I read on this Wikipedia page (about projective lines over rings) that "Similarly, a homography of $P(\mathbb{Q})$ corresponds to an element of the modular group, the automorphisms of $P(\mathbb{Z})$." Now, a homography of the projective line $P(\mathbb{Q})$ is an element of $\mathbf{PGL}_2(\mathbb{Q})$ (invertible $(2 \times 2)$-matrices over $\mathbb{Q}$, modulo scalars), while elements of the modular group are elements of $\mathbf{SL}_2(\mathbb{Z})$ ($(2 \times 2)$-matrices over the integers with determinant $1$).
Is this correct ?
As homographies are determined up to a scalar, their determinants are determined up to a square of an integer (well, actually of a rational number), so how can this statement work ?
This line in the Wikipedia article is incorrect and I have deleted it. You are correct that $PGL_2(\mathbb{Q})$ is the correct group of homographies and is strictly larger than the modular group $PSL_2(\mathbb{Z})$, e.g. the homography $x \mapsto 2x$ corresponding to the matrix $\left[ \begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array} \right]$ is not in $PSL_2(\mathbb{Z})$ because its determinant is not a square.
This whole Wikipedia article is strange. It feels like it was written by one person with idiosyncratic tastes and indeed if you scroll down to the external links you can find out who that person likely was...