Coverage for local/lib/python2.7/site-packages/sage/modular/buzzard.py : 67%

""" 

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 <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# http://www.gnu.org/licenses/ 

############################################################################# 

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. 

EXAMPLES:: 

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