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)]

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    $\begingroup$ 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. $\endgroup$ – Kevin Ventullo Jan 16 '14 at 20:26
  • $\begingroup$ 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... $\endgroup$ – stl Jan 16 '14 at 20:44

It depends rather what you mean by "compute". These are very big spaces (infinite-dimensional p-adic Banach spaces, and without a good theory of Hecke eigenforms). Can you be a bit more specific what it is you want to compute about them?

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).

(I can't help thinking that you started with a question about a space too big to be interesting, and edited it to a question about a subspace too small to be interesting; there is a very interesting question in between the two which you have whizzed past, which is to compute overconvergent weight 2 eigenforms of small but non-zero slope.)

  • $\begingroup$ Ah, okay. What about just the subspace of $p$-ordinary forms? $\endgroup$ – stl Jan 16 '14 at 17:33
  • $\begingroup$ I edited my reply in response to your edited question. $\endgroup$ – David Loeffler Jan 17 '14 at 9:08

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