Editing Congruence subgroup (section)

=== Properties of congruence subgroups ===

The congruence subgroups of the modular group and the associated Riemann surfaces are distinguished by some particularly nice geometric and topological properties. Here is a sample:
* There are only finitely many congruence covers of the modular surface that have genus zero;<ref>{{cite journal | last1=Long | first1=Darren D. | last2=Maclachlan | first2=Colin | last3=Reid | first3=Alan | title=Arithmetic Fuchsian groups of genus zero | journal=Pure and Applied Math Quarterly 2 | date=2006 | volume=Special issue to celebrate the 60th birthday of Professor J. H. Coates | issue=2 | pages=569&ndash;599| doi=10.4310/PAMQ.2006.v2.n2.a9 | doi-access=free }}</ref> 
* ([[Selberg's 1/4 conjecture|Selberg's 3/16 theorem]]) If <math>f</math> is a nonconstant eigenfunction of the [[Laplace-Beltrami operator]] on a congruence cover of the modular surface with eigenvalue <math>\lambda</math> then {{tmath|1= \lambda \geqslant \tfrac{3}{16} }}.

There is also a collection of distinguished operators called [[Hecke operator]]s on smooth functions on congruence covers, which commute with each other and with the Laplace–Beltrami operator and are diagonalisable in each eigenspace of the latter. Their common eigenfunctions are a fundamental example of [[automorphic form]]s. Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the associated Riemann surfaces via the [[Eichler-Shimura isomorphism]].