Talk:Congruence subgroup

The article does not make a distinction between principal congruence subgroups (which is what is discussed here apparently) and more general congruence subgroups. Katzmik 08:04, 2 August 2007 (UTC)


 * Yes, it would be useful to expand and clarify. Charles Matthews 02:42, 4 August 2007 (UTC)

Theta group
Different opinion on the definition of the theta group. Deltahedron redirected one of the defnitions of the theta group in the wiki page theta group to here. In section 2.4 Modular Group Λ, Charles Matthews claimed that the Modular Group Λ is also called the theta group, and that by his definition, it is exactly the same as Γ(2). However, from the literature that I can find, the theta group (of degree 1) is all defined to be the group of 2 by 2 matrices in SL(2, Z) with $$ac\equiv bd \equiv 0 \mod 2$$. That is, a 2 by 2 matrix in SL(2, Z) is in the theta group if and only if either the diagonal entries are even and off-diagonal entries are odd, or the off-diagonal entries are odd and diagonal entries even. Dapengzhang0 (talk) 03:07, 14 October 2014 (UTC)
 * Eichler, Martin. Introduction to the Theory of Algebraic Numbers and Fuctions. Vol. 23. Academic Press, 1966, pp. 36–39.
 * Petersson, Hans. "Über die Eisensteinschen Reihen der Thetagruppe." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 31. No. 3. Springer Berlin/Heidelberg, 1967
 * Friedberg, Solomon. "Theta function transformation formulas and the Weil representation." Journal of Number Theory 20.2 (1985): 121-127

Genus zero?
Reading the article, I see the following:


 * The normalizer Γ0(p)+ of Γ0(p) in SL(2,R) has been investigated; one result from the 1970s, due to Jean-Pierre Serre, Andrew Ogg and John G. Thompson is that the corresponding modular curve (the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+) has genus zero (the modular curve is an elliptic curve) if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71.

Similar text also appears at monstrous moonshine. Should this be "genus one"? Or is "genus" being used in some non-obvious sense here? (If so, it should be clarified.)  Sławomir Biały  (talk) 21:31, 16 April 2015 (UTC)

Rewrite
I expanded the article to make it more structured and add features of the congruence subgroups. There are some topics I did not write about but I think should be included at some point: jraimbau (talk) 08:17, 17 August 2016 (UTC)
 * congruence subgroups in solvable groups (there is some material on this in Sury's book);
 * congruence subgroups in some other families of discrete groups, in particular non-arithmetic lattices, mapping class group of a surface, Out(Fn).

Puzzling definition
The section Congruence groups and adèle groups begins as follows:

"''The ring of adeles $$\mathbb A$$ is the restricted product of all completions of $$\Q,$$ i.e.


 * $$\mathbb A = \R \times \prod_p' \Q _p$$

where the product is over all primes and $$\Q _p$$ is the field of p-adic numbers.''"

But the linked article "Restricted product" does not explain how this product should be restricted.

My guess: For each element of the product, for all but finitely many factors among $$\mathbb \R$$ and the $$\mathbb \Q _p$$, the coordinate of an element of the product is an algebraic integer of the field it belongs to.

Is this correct? Whatever the case, I hope that someone knowledgeable about this subject can fill in the correct definition here. 2601:200:C000:1A0:3DE4:B577:9B28:A000 (talk) 17:21, 5 November 2021 (UTC)


 * That's correct, as it does not take much space I added this explicit definition. It's also spelled out (in greater detail and with additional explanations) in the article about adèles linked there. jraimbau (talk) 09:06, 6 November 2021 (UTC)