Preprint C079/2009
On Supersingular Curves over Finite Fields

Saeed Tafazolian

**Keywords: **
finite field | Hasse-Weil bound | classical curves the genus of maximal curves

In this work we will discuss on minimal and maximal curves
over a finite field $k$. Our method is to consider the curve over
$\bar{k}$, the algebraic closure of $k$, and look at some
invariants of the curve which are unchanged with respect to
constant field extensions. For example, the $p$-adic Newton
polygon, the Hasse-Witt matrix and the $p$-rank of the curve.
Using these arguments, we characterize some classical maximal and
minimal curves, such as Fermat curves, Artin-Schreier curves and
also hyperelliptic curves.