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)[source]¶
- 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: from sage.modular.buzzard import buzzard_tpslopes sage: c = buzzard_tpslopes(2,1,50) ... sage: c[50] [4, 8, 13] - >>> 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] - 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] - >>> 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] - AUTHORS: - Kevin Buzzard: several PARI/GP scripts 
- William Stein (2006-03-17): small Sage wrapper of Buzzard’s scripts