Conjectural slopes of Hecke polynomials

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

sage.modular.buzzard.buzzard_tpslopes(p, N, kmax)

Return 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

sage: from sage.modular.buzzard import buzzard_tpslopes
sage: c = buzzard_tpslopes(2,1,50)
...
sage: c[50]
[4, 8, 13]

Python

>>> from sage.all import *
>>> from sage.modular.buzzard import buzzard_tpslopes
>>> c = buzzard_tpslopes(Integer(2),Integer(1),Integer(50))
...
>>> c[Integer(50)]
[4, 8, 13]

Sage Live

from sage.modular.buzzard import buzzard_tpslopes
c = buzzard_tpslopes(2,1,50)
c[50]

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

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]

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(1),Integer(50), sign=Integer(1)).cuspidal_submodule()
>>> T = M.hecke_operator(Integer(2))
>>> f = T.charpoly('x')
>>> f.newton_slopes(Integer(2))
[13, 8, 4]

Sage Live

M = ModularSymbols(1,50, sign=1).cuspidal_submodule()
T = M.hecke_operator(2)
f = T.charpoly('x')
f.newton_slopes(2)

AUTHORS:

• Kevin Buzzard: several PARI/GP scripts

• William Stein (2006-03-17): small Sage wrapper of Buzzard’s scripts

Make live