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
sage.modular.buzzard.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 GP/PARI scripts
  • William Stein (2006-03-17): small Sage wrapper of Buzzard’s scripts
sage.modular.buzzard.gp()

Return a copy of the GP interpreter with the appropriate files loaded.

EXAMPLE:

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

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