They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in 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 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.
[3] This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases.
[4][5] Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra.
[6] 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 1972.
[7] 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.
[8] Some limited results in this direction had been obtained earlier by Selberg, but Margulis' 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.
One of the main tool used there is the trace formula originating in Selberg's work[9] and developed in the most general setting by James Arthur.
[10] 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,[11] "...often seem to have special beauty."
In general it is not so obvious how to make precise sense of the notion of "integer points" of a
A lattice in a Lie group is usually defined as a discrete subgroup with finite covolume.
The terminology introduced above is coherent with this, as a theorem due to Borel and Harish-Chandra states that an arithmetic subgroup in a semisimple Lie group is of finite covolume (the discreteness is obvious).
will be co-compact in the associated orthogonal group if and only if the quadratic form does not vanish at any point in
The spectacular result that Margulis obtained is a partial converse to the Borel—Harish-Chandra theorem: for certain Lie groups any lattice is arithmetic.
This result is true for all irreducible lattices in semisimple Lie groups of real rank larger than two.
has a factor of real rank one (otherwise the theorem always holds) and is not simple: it means that for any product decomposition
The Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely
An arithmetic Fuchsian group is constructed from the following data: a totally real number field
This amounts to classifying the algebraic groups whose real points are isomorphic up to a compact factor to
It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
Such graphs are known to exist in abundance by probabilistic results but the explicit nature of these constructions makes them interesting.
[19] Likewise the Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak have large girth.
It is in fact known that the Ramanujan property itself implies that the local girths of the graph are almost always large.
This was first realised by Marie-France Vignéras[21] and numerous variations on her construction have appeared since.
The isospectrality problem is in fact particularly amenable to study in the restricted setting of arithmetic manifolds.
By work of Klingler (also proved independently by Yeung) all such are quotients of the 2-ball by arithmetic lattices in