  We consider a three-dimensional lattice Abelian Higgs gauge model for a
charged $N$-component scalar field ${\phi}$, which is invariant under $SO(N)$
global transformations for generic values of the parameters. We focus on the
strong-coupling regime, in which the kinetic Hamiltonian term for the gauge
field is a small perturbation, which is irrelevant for the critical behavior.
The Hamiltonian depends on a parameter $v$ which determines the global symmetry
of the model and the symmetry of the low-temperature phases. We present
renormalization-group predictions, based on a Landau-Ginzburg-Wilson effective
description that relies on the identification of the appropriate order
parameter and on the symmetry-breaking patterns that occur at the
strong-coupling phase transitions. For $v=0$, the global symmetry group of the
model is $SU(N)$; the corresponding model may undergo continuous transitions
only for $N=2$. For $v\not=0$, i.e., in the $SO(N)$ symmetric case, continuous
transitions (in the Heisenberg universality class) are possible also for $N=3$
and 4. We perform Monte Carlo simulations for $N=2,3,4,6$, to verify the
renormalization-group predictions. Finite-size scaling analyses of the
numerical data are in full agreement.
