View source for Linear algebraic groups, arithmetic theory of

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

One of the principal objects of study in the arithmetic theory of linear algebraic groups are arithmetic subgroups of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590801.png" /> (see [[Arithmetic group|Arithmetic group]]), and one of the principal technical instruments is the [[Adèle|adèle]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590802.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590803.png" /> one can define in a natural way a measure, called the [[Tamagawa measure|Tamagawa measure]]. One of the first questions that arises here is the following: When is the volume of the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590804.png" /> with respect to the principal adèle subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590805.png" /> finite? A complete answer to this was obtained by A. Borel. It turned out that the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590806.png" /> is always finite for a semi-simple group. The solution of this problem preceded the construction of a reduction theory for arithmetic groups (see [[#References|[5]]], [[#References|[6]]]).

Using the theory of reduction for principal adèle subgroups, it was possible in many cases to calculate the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590807.png" />, which is called the Tamagawa number of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590808.png" />. For example, for an orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l0590809.png" /> the Tamagawa number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908010.png" />, and this is actually equivalent to a fundamental result in the analytic theory of quadratic forms (see [[#References|[1]]]). The study of the structure of arithmetic groups (begun in [[#References|[6]]]) was then extended in various directions. First of all one should mention investigations on the [[Congruence problem|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908011.png" /> be the set of all inequivalent norms of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908012.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908013.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908014.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908015.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908016.png" /> be the ring of integral elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908017.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908018.png" /> be the maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908019.png" />. For an arbitrary finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908020.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908021.png" /> denote the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908022.png" /> in which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908023.png" />-components with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908024.png" /> are equal to the identity. The question is: When is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908025.png" />? (Here the bar denotes closure in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908026.png" />.) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908028.png" /> is the set of all Archimedean norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908029.png" />), then an equivalent formulation of this problem is the following: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908031.png" />, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908032.png" /> and positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908033.png" />, when does the system of congruences

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908036.png" />, have a solution in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908037.png" />?

M. Eichler [[#References|[13]]] solved the problem of strong approximation for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908038.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908039.png" /> is a [[Skew-field|skew-field]] of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908040.png" />-rank. Later, various special cases of this problem were investigated by M. Kneser, G. Shimura and A. Weil (see [[#References|[4]]]). The problem of strong approximation has been solved (see [[#References|[9]]], [[#References|[10]]]) for classical groups over number fields, and necessary conditions have been found for its affirmative solution in the general case, namely: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908041.png" /> must be simply connected as an algebraic group; and b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908042.png" /> is any simple component of the semi-simple part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908044.png" /> must be non-compact. The necessity of these conditions has been proved [[#References|[14]]] for a function field. Finally, the sufficiency of conditions a) and b) has been proved

(see also [[#References|[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|Kneser–Tits hypothesis]] on the structure of simply-connected groups over local fields: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908045.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908046.png" />-simple simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908047.png" />-isotropic group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908048.png" /> is generated by unipotent elements, or, equivalently, the quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908049.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908050.png" /> has the strong approximation property with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908051.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908052.png" /> be the ring of integral elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908053.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908054.png" /></td> </tr></table>

this shows that the arithmetic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908055.png" /> is determined to a significant extent by the arithmetic of the local components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908056.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908057.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908058.png" />, which consists of the fact that the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908059.png" /> under the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908060.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908061.png" />. 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 [[#References|[11]]], [[#References|[2]]]). Nevertheless, for a wide class of non-simply-connected semi-simple groups, in particular, for adjoint groups, the weak approximation property is satisfied [[#References|[12]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908062.png" /> is an [[Algebraic torus|algebraic torus]] and if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908063.png" /> the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908064.png" /> splits over a cyclic extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908066.png" /> has the weak approximation property with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908067.png" />. In certain cases this property is satisfied for algebraic groups over an arbitrary field (see [[#References|[11]]]). There is a conjecture (see [[#References|[11]]]) that the weak approximation property is satisfied for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908068.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908069.png" /> is a skew-field of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908070.png" />-rank over an arbitrary infinite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908071.png" />. However, the development of reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908072.png" />-theory has led to a negative answer (see [[#References|[15]]]): For the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908073.png" /> 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|Hasse principle]] (see [[Galois cohomology|Galois cohomology]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel (ed.)  G.D. Mostow (ed.) , ''Algebraic groups and discontinuous subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,   "Basic number theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Weil,   "Sur la formule de Siegel dans la théorie des groupes classiques"  ''Acta Math.'' , '''113'''  (1965)  pp. 1–87</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Borel,   "Arithmetic properties of linear algebraic groups" , ''Proc. Internat. Congress mathematicians (Stockholm, 1962)'' , Inst. Mittag-Leffler  (1963)  pp. 10–22</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Borel,   Harish-Chandra,   "Arithmetic subgroups of algebraic groups"  ''Ann. of Math.'' , '''75'''  (1962)  pp. 485–535</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  V.P. Platonov,   "The problem of strong approximation and the Kneser–Tits conjecture for algebraic groups"  ''Math. USSR Izv.'' , '''3'''  (1969)  pp. 1139–1148  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''33'''  (1969)  pp. 1211–1219</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  V.P. Platonov,   "Addendum to  "The problem of strong approximation and the Kneser–Tits conjecture for algebraic groups" "  ''Math. USSR Izv.'' , '''4'''  (1970)  pp. 784–786  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34'''  (1970)  pp. 775–777</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.P. Platonov,   "The arithmetic theory of linear algebraic groups and number theory"  ''Proc. Steklov Inst. Math.'' , '''132'''  (1973)  pp. 184–191  ''Trudy Mat. Inst. Steklov.'' , '''132'''  (1973)  pp. 162–168</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Kneser,   "Starke Approximation in algebraischen Gruppen I"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 190–203</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Kneser,   "Strong approximation"  G.D. Mostow (ed.)  A. Borel (ed.) , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)  pp. 187–196</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M. Kneser,   "Schwache Approximation in algebraischen Gruppen" , ''Colloq. Groupes Algébriques, Bruxelles'' , Gauthier-Villars  (1962)  pp. 41–52</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  G. Harder,   "Halbeinfache Gruppenschemeta über Dedekindringen"  ''Invent. Math.'' , '''4''' :  3  (1967)  pp. 165–191</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  M. Eichler,   "Algemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraische Zahlkörpern und ihre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908074.png" />-Reihen"  ''J. Reine Angew. Math.'' , '''179'''  (1938)  pp. 227–251</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  H. Behr,   "Zur starken Approximation in algebraischen Gruppen über globalen Körpern"  ''J. Reine Angew. Math.'' , '''229'''  (1968)  pp. 107–116</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  V.P. Platonov,   "Reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908075.png" />-theory and approximation in algebraic groups"  ''Proc. Steklov Inst. Math.'' , '''142''' :  3  (1979)  pp. 213–224  ''Trudy Mat. Inst. Steklov.'' , '''142'''  (1976)  pp. 198–207</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  G. Prasad,   "Strong approximation for semi-simple groups over function fields"  ''Ann. of Math. (2)'' , '''105'''  (1977)  pp. 553–572</TD></TR></table>



====Comments====
The [[Tamagawa number|Tamagawa number]] has been computed for every simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908076.png" />. In particular, Weil's conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908077.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059080/l05908078.png" /> is simply connected has been shown to be true.