A new family of maximal curves over a finite field.

It has been known for a long time that the Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2,\,^2B_2$ and $^2G_2$ defined over the finite field $\fn$ all have the maximum number of $\fn$-rational points allowed by the Weil ``explicit formulas'', and that these curves are $\fqq$-maximal curves over infinitely many algebraic extensions $\fqq$ of $\fn$. Serre showed that an $\fqq$-rational curve which is $\fqq$-covered by an $\fqq$-maximal curve is also $\fqq$-maximal. This has posed the problem of the existence of $\fqq$-maximal curves other than the Deligne-Lusztig curves and their $\fqq$-subcovers. In this paper, a positive answer to this problem is obtained. For every $q=n^3$ with $n=p^r>2$, $p\geq 2$ prime, we give a simple, explicit construction of an $\fqq$-maximal curve $\cX$ that is not $\fqq$-covered by any $\fqq$-maximal Deligne-Lusztig curve. Interestingly, $\cX$ has a very large $\fqq$-automorphism group with respect to its genus $g=\ha\,(n^3+1)(n^2-2)+1$, and it is the first known example of a curve with $p$-rank zero that has an automorphism group isomorphic to $\SU(3,n)$.