(since the reduced norm of a matrix algebra is just the determinant) and we can consider the Fuchsian group which is its image in
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on
It follows immediately from this definition that arithmetic Fuchsian groups are discrete and of finite covolume (this means that they are lattices in
A natural question is to identify those among arithmetic Fuchsian groups which are not strictly contained in a larger discrete subgroup.
These are called maximal Kleinian groups and it is possible to give a complete classification in a given arithmetic commensurability class.
commensurable to those obtained by this construction are called arithmetic Kleinian groups.
As in the Fuchsian case arithmetic Kleinian groups are discrete subgroups of finite covolume.
The invariant trace field of a Fuchsian group (or, through the monodromy image of the fundamental group, of a hyperbolic surface) is the field generated by the traces of the squares of its elements.
One can in fact characterise arithmetic manifolds through the traces of the elements of their fundamental group, a result known as Takeuchi's criterion.
the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals
Coupled with the description of maximal subgroups and finiteness results for number fields this formula allows to prove the following statement: Note that in dimensions four and more Wang's finiteness theorem (a consequence of local rigidity) asserts that this statement remains true by replacing "arithmetic" by "finite volume".
One can give an effective description of the set of such curves in an arithmetic surface or three—manifold: they correspond to certain units in certain quadratic extensions of the base field (the description is lengthy and shall not be given in full here).
For example, the length of primitive closed geodesics in the modular surface corresponds to the absolute value of units of norm one in real quadratic fields.
This description was used by Sarnak to establish a conjecture of Gauss on the mean order of class groups of real quadratic fields.
is compact it extends to an unbounded, essentially self-adjoint operator on the Hilbert space
The spectral theorem in Riemannian geometry states that there exists an orthonormal basis
are of interest for number theorists, as well as the distribution and nodal sets of the
is of finte volume is more complicated but a similar theory can be established via the notion of cusp form.
In general it can be made arbitrarily small (ref Randol) (however it has a positive lower bound for a surface with fixed volume).
The Selberg conjecture is the following statement providing a conjectural uniform lower bound in the arithmetic case: Note that the statement is only valid for a subclass of arithmetic surfaces and can be seen to be false for general subgroups of finite index in lattices derived from quaternion algebras.
Selberg's original statement[10] was made only for congruence covers of the modular surface and it has been verified for some small groups.
[13] Selberg's trace formula shows that for an hyperbolic surface of finite volume it is equivalent to know the length spectrum (the collection of lengths of all closed geodesics on
Another relation between spectrum and geometry is given by Cheeger's inequality, which in the case of a surface
states roughly that a positive lower bound on the spectral gap of
translates into a positive lower bound for the total length of a collection of smooth closed curves separating
The quantum ergodicity theorem of Shnirelman, Colin de Verdière and Zelditch states that on average, eigenfunctions equidistribute on
The unique quantum ergodicity conjecture of Rudnick and Sarnak asks whether the stronger statement that individual eigenfunctions equidistribure is true.
In the case of congruence covers of the modular some additional difficulties occur, which were dealt with by K.
was pointed out by M. F. Vignéras[16] and used by her to construct examples of isospectral compact hyperbolic surfaces.
The precise statement is as follows: Vignéras then constructed explicit instances for