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.

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