# On equality of two quotients of a congruence subgroup

Related question: Non-torsion part of the abelianisation of congruence subgroups

Let $$A = \mathbb{F}_q[T]$$ be the ring of polynomials with coefficients in a finite field, with $$N$$ a nonconstant element of $$A$$, and let $$\Gamma(N)$$ be the group of matrices in $$GL_2(A)$$ which are congruent to the identity matrix modulo $$N$$ (elementwise).

In section 5.4 of Jacobians of Drinfeld Modular Forms, the authors consider the quotient $$G = \Gamma(N)^{ab}/torsion$$, the torsion-free abelianisation of $$\Gamma(N)$$, which is a finitely generated free abelian group (they don't call it $$G$$, but for simplicity I am doing so here). Moreover, if one considers the logarithmic derivatives of the theta functions defined there, they live more uniquely on $$G/pG$$, where $$p$$ is the characteristic of $$\mathbb{F}_q$$, which is isomorphic to $$(\mathbb{Z}/p\mathbb{Z})^k$$ for some positive integer $$k$$.

On the other hand, the quotient $$\Gamma(N)/\Gamma(N^2)$$ is also isomorphic to $$(\mathbb{Z}/p\mathbb{Z})^l$$ for some positive integer $$l$$. Are these quotients $$G/pG$$ and $$\Gamma(N)/\Gamma(N^2)$$ the same? I.e. are $$k$$ and $$l$$ equal?