) for some t 2 IN.

17 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 17 For the proof repeat verbatim the same arguments as in 6.3, 6.4 and so we omit it The case n? rd = 1 The Tits K-index is as follows: (12) r ri r ri r : : : r ri r r 1 2 n?1 n Proposition 11. The variety G is rational over K. Proof. If = < h n?1 (?1)h n (?1) >, then G ' SU(D; h) and hence by Lemma 5, G is K-rational. Thus, we have to consider the simply connected and adjoint cases only. To do this we need the following claim. Lemma 7. Let T = (; ) be a quaternion algebra over K and = H 1 H t, where H i (K) ' SL 1 (T ) for all i = 1; : : : ; t. Then for any central subgroup A Z() the quotient =A is a rational variety over K. Proof. Let H i = H i =(1); i = 1; : : : ; t; and = H 1 H t. It is clear that H i is a K-rational variety and furthermore, its K-points can be parametrized in such a way that K(H i ) = K(x 1;i ; x 2;i ; x 3;i ) and K(H i ) = K(H i ) q 1? x 2 1;i? x2 2;i + x2 3;i where x 1;i ; x 2;i, x 3;i are algebraically independent variables over K. Put f 1 = 1? x 2 1;1? x2 2;1 + x 2 3;1 ; : : : ; f t = 1? x 2 1;t? x2 2;t + x 2 3;t : Denote the eld of rational functions K() by E and consider any intermediate subeld ; K 0 = K(x 1;1 ; x 2;1 ; x 3;1 ; : : : ; x 1;t ; x 2;t ; x 3;t ) F E Obviously, F can be obtained by adding to K 0 some square roots of the form p f i1 f is. We are going to show that F is a purely transcendental extension of K by induction on the number of these square roots. To begin with, let us consider rst the case F = K 0 ( p f i1 f is ). Then F is clearly the eld of K-rational functions on the variety given by the equation z 2 = f i1 f is or equivalently, z 2 f i1 = f i2 f is : Its left part is a 4-dimensional Pster form g = z 2? (x 1;i1 z) 2? (x 2;i1 z) 2 + (x 3;i1 z) 2 : and obviously it represents over the eld ek = K(x 1;i2 ; x 2;i2 ; x 3;i2 ; : : : ; x 1;is ; x 2;is ; x 3;is ) the elements f i2 ; : : : ; f is. This means that the K-dened e quadric g = f i2 f is has K-rational e points and hence it denes a K-rational e variety, as claimed. In the general case, suppose that F is obtained by adjoining to the eld p p K 0 the square roots f i1 f i ; : : : ; f j1 f j. Without loss of generality we may assume that the function f j occurs in one square root only. Then

18 18 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV our eld F is the eld of K-rational functions on the variety V given by the equations 8 >< >: z1 2 = f i1 f i : : : zs?1 2 = f l1 f l zs 2f j = f j1 f j?1 Arguing as above we may look at the last equation as a e K-dened quadric in the variables z s ; ~x 1 = z s x 1;j ; ~x 2 = z s x 2;j ; ~x 3 = z s x 3;j : As above, this denes a e K-rational variety and furthermore, the variables z s ; ~x 1 ; ~x ; ~x 3 do not occur in the other equations of the system under consideration. Therefore, to complete the proof it remains to apply the induction step. Let us proceed with the proof of Proposition 11 and rst consider the case of the simply connected group e G. Here we have (see (12)) dim S0?dim S = 1, where as usually S 0 denotes the center of C G (S). Then the quotient C G (S)=S is an almost direct product of the 1-dimensional central torus S 0 =S and the semisimple group = = \ S, where = H 1 H (n?1)=2 and H 1 = G 1 ; H 2 = G 3 ; : : : ; H (n?1)=2 = G n?2. Since H i (K) ' SL 1 (D) for all i = 1; 2; : : : ; (n? 1)=2; Lemma 7 means that the variety is rational over K. A straightforward check also shows that \ S is generated by the elements h 1 (?1)h n?2 (?1); h 3 (?1)h n?2 (?1); : : : ; h n?4 (?1)h n?2 (?1): If (S 0 =S) \ = 1, then C G (S) = S 0 =S is K-rational since all 1- dimensional K-tori are rational over K. Otherwise, the intersection (S 0 =S)\ must coincide with the center of for it has order 2. Then the group C G (S)=S is part of an exact sequence where 1?! (n?1)=2 2 '?! (S 0 =S) H 1 H (n?1)=2?! C G (S)=S?! 1 '(?1; 1; : : : ; 1) = (1;?1; 1; : : : ; 1;?1); '(1;?1; 1; : : : ; 1) = (1; 1;?1; 1; : : : ; 1;?1) : : : '(1; : : : ; 1;?1; 1) = (1; : : : ; 1;?1;?1); '(1; : : : ; 1;?1) = (?1; 1; : : : ; 1;?1) By construction, S 0 =S ' R (1) L=K (G m;l), where L is a quadratic extension of K such that G has inner type over L. Obviously, L splits D ; therefore, each H i is also split over L. Then arguing just as in x 3 one can easily see that Ker H 1 (E; (1 1 H (n?1)=2 ))?! H 1 (E; C G (S)=S) = 1

19 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 19 for any extension E=K. This means that C G (S)=S is birationally isomorphic to (1 1 H (n?1)=2 ) (C G (S)=S)= (1 1 H (n?1)=2 ) ' H (n?1)=2 H 1 = 2 H (n?3)=2 = 2 (S 1 =S)= 2 and so is a K-rational variety, yielding the required result. The adjoint case is completely analogous to the previous one and so we omit it. Proposition 11 is proved General theorem Combining previous results we get the following Theorem 1. Let G be a K-isotropic almost simple group of classical type 1;2 D n. If G is not K-rational, then in the notation above either G SL 1 (D)= d=2 IA m K or G PSU(D; h an) IA m K or G SL 1(D)= d=2 PSU(D; h an ) IA m K for some m 2 IN. In particular, G is K-rational if d 4 and either n? rd 3 and G has inner type or n? rd 2 and G has outer type. 8. Isotropic groups of types F 4, 3;6 D 4 Proposition 12. The variety of any K-isotropic group G of type F 4 is rational over K. Proof. For a K-split group the statement is trivial. So we may assume that G is K-isotropic, but not K-split. Then the Tits K-index (see [T] ) is of the form r r > r ri It follows that the semisimple K-anisotropic kernel has type B 3. Solving the system (2) one gets that \ S coincides with the centre of. This means that = = \ S ' SO(f) for some 7-dimensional quadratic form f and it remains only to apply Lemma 5. Now we turn to type 3;6 D 4. As above we may assume that G is not quasisplit over K. Then the only possibility (see [T] ) for the Tits K-index is to be of the form i r Let L be any cubic extension of K such that G has type 1;2 D 4 over L. Then the semisimple K-anisotropic kernel of the corresponding simply connected group is isomorphic to R L=K (SL 1 (D)), where D is a quaternion algebra over L. Lemma 8. D has a maximal subeld E of the form E = L( p ) for some 2 K. rr r

20 20 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV Proof. By [KMRT], Proposition 43.7, it suces to show that the algebra T = cor L=K (D) is trivial over K or equivalently, T L = T K L is trivial over L. Denote all K-embeddings of L into an algebraic closure K by 1 ; 2 ; 3 and let D 1 = 1 D; D 2 = 2 D; D 3 = 3 D be the corresponding twisting algebras. Then we have T L = D 1 L D 2 L D 3. To compute T L recall that the Tits L-index of G is of the form (13) r ri r r H HH and by construction, the quaternion algebras D 1 ; D 2 ; D 3 correspond to the simple components G 1 G 3 G 4 of the semisimple L-anisotropic kernel L of G. On the other hand, it follows from (13) that up to isogeny G can be realized over L as a spinor group Spin(D 0 ; h) associated to a non-degenerate 4-dimensional hermitian form h over a quaternion L-algebra D 0 with an involution of the rst type and of orthogonal type. Since h is L-isotropic it can be written as the direct sum h = h 1 h 2, where h 1 is a 2-dimensional L-isotropic subform. Then a straightforward computation shows that for a L-split torus S Spin(D 0 ; h 1 ) the commutator of Z G (S) is the direct product of three groups of type A 1, corresponding to D 0 and the components of the Cliord algebra C(D 0 ; h 2 ). Furthermore, it is known that the simple components of the Cliord algebras C(D 0 ; h); C(D 0 ; h 2 ) of the form h and its anisotropic part h 2 coincide in Br(L) and the tensor product of the components of C(D 0 ; h) equals D 0 in Br(L) (see [KMRT] ). Therefore, the tensor product of D 0 and the components of C(D 0 ; h 2 ) is a matrix algebra over L, as claimed. Proposition 13. The variety of any K-isotropic group G of type 3;6 D 4 is rational over K. The proof is purely computational. Let rst G be the simply connected group. Since \ S = (1), where (1)?! R L=K (SL 1 (D)) is the diagonal embedding, it suces to prove the K-rationality of the variety R L=K (SL 1 (D))=(1). Let D = (; ) L, L = K() for some ; 2 L. The group R L=K (SL 1 (D)) is described by the following system of equations: 4 (14) (x 1 + x 2 + x 3 2 ) 2? (y 1 + y 2 + y 3 2 ) 2? (u 1 + u 2 + u 3 2 ) 2 + (v 1 + v 2 + v 3 2 ) 2 = 1 Then the eld of K-rational functions P = K? R L=K (SL 1 (D)) is generated over K by the coordinate functions x 1 ; x 2 ; x 3 ; y 1 ; y 2 ; y 3 ; u 1 ; u 2 ; u 3 ; v 1 ; v 2 ; v 3 ; and the eld K? R L=K (SL 1 (D))=(1) can be identied with the subeld of invariants P of the automorphism mapping x i ; y i ; u i ; v i respectively to?x i ;?y i ;?u i ;?v i for all i = 1; 2; 3.

21 THE RATIONALITY PROBLEM FOR SEMISIMPLE GROUP VARIETIES 21 Making a substitution v 2 = v 2 =v 1 ; v 3 = v 3 =v 1 ; v 1 = 1=v 1 and x i = x i =v 1 ; y i = y i =v 1 ; u i = u i =v 1 ; i = 1; 2; 3; we reduce (14) to the form (15) (x 1 + x 2 + x 3 2 ) 2? (y 1 + y 2 + y 3 2 ) 2? (u 1 + u 2 + u 3 2 ) 2 + (1 + v 2 + v 3 2 ) 2 = v 2 1 ; and (x i ) = x i ; (y i ) = y i ; (u i ) = u i ; i = 1; 2; 3, (v i ) = v i ; i = 2; 3; (v 1 ) =?v 1. Expanding the left part of (15) along the basis 1; ; 2 we arrive at a system of the form 8 < : f 1 = v 2 1 f 2 = 0 f 3 = 0 where f 1 ; f 2 ; f 3 are quadratic forms which do not depend on the variable v 1. Furthermore, the variables x 1 ; y 1 occur in the quadratic forms f 2 ; f 3 in the rst degree only. This means that the rational functions x 1 ; y 1 ; w = v 2 1 can be expressed as rational functions of the variables x i ; y i ; v i ; i = 2; 3 and u j ; j = 1; 2; 3, implying P = K(x 2 ; x 3 ; y 2 ; y 3 ; v 2 ; v 3 ; u 1 ; u 2 ; u 3 ) is a purely transcendental extension of K, as claimed. Since the center of G does not contain K-dened subgroups of order two it remains only to consider the case of the adjoint group G = G=. In this case we have to prove the K-rationality of the quotient C G (S)= < S; >. But C G (S)= < S; > ' = \ < S; > and it is easy to see that the subgroup \ < S; > coincides with the center of. Therefore, keeping the notation of the simply connected case we have = \< S; > ' R L=K (SL 1 (D))=R L=K (1) ' R L=K (SL 1 (D)=(1)) ' R L=K (PSL 1 (D)); whence C G (S)= < S; > is clearly a K-rational variety. The proposition is completely proved. 9. Isotropic groups of type E The case of inner type groups Proposition 14. Let G be any almost simple K-isotropic group of type 1 E 6. Then G is rational over K. Proof. Keep the notation of the previous sections. In particular, let G be a simply connected group, S its maximal K-split torus, a semisimple K-anisotropic kernel, the center of G and G = G= the corresponding adjoint group. The non-split Tits K-index of G is one of the following: r 2 ri 2 a) ir r r r ri b) r r ri r r In case a) a straightforward computation shows that both the center 1 of = G f2 ; 3 ; 4 ; 5 g and lie in S. So both G and G are birationally isomorphic over K to the product IA 50 K G f 2 ; 3 ; 4 ; 5 g= 1. Futhermore, it is obvious that G f2 ; 3 ; 4 ; 5 g ' Spin (f) for some 8-dimensional quadratic form f and

22 22 VLADIMIR I. CHERNOUSOV AND VLADIMIR P. PLATONOV f posseses the following us property important for: if f is isotropic over some extension E=K then it is split over E. This fact implies immediately that f is a Pster form and hence the K-rationality of G and G follows from the following Proposition 15. If f is a K-dened Pster form, then the variety PSO(f) is rational over K. To the best of our knowledge this statement was not formulated before in such a form. To prove it one can use, for example, the same arguments as in either [C2] or [M2] and so we omit the proof. Let us proceed with the proof of the last case b). First we consider the case of a simply connected group G. It follows from b) that = H 1 H 2, where H 1 = G f1 ; 3 g and H 2 = G f5 ; 6 g. Since H i is the group of inner type A 2 we have H i ' SL 1 (D i ) for some division algebra D i of degree 3. As before, we have to prove the rationality of the variety = = \ S. Solving the system (2) one gets \ S has order 3 and so is a part of the exact sequence 1?! 3?! H 1 H 2?!?! 1 Clearly, any extension E=K splitting H 1 splits G and hence H 2 too. So the division algebras D 1 ; D 2 have the same set of splitting elds and hence we have either D 1 ' D 2 or D 1 ' D 2?1 ; in particular, the division algebras D 1 ; D 2 have the same reduced norms. Using this fact it is easy to show that for any extension E=K Ker H 1 (E; (1 H 2 ))?! H 1 (E; ) = 1; implying ' H 1 H 2, where H 2 is the corresponding adjoint group. It remains to observe that both H 1 and H 2 have rank 2 and hence are K- rational. Finally, one can make sure that and \ S 6=. So the K-rationality of G follows from the K-rationality of = \ < S; > ' H 1 H 2. Proposition 14 is completely proved The case of outer type groups Proposition 16. Let G be a K-group with Tits K-index of the form a) ri ri r r b) ri 2 ri 4 r 3 r 1 c) r 5 5 r r i 2 4 r Then G is rational over K. Proof. If G is adjoint then one can use exactly the same arguments as above. Since every 2-dimensional torus is K-rational, the variety G is K-rational in the quasi-split case a). Consider a simply connected group G corresponding to diagram b). For such a group G we are going to compare the K-anisotropic kernels of G and its subgroup G 1 = G X where X = f 1 ; 3 ; 4 ; 5 ; 6 g. It follows from picture b) that G 1 is a simply connected K-isotropic group of type 2 A 5. Let S (resp. S 1 ) be a maximal K-split torus in G (resp. G 1 ). 6