Maximal Curves With Respect To Quadratic Extensions Over Finite Fields

TL;DR

This paper investigates curves over finite fields that reach a canonical bound relating their rational points over the field and its quadratic extension, providing characterizations, classifications, and connections to known bounds and special curves.

Contribution

It introduces new characterizations of Diophantine-maximal curves, completes the inequality symmetrically, and classifies low-genus cases, including genus 2 Jacobians.

Findings

Characterization of Diophantine-maximal curves

Complete symmetry of the rational point bound

Classification of low-genus maximal curves

Abstract

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which relate to the variance enable to complete the inequality in a symmetrical way and to obtain optimal refinements. We focus on the curves reaching the bound, which we call Diophantine-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. At last we determine which polynomials correspond to the Jacobian of a Diophantine-maximal curve of genus 2.