turkmath.org

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and by $K$ its algebraic closure. By a plane curve over $\mathbb{F}_q$ (more precisely: an irreducible affine plane curve over $\mathbb{F}_q$) we mean the set $${\cal C}=\{P=(a,b)\, |\, a,b\in K \, \text{and} \, f(a,b)=0\},$$ where $f(x,y)\in \mathbb{F}_q[x,y]$ is a given irreducible polynomial with coefficients in $\mathbb{F}_q$. We are interested in the set of $\mathbb{F}_q$- rational points of ${\cal C}$, $${\cal C }(\mathbb{F}_q)=\{(a,b)\in {\cal C} \, |\, a,b\in \mathbb{F}_q\}$$ and in particular its cardinality $$N({\cal C})=\#{\cal C}(\mathbb{F}_q).$$ An important numerical invariant of a curve is its genus. The famous Hasse-Weil theorem gives upper and lower bounds for $N({\cal C})$ in terms of the genus. In this talk I will introduce these notions and discuss some recent results about $N({\cal C})$ for curves over $\mathbb{F}_q$ of large genus (here we will also consider non-plane curves). Such results have interesting applications, not only in Number Theory but also in Coding Theory, Cryptography and Theoretical Computer Science.