We construct curves over finite fields with properties similar to those of classical elliptic or Drinfeld modular curves (as far as elliptic points, cusps, ramification, ... are concerned), but whose coverings have Galois groups of type $\mathbf{GL}(r)$ over finite rings $(r\ge 3)$ instead of $\mathbf{GL}(2)$. In the case where the finite field is non-prime, there results an abundance of series or towers with a large ratio "number of rational points/genus". The construction relies on higher-rank Drinfeld modular varieties and the supersingular trick and uses mainly rigid- analytic techniques.[-]