Encyclopedia topics
For which groups does the p-adic L-function exist?
Iwasawa main conjecture (relationship between p-adic L-function and Selmer groups); what is known in various cases?
- Mazur-Tate-Teitelbaum conjecture
- Constructions of the L-invariant (Coleman, Colmez, Teitelbaum)
A table we'd like
See what we can compute about a p-adic L-function associated to a classical weight-2 newform over \mathbb{Q}. (Possibly this should be attached to the entry for that form.)
- prime
Fourier coefficients (truncated in p and T)
- order of vanishing at the critical point
- first non-zero coefficient of the expansion around the critical point
algebraic/transcendental part (p-adic regulator)
What exists
All of the data can be computed in SAGE in the case of good ordinary reduction. For the Fourier coefficients and related data, the dependence on conductor is not good. However, we think we can tabulate p-adic regulators for p = 5,7,11,13,17,19 for elliptic curves of conductor \leq 100000.
Also, here's a list of what we've found online:
Gouvea has a collection of data about slopes for classical forms: Gouvea's tables
Lloyd Kilford has some code to compute U_p and T_\ell for quaternionic p-adic automorphic forms; we should be able to get some valuable data out of this (via Jacquet-Langlands).
Coleman, Stevens, and Teitelbaum have a paper called "Numerical experiments on families of p-adic modular forms" in _Computational Perspectives on Number Theory_, Buell & Teitelbaum, eds. This has code for computing a handful of interesting data.
Wish list
What about higher weight, larger fields of definition, or other reduction types? The coefficients should still be computable. The p-adic regulator may need a new method; one possibility is the method of Coleman-Gross, using Coleman integration (see some recent papers of Besser).
Can one use Kato's inequality (the easy direction of Mazur-Tate-Teitelbaum's p-adic BSD conjecture) plus p-adic computations to verify the rank of a typical elliptic curve over \mathbb{Q}?
Is there a p-adic analogue of the Selberg class? (We'd like to have standard parameters for p-adic L-functions, as in the holomorphic case.)
References
For p-adic L-functions, including computations: William Stein's spring 2007 course notes.
For computing p-adic regulators: B. Mazur, W. Stein, and J. Tate, Computation of $p$-adic heights and log convergence, Documenta Mathematica 2006, Extra Vol., 577-614.
For constructing Coleman integrals: R.F. Coleman, Torsion points on curves and $p$-adic abelian integrals, Annals of Math. (2) 121 (1985), 111-168.
For computing Coleman integrals: Kiran Kedlaya's 2007 lecture slides.
For expressing p-adic heights in terms of Coleman integrals: R.F. Coleman and B.H. Gross, p-adic heights on curves, Algebraic number theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, MA, 1989, 73-81. (Besser's 2007 lecture slides do not appear to be online.)