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 \[ {\mathcal R}(\gamma)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ \gamma=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\|\gamma\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq X$ 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 $N(X)=\#\{\gamma\in{\mathrm{SL}}_2({\mathbb{Z}}):\|\gamma\|_\infty\leq X\}$ 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 $\|\gamma\|_\infty\le X$, where $c>0$ is an absolute constant.