Number of points on a family of curves over a finite field

TL;DR

This paper constructs specific algebraic curves over finite fields with many rational points by using fiber products of hyperelliptic curves, leading to improved bounds for certain genus and field size pairs.

Contribution

It introduces a novel family of curves via fiber products of hyperelliptic curves to produce curves with many rational points, improving existing bounds.

Findings

Constructed new curves with more rational points than previously known.

Achieved improved bounds for specific pairs of (q, g).

Demonstrated the effectiveness of fiber products in curve construction.

Abstract

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results obtained improve the known bounds for a few pairs (q, g).