I am familiar with the classical thoery of modular forms.
I've heard that a modular form on gives rise to a function on $SL_2(\mathbb{Q}_p)$ (or maybe $GL_2(\mathbb{Q}_p)$, or even $GL_2(\mathbb{A})$ - matrices over the adeles).
I'm confused, and googling about it seems to suggest a relation to Strong Approximation (which I'm not familiar with). Maybe it's stronger than what I need, because I'm curious mainly about getting a function on $SL_2(\mathbb{Q}_p)$ and not the adeles, but I can always restrict such map on adelic matrices.
Given a modular (or even cusp) form $f$, what is the corresponding function on $SL_2(\mathbb{Q}_p)$. What does it remember about $f$ and what does it forget? Is it left/right invariant under some subgroup of $SL_2(\mathbb{Q}_p)$?