| Résume | The notion of $D$-elliptic sheaf is a generalization of thenotion of Drinfeld module. $D$-elliptic sheaves and their modulischemes were introduced by Laumon, Rapoport and Stuhler in theirproof of certain cases of Langlands conjecture over function fields.We discuss basic arithmetic properties of modular curves of$D$-elliptic sheaves and draw parallels with the theory of Shimuracurves. In particular, we produce a genus formula for modular curvesof $D$-elliptic sheaves, examine the existence of rational points onthese curve, compute their fundamental domains in Bruhat-Tits trees,and determine the cases when these curves are hyperelliptic. Asapplications of previous results, we construct new asymptoticallyoptimal sequences of curves over finite fields (such sequences areimportant in coding theory), and find presentations for certainarithmetic groups arising from quaternion algebras over functionfields. |
