The relation between two terminology cusp (group & algebraic curve)

[Fangyang Tian]

The relation between two terminology "cusp" (group & algebraic curve)

Dear Folks:
I come across the word "cusp" in two different fields and I think they are related. Could anyone specify their relationship for me?? Many thanks!
the cusp of an algebraic curve: for example: (0,0) is the cusp of the complex algebraic curve y²=x³;
the cusp point of a discrete group of SL(2,R) , where SL(2,R) acts on the upper half plane by linear fractional transformation. This terminology usually appears when we talk about modular forms.

 

[mathwonk]

I do not know the answer but I will guess. One passes from a discrete group acting on the upper half plane, to a Riemann surface, by making a quotient of the half plane by the group action. This quotient however is not compact, and to render it compact one must add in some points at infinity which seem to be called cusps. Thus in some sense, the cusps coming from the theory of modular forms do correspond to isolated points on a Riemann surface or algebraic curve. What I do not know is whether there is also some natural way to render that quotient compact by adding in those points as if they were actually cusps in the sense of algebraic geometry, i.e. singularities resembling ones defined by equations like y^n = x^m.