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Given an [[Algebraic curve|algebraic curve]]  $  X/K $, 
where  $  K $
is a [[Field|field]] of characteristic  $  p > 0 $, 
a [[Covering|covering]]  $  Y \rightarrow X $
is called an Artin–Schreier curve over  $  X $
if the corresponding extension of function fields  $  K ( Y ) /K ( X ) $
is generated by some function  $  y \in K ( Y ) $
such that  $  y  ^ {l} \pm y = f \in K ( X ) $(
where  $  l $
is a power of  $  p $, 
cf. also [[Extension of a field|Extension of a field]]). If  $  K = \mathbf F _ {q} $
is a [[Finite field|finite field]], it turns out that Artin–Schreier curves often have many rational points.

To be precise, let  $  N ( Y ) $(
respectively,  $  g ( Y ) $) 
denote the number of  $  \mathbf F _ {q} $-
rational points (respectively, the genus) of a curve  $  Y/ \mathbf F _ {q} $. 
The Hasse–Weil theorem states that

$$ 
N ( Y ) \leq  q + 1 + 2g ( Y ) \sqrt q .
$$

If the genus is large with respect to  $  q $, 
this bound can be improved as follows. Let  $  ( Y _ {i} ) _ {i \in \mathbf N }  $
be a sequence of curves over  $  \mathbf F _ {q} $
such that  $  g ( Y _ {i} ) \rightarrow \infty $. 
Then

$$ 
{\lim\limits }  \sup  {
\frac{N ( Y _ {i} ) }{g ( Y _ {i} ) }
 } \leq  \sqrt q - 1
$$

(the Drinfel'd–Vladut bound).

Curves over  $  \mathbf F _ {q} $
can be used to construct error-correcting linear codes, so-called geometric Goppa codes or algebraic-geometric codes (cf. [[Error-correcting code|Error-correcting code]]; [[Goppa code|Goppa code]]; [[Algebraic-geometric code|Algebraic-geometric code]]; [[#References|[a4]]], [[#References|[a5]]]). If the curves have sufficiently may rational points, these codes have very good error-correcting properties. Hence, one is interested in explicit constructions of curves with many rational points.

==Examples of Artin–Schreier curves.==
The Hermitian curve over  $  \mathbf F _ {q} $, 
for  $  q = l  ^ {2} $, 
is given by the equation  $  y  ^ {l} + y = x ^ {l + 1 } $. 
It has  $  N = l  ^ {3} + 1 $
rational points and its genus is  $  g = { {l ( l - 1 ) } / 2 } $. 
Hence, for it the Hasse–Weil bound  $  N = q + 1 + 2g \sqrt q $
is attained, see [[#References|[a4]]].

Again, let  $  q = l  ^ {2} $
be a square. Define a tower of function fields  $  F _ {1} \subseteq F _ {2} \subseteq \dots $
over  $  \mathbf F _ {q} $(
cf. [[Tower of fields|Tower of fields]]) by  $  F _ {1} = \mathbf F _ {q} ( x _ {1} ) $, 
$  F _ {n + 1 }  = F _ {n} ( z _ {n} ) $, 
where

$$ 
z _ {n + 1 }   ^ {l} + z _ {n + 1 }  = x _ {n} ^ {l + 1 }  \textrm{ and  }  x _ {n} = {
\frac{z _ {n} }{x _ {n -1 }  }
 }  ( \textrm{ for  }  n \geq 2 ) .
$$

For the corresponding algebraic curves  $  Y _ {1} ,Y _ {2} , \dots $, 
the coverings  $  Y _ {n + 1 }  \rightarrow Y _ {n} $
are Artin–Schreier curves. This sequence  $  ( Y _ {n} ) _ {n \in \mathbf N }  $
attains the Drinfel'd–Vladut bound, i.e.,  $  {\lim\limits } _ {i \rightarrow \infty }  { {N ( Y _ {i} ) } / {g ( Y _ {i} ) } } = l - 1 $(
see [[#References|[a1]]]).

The geometric Goppa codes constructed using these curves  $  Y _ {n} $
beat the Gilbert–Varshamov bound (cf. also [[Error-correcting code|Error-correcting code]]; [[#References|[a3]]]) for all  $  q \geq 49 $. 
This construction is simpler and more explicit than the construction based on modular curves (the Tsfasman–Vladut–Zink theorem, [[#References|[a5]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Garcia, H. Stichtenoth, "A tower of Artin–Schreier extensions of function fields attaining the Drinfeld–Vladut bound"  ''Invent. Math.'' , '''121'''  (1995)  pp. 211–222</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> G. van der Geer, M. van der Vlugt, "Curves over finite fields of characteristic two with many rational points"  ''C.R. Acad. Sci. Paris'' , '''317'''  (1993)  pp. 693–697</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> J.H. van Lint, "Introduction to coding theory" , Springer  (1992)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Stichtenoth, "Algebraic function fields and codes" , Springer  (1993)  {{ISBN|3-540-58469-6}} {{ZBL|0816.14011}}</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> M.A. Tsfasman, S.G. Vladut, "Algebraic geometric codes" , Kluwer Acad. Publ.  (1991)</TD></TR>
</table>

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