Arithmetic structure of $L_2$-norms of ${\mathrm{SL}}_2(\mathbb{Z})$ matrices

Abstract

For a matrix $\gamma \in {\mathrm{SL}}_2(\mathbb{Z})$, we define \[ {\mathcal R}(γ)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ γ=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\Vert \gamma {\Vert }_{\infty }=\max \{|a_1|,|a_2|,|a_3|,|a_4|\}\le X$ and such that $\mathcal{R}(\gamma )$ is squarefree. We prove that \[ S_{\mathrm{sq}}(X) = {\mathfrak S}_{\mathcal R}^{\mathrm{sq}}N(X) +O(X^{19/10+o(1)}), \quad \text{as}\ X\to \infty, \] where $N(X)=\#\{\gamma \in {\mathrm{SL}}_2(\mathbb{Z}):\Vert \gamma {\Vert }_{\infty }\le X\}$ and ${\mathfrak{S}}_{\mathcal{R}}^{\mathrm{sq}}$ is an explicit positive Euler product of local $p^2$-densities. The proof combines the $\delta$-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of $\mathcal{R}(\gamma )$, which, however, is conditional on a very strong form of the Elliott--Halberstam conjecture. We also show that $\mathcal{R}(\gamma )$ is squarefree and has at most $9$ prime divisors for at least $cN(X)/\log X$ matrices $\gamma \in {\mathrm{SL}}_2(\mathbb{Z})$ with $\Vert \gamma {\Vert }_{\infty }\le X$, where $c>0$ is an absolute constant.