On Fibre Products of Kummer Curves with Many Rational Points over Finite Fields

Abstract

We determined the number of rational points of fibre products of two Kummer covers over a rational point of the projective line in a recent work of F. Özbudak and B.G. Temür (Des Codes Cryptogr 70(3):385–404, 2014), where we also constructed explicit examples, including a record and two new entries for the current Table of Curves with Many Points (manYPoints: Table of curves with many points. http://www.manypoints.org (2014). Accessed 30 Sep 2014). Using the methods given in Özbudak and Gülmez Temür (Des Codes Cryptogr 70(3):385–404, 2014), we made an exhaustive computer search over $\mathbb{F}_{5}$ and $\mathbb{F}_{7}$ by the contributions of O. Yayla and at the end of this search we obtained 12 records and 6 new entries for the current table; in particular, we observed that the fibre product with genus 7 and 36 rational points coincides with the Ihara bound, thus we concluded that the maximum number N ₇(7) of $\mathbb{F}_{7}$-rational points among all curves of genus 7 is 36 (Özbudak et al., Turkish J Math 37(6):908–913, 2013). Recently, we made another exhaustive computer search over $\mathbb{F}_{11}$. In this paper we are representing the results as three records and three new entries for the current table.