Method of Graphs¶

The Mestre Method of Graphs is an intriguing algorithm for computing the action of Hecke operators on yet another module \(X\) that is isomorphic to \(M_2(\Gamma_0(N))\). The implementation in Sage unfortunately only works when \(N\) is prime; in contrast, my implementation in Magma works when \(N=pM\) and \(S_2(\Gamma_0(M))=0\).

The matrices of Hecke operators on \(X\) are vastly sparser than on any basis of \(M_2(\Gamma_0(N))\) that you are likely to use.

Sage

sage: X = SupersingularModule(389); X
Module of supersingular points on X_0(1)/F_389 over Integer Ring
sage: t2_mestre = X.T(2).matrix()
sage: p_mestre = t2_mestre.charpoly()
sage: factor(p_mestre)
(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * ...

sage: t2 = ModularSymbols(389,sign=1).hecke_matrix(2)
sage: p = t2.charpoly()
sage: factor(p)
(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * ...

sage: p_mestre == p
True

Python

>>> from sage.all import *
>>> X = SupersingularModule(Integer(389)); X
Module of supersingular points on X_0(1)/F_389 over Integer Ring
>>> t2_mestre = X.T(Integer(2)).matrix()
>>> p_mestre = t2_mestre.charpoly()
>>> factor(p_mestre)
(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * ...

>>> t2 = ModularSymbols(Integer(389),sign=Integer(1)).hecke_matrix(Integer(2))
>>> p = t2.charpoly()
>>> factor(p)
(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * ...

>>> p_mestre == p
True

Sage Live

X = SupersingularModule(389); X
t2_mestre = X.T(2).matrix()
p_mestre = t2_mestre.charpoly()
factor(p_mestre)
t2 = ModularSymbols(389,sign=1).hecke_matrix(2)
p = t2.charpoly()
factor(p)
p_mestre == p

The method of graphs is also used in computer science to construct expander graphs with good properties. And it is important in my algorithm for computing Tamagawa numbers of purely toric modular abelian varieties. This algorithm is not implemented in Sage yet, since it is only interesting in the case of non-prime level, as it turns out.

REFERENCE:

Jean-François Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), 217-242