# computing spaces of $p$-adic modular forms

Let $p$ be a prime, and $\alpha$ a positive integer. How do you compute the space of $p$-ordinary $p$-adic modular forms (in the sense of Serre) of weight 2 on $\Gamma_0(p^\alpha)$? I'm really only concerned with the cases where $\Gamma_0(p^\alpha)$ is genus zero and $\alpha = 1, 2$.

[Edit: Changed from space of $p$-adic modular forms (which is really big) to the subspace of $p$-ordinary forms (which I think should be finite dimensional)]

• If you're only looking at $p$-ordinary forms, this is the same as the space of classical $p$-ordinary modular forms by work of Hida. Jan 16, 2014 at 20:26
• Thanks! That's exactly what I wanted. I've looked at Hida's book, but it's not exactly easy to read for a beginning grad student...
– stl
Jan 16, 2014 at 20:44

EDIT: You've edited your question to focus attention on the ordinary cusp forms instead. But in weight 2 these are just the p-ordinary classical forms of level $\Gamma_0(p^{\beta})$, where $\beta = \max(1, \alpha)$, and there are extremely efficient algorithms for computing classical weight 2 modular forms (e.g. using modular symbols).
• Ah, okay. What about just the subspace of $p$-ordinary forms?