Maximal curves over finite fields: examples

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A (smooth, complete, geometrically irreducible) curve $LaTeX: C$ defined over a finite field $LaTeX: {\mathbb F}_q$ is called maximal, if the cardinality of the set of $LaTeX: {\mathbb F}_q$ -rational points on $LaTeX: C$ equals $LaTeX: q+1+gm$ . Here $LaTeX: g$ denotes the genus of the curve, and $LaTeX: m$ is the largest integer not exceeding $LaTeX: \sqrt{4q}$ .

We consider the following question: given a curve over $LaTeX: {\mathbb Z}[\frac{1}{N}]$ , for which prime powers $LaTeX: q$ coprime with $LaTeX: N$ , is $LaTeX: C\otimes {\mathbb F}_q$ maximal over $LaTeX: {\mathbb F}_q}$ ?

Although in general this is a difficult problem (already for genus 1!), it turns out that in certain examples one can give a complete answer. In the talk this is done for a particular hyperelliptic curve of genus 3.

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Maximal curves over finite fields: examples

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