Linear algebraic groups, arithmetic theory of

The theory that studies arithmetic properties of linear algebraic groups (cf. Linear algebraic group), defined, as a rule, over a global field.

One of the principal objects of study in the arithmetic theory of linear algebraic groups are arithmetic subgroups of an algebraic group $G$ (see Arithmetic group), and one of the principal technical instruments is the adèle group $G_{\mathbf{A}}$. On $G_{\mathbf{A}}$ one can define in a natural way a measure, called the Tamagawa measure. One of the first questions that arises here is the following: When is the volume of the quotient space of $G_{\mathbf{A}}$ with respect to the principal adèle subgroup $G_k$ finite? A complete answer to this was obtained by A. Borel. It turned out that the volume of $G_{\mathbf{A}}/G_k$ is always finite for a semi-simple group. The solution of this problem preceded the construction of a reduction theory for arithmetic groups (see [5], [6]).

Using the theory of reduction for principal adèle subgroups, it was possible in many cases to calculate the volume of $G_{\mathbf{A}}/G_k$, which is called the Tamagawa number of the group $G$. For example, for an orthogonal group $G$ the Tamagawa number $\tau(G)$, and this is actually equivalent to a fundamental result in the analytic theory of quadratic forms (see [1]). The study of the structure of arithmetic groups (begun in [6]) was then extended in various directions. First of all one should mention investigations on the congruence problem, the problem of maximality of arithmetic subgroups and the problem of the genus of arithmetic groups.

In all basic questions in the arithmetic theory of linear algebraic groups an essential role is played by approximation theorems, which reduce the investigation of arithmetic properties of algebraic groups defined over global fields to the investigation of arithmetic properties of algebraic groups defined over local fields. Of greatest significance is the problem of strong approximation in algebraic groups, which consists of the following. Let $V = \{v\}$ be the set of all inequivalent norms of a field $k$, let $k_v$ be the completion of $k$ with respect to $v$, let $O_v$ be the ring of integral elements of $k_v$, and let $\mathfrak{P}_v$ be the maximal ideal of $O_v$. For an arbitrary finite subset $S \subset V$, let $G_S$ denote the subgroup of $G_{\mathbf{A}}$ in which all $v$-components with $v \notin S$ are equal to the identity. The question is: When is $\overline{G_S G_k} = G_{\mathbf{A}}$? (Here the bar denotes closure in the topology of $G_{\mathbf{A}}$.) If $S = \infty$ ($\infty$ is the set of all Archimedean norms of $k$), then an equivalent formulation of this problem is the following: For any $v_i \in S$, $i=1,\ldots,n$, any $a_{v_i} \in G_{k_{v_i}}$ and positive integers $m_i$, when does the system of congruences $$ x \equiv a_{v_i} \pmod{\mathfrak{P}_{v_i}^{m_i}} $$ where $x \in G_{O_v}$ for $v \in \cup \{v_1,\ldots,v_n\}$, have a solution in the group $G_k$?

M. Eichler [13] solved the problem of strong approximation for the groups $\textrm{SL}(n, D)$ where $D$ is a skew-field of finite $r$-rank. Later, various special cases of this problem were investigated by M. Kneser, G. Shimura and A. Weil (see [4]). The problem of strong approximation has been solved (see [9], [10]) for classical groups over number fields, and necessary conditions have been found for its affirmative solution in the general case, namely: a) $G$ must be simply connected as an algebraic group; and b) if $F$ is any simple component of the semi-simple part of $G$, then $F_S$ must be non-compact. The necessity of these conditions has been proved [14] for a function field. Finally, the sufficiency of conditions a) and b) has been proved

(see also [16]) over both number fields and function fields, which gives a complete solution of the strong approximation problem. At the base of the method of proof lies the reduction of this problem to the proof of the Kneser–Tits hypothesis on the structure of simply-connected groups over local fields: If $G$ is a $k_v$-simple simply-connected $k_v$-isotropic group, then $G_{k_v}$ is generated by unipotent elements, or, equivalently, the quotient group of $G_{k_v}$ with respect to its centre is simple in the abstract sense. As the simplest application of the strong approximation theorem one obtains the following fact: Suppose that $G$ has the strong approximation property with respect to $S=\infty$ and let $O$ be the ring of integral elements of $k$, then

$$\overline{G_O} = \prod_{v\notin S} G_{O_v};$$ this shows that the arithmetic of $G_O$ is determined to a significant extent by the arithmetic of the local components $G_{O_v}$.

Together with strong approximation, an important role in the arithmetic theory of linear algebraic groups is played by the property of weak approximation of an algebraic group $G$ with respect to $S$, which consists of the fact that the image of $G_k$ under the canonical projection $G_k \to G_S$ is dense in $G_S$. All simply-connected groups have the weak approximation property. On the other hand, there are examples of semi-simple groups and algebraic tori that do not have the weak approximation property (see [11], [2]). Nevertheless, for a wide class of non-simply-connected semi-simple groups, in particular, for adjoint groups, the weak approximation property is satisfied [12]. If $G$ is an algebraic torus and if for every $v \in S$ the torus $\textrm{SL}(n, D)$ splits over a cyclic extension of the field $D$, then $k$ has the weak approximation property with respect to $k$. In certain cases this property is satisfied for algebraic groups over an arbitrary field (see [11]). There is a conjecture (see [11]) that the weak approximation property is satisfied for the groups $K$ where $\textrm{SL}(n, D)$ is a skew-field of finite $k$-rank over an arbitrary infinite field $k$. However, the development of reduced $K$-theory has led to a negative answer (see [15]): For the groups $\textrm{SL}(n, D)$ the deviation from weak approximation can be arbitrary large.

An important role in the arithmetic theory of linear algebraic groups is played by cohomology methods, in particular the Hasse principle (see Galois cohomology).

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The Tamagawa number has been computed for every simple $G$. In particular, Weil's conjecture that $\tau(G) = 1$ if $G$ is simply connected has been shown to be true.