On page 20 of Merel’s paper, he proves that Hecke operators can be computed using any element ∑_{M} a_{M}[M] ∈ Z[M_{2}(Z)_{p}] that satisfies the following condition:

For all classes K ∈ M_{2}(Z)_{p}/ SL_{2}(Z), we have the following equality in Z[P^{1}(Q)]:

∑_{M∈K} a_{M}([M(∞)] − [M(0)]) = [∞] − [0].

I propose to replace this condition by a condition that allows computation of the Hecke operators on modular symbols of level N.

Condition: An element ∑_{M} a_{M}[M] ∈ Z[M_{2}(Z/NZ)_{p}] can be used to compute the Hecke operators on M_{2}(Γ_{1}(N)) if it satisfies the following condition:

For all classes K ∈ M_{2}(Z)_{p}/ SL_{2}(Z), we have the following equality in Z[P^{1}(Z/NZ)]:

∑_{M∈\overline{K}} a_{M}([M(∞)] − [M(0)]) = [∞] − [0],

where \overline{K} is the image of K in M_{2}(Z/NZ). (We view K as a subset of M_{2}(Z).)

Remark 0.1. The set of coset M_{2}(Z)_{p}/ SL_{2}(Z) is finite. Representatives are

( \begin{matrix}1 & r \\ 0 & p \end{matrix} ) for 0 ≤ r < p and ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ).

Even if this is right, there are p + 1 linear conditions, so maybe it is still too difficult to solve. If the solution to the linear conditions were unique, then there is no problem, because we just add conditions until the dimension of the solution space is small enough. A second problem is that the definition only seems to depend on p modulo N, which is clearly totally wrong.