The Arithmetic Module on the Singular Local Ring

Singular projector

Using the exact idempotent

e_sing(T_2) = h_other(T_2) * h_other(T_2)^{-1} mod h_sing(T_2)

the integral modular-symbol lattice projects to a rank-3 Z_3-lattice

L_sing = e_sing * Z_3^13.

A basis is obtained from the projected standard basis columns [0, 1, 2].

Main result

The singular arithmetic module is cyclic over the singular Hecke order.

A generator is the projected basis vector with mod-3 coordinates [1, 0, 0] in the chosen basis. The determinant of the spanning matrix

[ w , T_3 w , T_11 w ]

is a 3-adic unit (valuation 0), so

L_sing ≅ O_sing

as Z_3-lattices with Hecke action.

So the arithmetic chooses the regular module of the singular order itself, not the conductor or the maximal ideal.

Consequence for the local factor

Since O_sing ≅ Z_3 × R_loc and the singular module is free rank 1 over O_sing, the local part chosen by arithmetic is the free rank-1 R_loc-module. The node is therefore not just a ring defect: the cusp-form lattice realizes the defect in its regular representation.