buzzard.py

"""
Conjectural Slopes of Hecke Polynomial

Interface to Kevin Buzzard's PARI program for computing conjectural
slopes of characteristic polynomials of Hecke operators.

AUTHORS:

- William Stein (2006-03-05): Sage interface

- Kevin Buzzard: PARI program that implements underlying functionality
"""

#############################################################################
#       Copyright (C) 2006 William Stein <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#                  http://www.gnu.org/licenses/
#############################################################################

__doc_exclude = ['gp', '_gp', 'Gp', 'Integer', 'sage_eval']

from sage.interfaces.gp import Gp
from sage.misc.all import sage_eval

_gp = None

def gp():
    r"""
    Return a copy of the GP interpreter with the appropriate files loaded.

    EXAMPLE::

        sage: sage.modular.buzzard.gp()
        PARI/GP interpreter
    """

    global _gp
    if _gp is None:
        _gp = Gp(script_subdirectory='buzzard')
        _gp.read("DimensionSk.g")
        _gp.read("genusn.g")
        _gp.read("Tpprog.g")
    return _gp

## def buzzard_dimension_cusp_forms(eps, k):
##     r"""
##     eps is [N, i x 3 matrix], where eps[2][,1] is the primes dividing
##     N, eps[2][,2] is the powers of these primes that divide N, and eps[2][,3]
##     is the following: for p odd, p^n||N, it's t such that znprimroot(p^n)
##     gets sent to exp(2*pi*i/phi(p^n))^t. And for p=2, it's
##     0 for 2^1, it's 0 (trivial) or -1 (non-trivial) for 2^2, and for p^n>=8
##     it's either t>=0 for the even char sending 5 to exp(2*pi*i/p^(n-2))^t,
##     or t<=-1 for the odd char sending 5 to exp(2*pi*i/p^(n-2))^(-1-t).
##     (so either 0<=t<2^(n-2) or -1>=t>-1-2^(n-2) )

##     EXAMPLES:
##         sage: buzzard_dimension_cusp_forms('TrivialCharacter(100)', 4)

##     Next we compute a dimension for the character of level 45 which is
##     the product of the character of level 9 sending znprimroot(9)=2 to
##     $e^{2 \pi i/6}^1$ and the character of level 5 sending
##     \code{znprimroot(5)=2} to $e^{2 \pi i/4}^2=-1$.

##         sage: buzzard_dimension_cusp_forms('DirichletCharacter(45,[1,2])', 4)
##         <boom!>  which is why this is commented out!
##     """
##     s = gp().eval('DimensionCuspForms(%s, %s)'%(eps,k))
##     print s
##     return Integer(s)


def buzzard_tpslopes(p, N, kmax):
    """
    Returns a vector of length kmax, whose `k`'th entry
    (`0 \leq k \leq k_{max}`) is the conjectural sequence
    of valuations of eigenvalues of `T_p` on forms of level
    `N`, weight `k`, and trivial character.
    
    This conjecture is due to Kevin Buzzard, and is only made assuming
    that `p` does not divide `N` and if `p` is
    `\Gamma_0(N)`-regular.
    
    EXAMPLES::
    
        sage: c = buzzard_tpslopes(2,1,50)
        sage: c[50]
        [4, 8, 13]
    
    Hence Buzzard would conjecture that the `2`-adic valuations
    of the eigenvalues of `T_2` on cusp forms of level 1 and
    weight `50` are `[4,8,13]`, which indeed they are,
    as one can verify by an explicit computation using, e.g., modular
    symbols::
    
        sage: M = ModularSymbols(1,50, sign=1).cuspidal_submodule()
        sage: T = M.hecke_operator(2)
        sage: f = T.charpoly('x')
        sage: f.newton_slopes(2)
        [13, 8, 4]
    
    AUTHORS:

    - Kevin Buzzard: several PARI/GP scripts

    - William Stein (2006-03-17): small Sage wrapper of Buzzard's
      scripts
    """
    v = gp().eval('tpslopes(%s, %s, %s)'%(p,N,kmax))
    v = sage_eval(v)
    v.insert(0, [])   # so v[k] = info about weight k (since python is 0-based)
    return v