Arithmetic group

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm{SL}_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm{SL}_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory. ==History== One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces. The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariants of number fields such as the discriminant. Arithmetic groups can be thought of as a vast generalisation of the unit groups of number fields to a noncommutative setting. The same groups also appeared in analytic number theory as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in Langlands' computation of the volume of certain fundamental domains using analytic methods. This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases. For the modern theory to begin foundational work was needed, and was provided by the work of Armand Borel, André Weil, Jacques Tits and others on algebraic groups. Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra. Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972¹. In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group. Some limited results in this direction had been obtained earlier by Selberg, but Margulis's methods (the use of ergodic-theoretical tools for actions on homogeneous spaces) were completely new in this context and were to be extremely influential on later developments, effectively renewing the old subject of geometry of numbers and allowing Margulis himself to prove the Oppenheim conjecture; stronger results (Ratner's theorems) were later obtained by Marina Ratner. In another direction the classical topic of modular forms has blossomed into the modern theory of automorphic forms. The driving force behind this effort is mainly the Langlands program initiated by Robert Langlands. One of the main tools used there is the trace formula originating in Selberg's work and developed in the most general setting by James Arthur. Finally arithmetic groups are often used to construct interesting examples of locally symmetric Riemannian manifolds. A particularly active research topic has been arithmetic hyperbolic 3-manifolds, which as William Thurston wrote, "...often seem to have special beauty." If \mathrm G is an algebraic subgroup of \mathrm{GL}_n(\Q) for some n then we can define an arithmetic subgroup of \mathrm G(\Q) as the group of integer points \Gamma = \mathrm{GL}_n(\Z) \cap \mathrm G(\Q). In general it is not so obvious how to make precise sense of the notion of "integer points" of a \Q -group, and the subgroup defined above can change when we take different embeddings \mathrm G \to \mathrm{GL}_n(\Q). Thus a better notion is to take for definition of an arithmetic subgroup of \mathrm G(\Q) any group \Lambda which is commensurable (this means that both \Gamma/(\Gamma\cap \Lambda) and \Lambda/(\Gamma \cap \Lambda) are finite sets) with a group \Gamma defined as above (with respect to any embedding into \mathrm{GL}_n ). With this definition, to the algebraic group \mathrm G is associated a collection of "discrete" subgroups all commensurable to each other. On the other hand, the class of groups thus obtained is not larger than the class of arithmetic groups as defined above. Indeed, if we consider the algebraic group \mathrm G' over \Q obtained by restricting scalars from F to \Q and the \Q -embedding \rho' : \mathrm G' \to \mathrm{GL}_{dn} induced by \rho (where d = [F:\Q ] ) then the group constructed above is equal to (\rho')^{-1}(\mathrm{GL}_{nd}(\Z)) . ==Examples== The classical example of an arithmetic group is \mathrm{SL}_n(\Z) , or the closely related groups \mathrm{PSL}_n(\Z) , \mathrm{GL}_n(\Z) and \mathrm{PGL}_n(\Z) . For n = 2 the group \mathrm{PSL}_2(\Z) , or sometimes \mathrm{SL}_2(\Z) , is called the modular group as it is related to the modular curve. Similar examples are the Siegel modular groups \mathrm{Sp}_{2g}(\Z) . Other well-known and studied examples include the Bianchi groups \mathrm{SL}_2(O_{-m}), where m > 0 is a square-free integer and O_{-m} is the ring of integers in the field \Q(\sqrt{-m}), and the Hilbert–Blumenthal modular groups \mathrm{SL}_2(O_m) . Another classical example is given by the integral elements in the orthogonal group of a quadratic form defined over a number field, for example \mathrm{SO}(n,1)(\Z ) . A related construction is by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed with unitary groups of hermitian forms, a well-known example is the Picard modular group. When G is a Lie group one can define an arithmetic lattice in G as follows: for any algebraic group \mathrm G defined over \Q such that there is a morphism \mathrm G(\R) \to G with compact kernel, the image of an arithmetic subgroup in \mathrm G(\Q) is an arithmetic lattice in G . Thus, for example, if G = \mathrm G(\R) and G is a subgroup of \mathrm{GL}_n then G \cap \mathrm{GL}_n(\Z) is an arithmetic lattice in G (but there are many more, corresponding to other embeddings); for instance, \mathrm{SL}_n(\Z) is an arithmetic lattice in \mathrm{SL}_n(\R ) . Irreducibility only plays a role when G has a factor of real rank one (otherwise the theorem always holds) and is not simple: it means that for any product decomposition G = G_1\times G_2 the lattice is not commensurable to a product of lattices in each of the factors G_i . For example, the lattice \mathrm{SL}_2(\Z [\sqrt 2]) in \mathrm{SL}_2(\R) \times \mathrm{SL}_2(\R) is irreducible, while \mathrm{SL}_2(\Z) \times \mathrm{SL}_2(\Z) is not. The Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely \mathrm{Sp}(n,1) for n \geqslant 1 and the exceptional group F_4^{-20} . It is known not to hold in all groups \mathrm{SO}(n,1) for n \geqslant 2 (ref to GPS) and for \mathrm{SU}(n, 1) when n = 1,2,3 . There are no known non-arithmetic lattices in the groups \mathrm{SU}(n,1) when n \geqslant 4 . For every semisimple Lie group G it is in theory possible to classify (up to commensurability) all arithmetic lattices in G , in a manner similar to the cases G = \mathrm{SL}_2(\R), \mathrm{SL}_2(\Complex) explained above. This amounts to classifying the algebraic groups whose real points are isomorphic up to a compact factor to G . G = \mathrm G(\R) \times \prod_{p\in S} \mathrm G(\Q_p). ==References==

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