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 ($0\le k\le k_{max}$) 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 ${\mathrm{\Gamma }}_0(N)$-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