2
votes
1answer
43 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
5
votes
2answers
108 views

Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$?

In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic ...
1
vote
1answer
96 views

Cusps for higher dimensional hyperbolic spaces

Take your favourite model of the hyperbolic plane $\mathfrak h$. It is well known how to define cusps, and we know those are $\mathbb Q \cup \{ \infty \}$. Now pick a higher dimensional hyperbolic ...
0
votes
1answer
229 views

solving this laplacian

given the laplacian $ -y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y) $ can we find a solution in the form $ f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$ if we impose the extra ...
0
votes
1answer
234 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
0
votes
0answers
90 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
2
votes
1answer
114 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...