Questions tagged [modular-group]

is there a fourier series analogue to the modular group ?=? $$ z\mapsto\frac{az+b}{cz+d},$$ for example we could expand any function that satisfies the modular equation $$ f(\frac{az+b}{cz+d}) =f(z) $$ in terms of simpler function which are modular…

I‘m working on an exercise of a book over modular forms (Henri Cohen, Modular Forms, a classical approach) and I’m confused by the frasing of the question: Show that the map $\gamma \mapsto \gamma$i induces a bijection from…

The principal congruence subgroup $\Gamma(n)$ of $SL(2,\mathbb Z)$ is a normal subgroup of $SL(2,\mathbb Z)$ with index $$[SL(2,\mathbb Z):\Gamma(n)]=n^3\prod_{p|n}(1-\tfrac{1}{p^2}),$$ where the product is over all primes dividing $n$. The quotient…

Wikipedia states that the boundary of any fundamental domain should have some restrictions, such as smoothness of polyhedrality. Further, this thesis on Page 8, imposes the condition that the boundary should be piecewise smooth. My question is why…