UDC 513.6

MATHEMATICS

V. E. VOSKRESENSKII

ON THE BIRATIONAL EQUIVALENCE OF LINEAR ALGEBRAIC GROUPS

(Presented by Academician Yu. V. Linnik on 10 III 1969)

1. Let \(k\) be a field of characteristic zero, \(\bar{k}\) its algebraic closure, and \(\mathscr G\) the Galois group of the extension \(\bar{k}/k\). Let \(X\) be an algebraic variety over the field \(k\), i.e. a geometrically irreducible reduced scheme of finite type over the field \(k\); \(\bar X=X\otimes_k \bar k\) is the scheme obtained from \(X\) by extension of the ground field. The group \(\mathscr G\) acts naturally on \(\bar X\) and on the objects determined by the scheme \(\bar X\); in particular, the group \(\operatorname{Pic}\bar X=H^1(\bar X,O_{\bar X}^{*})\) is a \(\mathscr G\)-module.

Every \(\mathscr G\)-module isomorphic to a finite direct sum of modules of the form \(\mathbf Z[\mathscr G]\otimes_{\mathbf Z[U]}\mathbf Z\), where \(U\) runs through some set of subgroups of finite index in the group \(\mathscr G\), will be called, following Yu. I. Manin, trivial. If \(\bar X\) is a nonsingular variety, then every submodule of the divisor group generated by an invariant finite set of prime divisors is trivial.

Two \(\mathscr G\)-modules \(A\) and \(B\) will be called similar if there exist trivial \(\mathscr G\)-modules \(M_1\) and \(M_2\) such that there is an isomorphism of direct sums \(A+M_1\simeq B+M_2\). The following assertion is apparently known; for surfaces it was proved in \((^1)\), and over an arbitrary perfect field.

Theorem 1. Let \(X\) and \(Y\) be nonsingular projective varieties over the field \(k\), birationally equivalent over this field. Then the \(\mathscr G\)-modules \(\operatorname{Pic}\bar X\) and \(\operatorname{Pic}\bar Y\) are similar.

The proof is based on the existence of a nonsingular model \(Z\) and birational morphisms
\[ X \xleftarrow{\varphi} Z \xrightarrow{\psi} Y \]
with subsequent use of Leray’s spectral sequence for the maps \(\varphi\) and \(\psi\).

Corollary 1. If the variety \(X\) is rational, i.e. \(\bar X\) is birational to \(\bar{\mathbf P}^{\,n}\), then \(\operatorname{Pic}\bar X\) is a \(\mathscr G\)-module without torsion of finite \(\mathbf Z\)-rank.

Corollary 2. If the variety \(X\) is rational over the field \(k\), i.e. \(X\) is birational to \(\mathbf P^n\), then the \(\mathscr G\)-module \(\operatorname{Pic}\bar X\) is similar to a trivial one; in particular, \(H^1(k,\operatorname{Pic}\bar X)=0\).

1. Let \(G\) be a connected linear algebraic group defined over a field \(k\) of characteristic zero. Consider some nonsingular projective variety \(V\), defined over the field \(k\), such that the variety \(\operatorname{Spec} k[G]\) is mapped biregularly onto an open subvariety of the variety \(V\). We shall call the variety \(V=V(G)\) a projective model of the group \(G\) over \(k\).

It is easy to see that such models always exist (the field \(k\) has characteristic zero). Let \(\bar G=\operatorname{Spec}\bar k[G]\), and let \(F=\bar V\setminus \bar G\) be a closed subset invariant with respect to the group \(\mathscr G\). Considering the exact sequence of local cohomology of the sheaf \(O_{\bar V}^{*}\), defined by the embedding \(\bar G\subset \bar V\), we obtain the exact sequence of \(\mathscr G\)-modules
\[ 0\to \hat G\to M\to \operatorname{Pic}\bar V\to \operatorname{Pic}\bar G\to 0, \tag{1} \]
where \(\hat G\) is the \(\mathscr G\)-module of rational characters of the group \(G\), and \(M\) is a trivial \(\mathscr G\)-module generated by divisors whose supports lie in \(F\).

1. Let us compute \(\operatorname{Pic}\overline{G}\), or, more generally, \(\operatorname{Pic}G=\operatorname{Pic}(\operatorname{Spec} k[G])\).

Theorem 2. Let \(G\) be a connected solvable algebraic group over \(k\). Then
\[ \operatorname{Pic}G=H^{1}(k,\hat{G}). \]

Theorem 3. Let \(G\) be a connected semisimple algebraic group defined over a field \(k\); let \(\pi\) be the fundamental group on which the Galois group \(\mathfrak G\) acts naturally. Then
\[ \operatorname{Pic}G=\operatorname{Hom}(\pi,\bar{k}^{*})^{\mathfrak G}. \]
In particular, if the group \(G\) is simply connected, then
\[ \operatorname{Pic}G=0. \]

It follows from these theorems that, for any algebraic linear group \(G\), the group \(\operatorname{Pic}G\) is finite. The proof of Theorem 2 is based on the fact that \(\operatorname{Pic}\overline{G}=0\) and the group \(\operatorname{Pic}G\) is computed as the kernel of the map \(\operatorname{Pic}G\to\operatorname{Pic}\overline{G}\). Similarly, in Theorem 3 it is first proved that \(\operatorname{Pic}\overline{G}=0\) for simply connected groups \(G\), and then the kernel of the map \(\operatorname{Pic}G\to\operatorname{Pic}\overline{G}\) is computed.

1. Let us return to the sequence (1). Let \(T\) be a torus defined over the field \(k\). Then \(\operatorname{Pic}\overline{T}=0\), and from (1) we obtain the exact sequence
   \[ 0\to \hat{T}\to \hat{M}\to \operatorname{Pic}\overline{V}\to 0. \tag{1} \]

All \(\mathfrak G\)-modules in the sequence (2) are, up to torsion, of finite rank, and the module \(M\) is trivial. If the torus \(T\) is rational over the field \(k\), then the module \(\operatorname{Pic}\overline{V}\) is similar to a trivial one. Adding in this case a trivial module to the modules \(M\) and \(\operatorname{Pic}\overline{V}\), one can arrange that in the sequence (2) the modules \(M\) and \(\operatorname{Pic}\overline{V}\) will be trivial. Hence

Theorem 4. In order that the torus \(T\) be rational over the field of definition \(k\), it is necessary that its character module \(\hat{T}\) can be included in an exact sequence
\[ 0\to \hat{T}\to \hat{M}\to \hat{N}\to 0, \]
where \(\hat{M}\) and \(\hat{N}\) are trivial \(\mathfrak G\)-modules.

It would be interesting to determine whether this necessary condition for rationality is also sufficient.

Let now \(k\) be a number field, \(k_p\) its \(p\)-adic completion (\(p\) a finite or infinite place of the field \(k\)). Consider two sets of arithmetic origin associated with the group \(G\) over \(k\). The first is \(\Sha(G)\), i.e. the kernel of the map
\[ H^{1}(k,G(\bar{k}))\to \prod_p H^{1}(k_p,G(\bar{k}_p)). \]
The second is constructed as follows. Take the direct product \(\prod_p G(k_p)\) with the direct-product topology. The group \(G(k)\) is embedded in \(\prod_p G(k_p)\) by means of the diagonal map. Let \(\overline{G(k)}\) be the closure of \(G(k)\) in \(\prod_p G(k_p)\). Denote by \(A(G)\) the quotient set
\[ \prod_p G(k_p)/\overline{G(k)}. \]
The set \(A(G)\) measures the failure of weak approximation in the group \(G\).

Theorem 5. Let \(T\) be a torus defined over a number field \(k\). Then there exists an exact sequence of groups
\[ 0\to A(T)\to H^{1}(k,\operatorname{Pic}\overline{V})\to \Sha(T)\to 0. \]

Corollary. If \(H^{1}(k,\operatorname{Pic}\overline{V})=0\), then
\[ A(T)=\Sha(T)=0, \]
and conversely. In particular, if the torus \(T\) is rational over the field \(k\), then \(\Sha(T)=0\) and the Tamagawa number \(\tau(T)\) is an integer.

The validity of the theorem on weak approximation in this case had been well known earlier.

Theorem 6. Let \(k\) be an arbitrary field of characteristic zero; let \(T\) be a torus over \(k\) that splits over a cyclic extension of the field \(k\); and let \(V\) be a projective model of the torus \(T\). Then
\[ H^{1}(k,\operatorname{Pic}\overline{V})=0. \]

Corollary 1. Let \(T\) be a torus defined over a number field with cyclic splitting field. Then
\[ A(T)=\Sha(T)=0. \]

These facts are known: the equality \(\Sha(T)=0\) was proved by T. Ono [2], and the equality \(A(T)=0\) was proved by J.-P. Serre (unpublished).

Corollary 2. Let \(T\) be a torus defined over the field of algebraic numbers and split over a field \(L\) whose decomposition groups are cyclic for all primes. Then \(A(T)=0\). (J.-P. Serre, unpublished).

1. We shall show that there exist tori with cyclic splitting field which are not rational over the field of definition. Let \(L\) be a cyclic extension of a field \(k\) of degree 23; let \(H\) be the Galois group of \(L/k\). It is known that there exists a projective \(H\)-module \(\hat T\), not free, such that \(\hat T\otimes Q\simeq Q[H]\). Suppose that the torus \(T\) is rational over the field \(k\). Then, by Theorem 4, there exists an exact sequence of \(\mathscr G\)-modules \(0\to \hat T\to \hat M\to \hat N\to 0\), where \(\hat M\) and \(\hat N\) are trivial \(\mathscr L\)-modules. Simple computations show that then \(\operatorname{Ext}_{\mathscr G}(\hat N,\hat T)=0\), i.e. \(\hat M\simeq \hat T+\hat N\). It is proved that \(\hat M\) and \(\hat N\) may be regarded as \(H\)-modules. But trivial \(H\)-modules are \(Z\) and \(Z[H]\), and Reiner’s theorem 3 shows that from the equality \(\hat T+\hat N=\hat M\) it follows that \(\hat T\simeq Z[H]\). This contradiction shows that the torus \(T\) is not rational over \(k\). It is known that one can choose projective nonfree \(H\)-modules \(\hat T_1\) and \(\hat T_2\) so that \(\hat T_1+\hat T_2\simeq Z[H]+Z[H]\). Hence
   \[ T_1\times T_2\simeq R_{L/k}(G_m)\times R_{L/k}(G_m). \]
   On the right is a direct product of tori rational over \(k\). This gives an interesting example of the rationality of the direct product of two varieties which are not rational over \(k\).

2. Let \(Q\) be the field of rational numbers, \(K=Q(\sqrt a,\sqrt b)\) an extension of degree 4. Consider the tori
   \[ T(a,b)=R^{(1)}_{K/Q}(G_m) \]
   of dimension 3 for different fields \(K\). Let \(V\) be a projective model of the torus \(T\). Then
   \[ H^1(Q,\operatorname{Pic}\bar V)=Z_2 \]
   for any field \(K\).

Consider two cases:

a) All decomposition groups of the field \(K\) are cyclic. Then
\[ A(T)=0,\qquad Ш(T)=Z_2. \]

b) There exists a decomposition group of order 4. Take, for example, a prime number \(p>2\) such that the Legendre symbol
\[ \left(\frac{-1}{p}\right)=-1. \]
There are infinitely many such primes \(p\). Then
\[ [Q_p(\sqrt p,\sqrt{-1}):Q_p]=4 \]
and the tori \(T(p,-1)\) and \(T(p',-1)\), for different such \(p\) and \(p'\), are birationally nonequivalent over the field \(Q\), since the torus \(T(p',-1)\) is rational over the field \(Q_p\) for all \(p\ne p'\) and is not rational over the field \(Q_{p'}\). This shows that in dimension 3 there is an infinite set of birationally nonequivalent tori over the field \(Q\). In dimensions 1 and 2 all tori are rational over \(Q\).

1. Let \(G\) be a connected semisimple algebraic group defined over a field \(k\) of characteristic zero. Since the character group \(\hat G\) is trivial, the sequence (1) takes the form
   \[ 0\to M\to \operatorname{Pic}\bar V\to \operatorname{Pic}\bar G\to 0, \]
   where \(M\) is a trivial \(\mathscr G\)-module. A theorem, analogous in form to Theorem 5 for tori, is proved.

Theorem 7. Let \(G\) be a connected semisimple algebraic group defined over a field of algebraic numbers \(k\); let \(\pi\) be the fundamental group of the group \(G\); let \(V\) be a projective model of the group \(G\).

Then there exists an exact sequence
\[ 0\to A\to H^1(k,\operatorname{Pic}\bar V)\to B\to 0, \]
where
\[ A=\operatorname{Coker}\left[H^1(k,\pi)\to\prod_p H^1(k_p,\pi)\right], \]
\[ B=\operatorname{Ker}\left[H^2(k,\pi)\to\prod_p H^2(k_p,\pi)\right] =\operatorname{Ker}\left[H^1(k,\operatorname{Pic}\bar G)\to\prod_p H^1(k_p,\operatorname{Pic}\bar G)\right]. \]

If for the simply connected covering \(\tilde G\) of the group \(G\) the sets \(A(\tilde G)\) and \(Ш(\tilde G)\) are equal to zero (there are conjectures that this is always so), then \(A(G)\)—

an abelian group that is a subgroup of the group \(A\). I have not been able to relate the group \(B\) to the set \(\Sha(G)\), as in the case of tori. However, the following assertion is true under the condition \(A(\widetilde G)=0=\Sha(\widetilde G)\):

Corollary. If \(G\) is a semisimple group defined and rational over the field of algebraic numbers, then \(\Sha(G)=0\).

Furthermore, the order of the group \(B\) enters the denominator in the formula obtained by T. Ono (4) for the ratio of the Tamagawa numbers of the groups \(G\) and \(\widetilde G\). In our notation it has the following form:
\[ \tau(G)/\tau(\widetilde G)=[\operatorname{Pic}G]/[B]. \]
Suppose that A. Weil’s hypothesis holds, i.e. \(\tau(\widetilde G)=1\). Then
\[ \tau(G)=[\operatorname{Pic}G]/[B]. \]
If the group \(G\) is rational over \(k\), then \(B=0\) and
\[ \tau(G)=[\operatorname{Pic}G] \]
is an integer. It is curious that for tori \(T\) the Tamagawa number is expressed by a very similar formula
\[ \tau(T)=[\operatorname{Pic}T]/[\Sha(T)]. \]

The corollaries of Theorems 5 and 7 make it possible to state the following conjecture (a necessary criterion for the rationality of algebraic groups).

Hypothesis. Let \(G\) be a connected algebraic linear group defined and rational over the field of algebraic numbers. Then
\[ \Sha(G)=0 \]
and the Tamagawa number
\[ \tau(G)=[\operatorname{Pic}G]. \]

The author expresses his sincere gratitude to Yu. I. Manin and I. R. Shafarevich for important advice.

Saratov State University
named after N. G. Chernyshevsky

Received
4 III 1969

CITED LITERATURE

1. Yu. I. Manin, Publ. Math. IHES, No. 30, 55 (1966).
2. T. Ono, Ann. Math., 78, No. 1, 47 (1963).
3. I. Reiner, Proc. Am. Math. Soc., 8, 142 (1957).
4. T. Ono, Ann. Math., 82, No. 1, 88 (1965).