On curves covered by the Hermitian curve. (Q1293103)

A maximal curve over the finite field \(\mathbb F_q\) is a projective geometrically irreducible nonsingular algebraic curve defined over \(\mathbb F_q\), whose number of \(\mathbb F_q\)-rational points attains the Hasse-Weil upper bound. Such curves play an important role in coding theory. A well-known example of such a curve is the Hermitian curve \(\mathcal H\) defined by \(Y^{\sqrt q}Z+YZ^{\sqrt q}=X^{\sqrt q+1}\). (The authors should assume that \(q\) is an even power of a prime). By a result of \textit{G. Lachaud} [C. R. Acad. Sci. Paris Sér. I 305, 729--732 (1987; Zbl 0639.14013)], any nonsingular curve \(\mathbb F_q\)-covered by \(\mathcal H\) is also maximal. In the paper under review, the authors present a new infinite class of maximal curves \(\mathbb F_q\)-covered by \(\mathcal H\), which arises from a cyclic automorphism group of \(\mathcal H\) of order \(q-\sqrt q+1\) and contains a curve of genus \(\frac 12(\frac{q-\sqrt q+1}d-1)\) for each proper divisor \(d\) of \(q-\sqrt q+1\). These curves are nonclassical (for the canonical morphism) whenever \(d\leq(q-2\sqrt q+1)/(2\sqrt q-1)\). The results in the paper are also related to recent investigations on maximal curves having many rational points for small values of \(d\), especially for \(d=3\). The authors use Stöhr-Voloch's approach to the Hasse-Weil bound to carry their research.