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?
