While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational numbers? I am talking about how cusps are defined. And in particular, if $f$ is modular form, what does $f(q)$ (where $q \in \mathbb{Q}$) mean?
2026-03-31 04:05:34.1774929934
Why are rational numbers required in cusps of congruence subgroups?
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