For a given prime p, integral weight k, tame level N, finite prime-to-p order character \chi, and a constant s, consider the overconvergent p-adic modular forms of weight k, tame level N, character \chi, which are Hecke eigenforms of slope (valuation of the U_p-eigenvalue) at most s. This is a finite set; some of its members may correspond to classical modular forms.
Data
For such a form, we would like to list:
Slope and U_p-eigenvalue
Fourier coefficients (to some p-adic accuracy)
- Is this form classical? (If so, link to the corresponding entry.)
p-adic Hodge theoretic data (e.g., the crystalline period)
Link to mod p info
The slope, U_p-eigenvalue, and Fourier coefficients should be computable using the method of overconvergent modular symbols, introduced by Stevens.
Wish list
We're not sure how to treat forms of infinite slope, but we definitely want to consider at least classical forms of infinite slope.
There are expected to be analogous constructions for some higher rank groups.
References
Unfortunately, Stevens has not yet written up the theory of overconvergent modular symbols. Try the Darmon-Pollack paper instead. See also Darmon. Trace formula can be used for a lot of calculations too (Koike).
For an application of overconvergent modular symbols to computing Stark-Heegner points (a p-adic analogue of the Heegner point construction over real quadratic fields): H. Darmon and R. Pollack, The efficient calculation of Stark-Heegner points via overconvergent modular symbols, currently on Darmon's web page (dvi, ps), to appear in Israel Journal of Mathematics. They also provide associated software.