turkmath.org

By a curve we mean a smooth geometrically irreducible projective curve. Explicit curves (i.e., curves given by explicit equations) over finite fields with many rational points with respect to their genera have attracted a lot of attention, after Goppa discovered that they can be used to construct good linear error-correcting codes. For the number of $\mathbb { F } _ { q }$ -rational points on the curve C of genus g(C) over Fq we have the following bound $\# \mathrm { C } \left( \mathbb { F } _ { q } \right) \leq 1 + q + 2 \sqrt { q } g ( \mathrm { C } )$ which is well-known as the Hasse-Weil bound. This is a deep result due to Hasse for elliptic curves, and for general curves is due to A. Weil. When the cardinality of the finite field is square, a curve C over $ \mathbb { F } _ { q ^ { 2 } } $ is called maximal if it attains the Hasse-Weil bound, i.e., if we have the equality $\# \mathrm { C } \left( \mathbb { F } _ { q ^ { 2 } } \right) = 1 + q ^ { 2 } + 2 q g ( \mathrm { C } )$ We introduce some geometric properties of curves with many rational points to classify certain maximal curves.