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We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for$n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimalcovolume. More precisely, let $\mu$ be the Euler-Poincar\'e measure on$PSL_2(\mathbb R)^n$ and $\chi=\mu/2^n$. We show that the Hilbert modular group$PSL_2(\mathfrak o_{k_{49}})\subset PSL_2(\mathbb R)^3$, with $k_{49}$ thetotally real cubic field of discriminant $49$ has the minimal covolume withrespect to $\chi$ among all irreducible lattices in $PSL_2(\mathbb R)^n$ for$n\geq 2$ and is unique such lattice up to conjugation. The uniform lattice ofminimal covolume with respect to $\chi$ is the normalizer $\Delta_{k_{725}}^u$of the norm-1 group of a maximal order in the quaternion algebra over theunique totally real quartic field with discriminant $725$ ramified exactly attwo infinite places, which is a lattice in $PSL_2(\mathbb R)^2$. There isexactly one more lattice in $PSL_2(\mathbb R)^2$ and exactly one in$PSL_2(\mathbb R)^4$ with the same covolume as $\Delta_{k_{725}}^u$, which arethe Hilbert modular groups corresponding to $\mathbb Q(\sqrt{5})$ and$k_{725}$. The two lattices $\Delta_{k_{725}}^u$ and $PSL_2(\mathfrako_{\mathbb Q(\sqrt{5})})$ have the smallest covolume with respect to theEuler-Poincar\'e measure among all arithmetic lattices in $G_n$ for all $n\geq2$. These results are in analogy with Siegel's theorem on the unique minimalcovolume (uniform and non-uniform) Fuchsian groups and its generalizations tovarious higher dimensional hyperbolic spaces due to Belolipetsky,Belolipetsky-Emery, Stover and Emery-Stover.

Lower bounds for covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$