  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 minimal
covolume. 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 $k_{49}$ the
totally real cubic field of discriminant $49$ has the minimal covolume with
respect 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 of
minimal 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 the
unique totally real quartic field with discriminant $725$ ramified exactly at
two infinite places, which is a lattice in $PSL_2(\mathbb R)^2$. There is
exactly 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 are
the Hilbert modular groups corresponding to $\mathbb Q(\sqrt{5})$ and
$k_{725}$. The two lattices $\Delta_{k_{725}}^u$ and $PSL_2(\mathfrak
o_{\mathbb Q(\sqrt{5})})$ have the smallest covolume with respect to the
Euler-Poincar\'e measure among all arithmetic lattices in $G_n$ for all $n\geq
2$. These results are in analogy with Siegel's theorem on the unique minimal
covolume (uniform and non-uniform) Fuchsian groups and its generalizations to
various higher dimensional hyperbolic spaces due to Belolipetsky,
Belolipetsky-Emery, Stover and Emery-Stover.
