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.

Anexos:

• thesis.pdf