A prime geodesic theorem for \(\mathrm{SL}_3(\mathbb{Z})\) (Q2335853)

The Lie group \(G=\mathrm{SL}_3(\mathbb{R})\) is of split rank 2, with associated symmetric space \(X=G/K\) for \(K=\mathrm{SO}_3(\mathbb{R})\). The discrete subgroup \(\Gamma=\mathrm{SL}_3(\mathbb{Z})\) is a cofinite, non cocompact, lattice in \(G\) and determines the locally symmetric space \(X_\Gamma=\Gamma\backslash X\). The authors recall some properties of the closed geodesics in \(X_\Gamma\), geodesics whose asymptotic length distribution is at the heart of this paper. Every geodesic loop is represented by a conjugacy class \([\gamma]\) in \(\mathrm{SL}_3(\mathbb{Z})\): up to a conjugacy in \(\mathrm{SL}_3(\mathbb{R})\), the element \(\gamma\) is reduced to an element \(a_\gamma\) in a Weyl chamber of the split Cartan subgroup \(A_{\mathrm{sp}}\) of diagonal matrices with positive entries: only regular closed geodesics will be considered, i.e., the element \(a_\gamma\) is in the interior of the Weyl chamber. A closed geodesic \([\gamma]\) lies in a unique 2d flat sub-manifold of \(\Gamma\backslash X\) with volume \(\lambda_\gamma\). Let us denote \(\mathcal{E}_\Gamma(T_1,T_2)\) the set of geodesic loops in \(\Gamma\backslash X\) such that \(a_\gamma^{\alpha-\beta}\le T_1\) and \(a_\gamma^{2\beta}\le T_2\) where \(\alpha,\beta\) are well chosen root vectors. Then the first prime geodesic theorem [PGT] of this paper proves the asymptotics \[ \sum_{[\gamma]\in\mathcal{E}_\Gamma(T_1,T_2)} \lambda_\gamma \sim T_1T_2,\quad T_1,T_2\to +\infty. \] Similar PGT have been proved in the rank one cofinite case along the asymptotic \(\sum_{\gamma:\lambda_\gamma\le T}\lambda_\gamma\sim T\). In higher rank, the cocompact case has been established by the first author in [Geom. Funct. Anal. 14, No. 6, 1238--1266 (2004; Zbl 1102.11028)] and [Rocky Mt. J. Math. 39, No. 2, 485--507 (2009; Zbl 1271.22007)]. This asymptotic gives the first PGT in rank at least 2 for a non-cocompact lattice. Using the correspondence between geodesics in \(X_\Gamma\) and orders \(\mathcal{O}\) in real cubic extensions (whose real embedding \((\rho_1,\rho_2,\rho_3)\) is ordered such that \(|\rho_1(\lambda)|\ge|\rho_2(\lambda)|\ge|\rho_3(\lambda)|\)), the authors prove an arithmetic PGT for orders of totally real cubic extensions, as a corollary of the first PGT: \[ \sum_{\substack{\mathcal{O}\in O_{\mathbb R}(3), \lambda\in\mathcal{O}^{\times}/\pm1, \\ \lambda\in \mathcal{C}_{\mathcal{O}}(T_1,T_2)}} R(\mathcal{O})h(\mathcal{O}) \sim \frac{16}{\sqrt3}T_1T_2,\quad T_1,T_2\to +\infty \] Here the order \(\mathcal O\) has class number \(h(\mathcal{O})\) and regulator \(R(\mathcal{O})\) while the unit \(\lambda\in\mathcal{O}^{\times}\) lies in the set \(\mathcal{C}_{\mathcal{O}}(T_1,T_2)=\{\lambda\in\mathcal{O}|1\alpha_1(\lambda)\le T_1, 1\alpha_2(\lambda)\le T_2\}\), where \(\alpha_1\) is defined by \(\alpha_1=|\rho_1\rho_3\rho_2^{-2}|\) and \(\alpha_2\) by \(\alpha_2=|\rho_2\rho_3^{-1}|^2\). We can say that this asymptotic is in a arithmetic PGT family, whose first example is due to \textit{P. Sarnak} [J. Number Theory 15, 229--247 (1982; Zbl 0499.10021)]: \(\sum_{R(\mathcal O)\le T}h(\mathcal O)\sim e^{2T}/(2T)\), the sum ranging over all real quadratic orders. The proof of the first PGT is based on the Arthur's adelic trace formula applied to pseudo-cusp forms. The introduction of these specific test functions null on the walls of the Weyl chambers causes the cancellation of some parabolic orbital integrals in the geometric side of the trace formula. The adapted trace formula introduces a Dirichlet series meromorphic function in 2 variables, an analytic continuation of the logarithmic derivative of a Selberg-like zeta function. Application of a well designed tauberian Wiener-Ikehara theorem in 2 variables gives the final argument. The authors conjecture that similar results hold for other (non arithmetic) lattices in \(\mathrm{SL}_3(\mathbb Z)\) or lattices of \(G\) of split rank at least 3 (which has very peculiar cancellations). They introduce a conjectural Lefschetz formula which would permit a proof for the general case. The valid cases of cocompact lattices (as done by the first author in [J. Fixed Point Theory Appl. 2, No. 1, 1--40 (2007; Zbl 1205.58014)]) or lattices in split rank one (e.g., \(\mathrm{SL}_2(\mathbb{Z})\), case checked in the last section of this paper) support the conjecture.