Slopes of modular forms
Fredholm series for
Up (data and code)
We implement a formula for the characteristic series of Up acting on
spaces of overconvergent cuspforms as a two-variable power series over the
p-adic integers. We used this implementation to produce the examples in
the paper "Arithmetic properties of Fredholm series for p-adic modular
forms". A README is included describing those examples. Also included is a
pre-computed sage object containing the first 20 (!) coefficients of the
characteristic series U2 acting in tame level one.
Code: The below SAGE code allows one to compute the ghost series, ghost
slopes at all weights (even in characteristic p) and to draw
pictures of the ghost spectral halo. The top of the code contains a brief explanation of the main functions and how
they are used.
SAGE code here
Code: The below SAGE code allows one to compute with abstract ghost series and in particular
with ghost series attached to a fixed mod p representation. See the sample run for an explicit
Ghost zeroes vs. true zeros: The below pdf files give tables
which compare the
location of the true zeroes of coefficients of the Fredholm series of
Up to that of the ghost zeroes. From these tables one can
see how strikely close the zeroes match. Further, the multiplicity
pattern of the ghost zeroes is explained by this data.
p=2 and N=1
p=3 and N=1
p=5 and N=1
Click here for a list of the slopes
of the Hecke operator T59 acting on cuspforms of level one,
in even weights between 2 and 1640.
Compare with the data on Gouvêa's page
Click here for this data in a SAGE readable format.
(The format for our file is (k,list) and list is another list containing pairs
[a,ma] meaning the slope a appear ma-many times.)
Code: The below MAGMA code allows one to compute the slopes of forms with a fixed residual
representation. See the sample run for an explicit example.
MAGMA code here