Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic p >= 3, in particular their standard forms C : x4+y4+z4+rx2y2+ sy2z2 + tz2x2 = 0. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then r, s, t belong to Fp2 and C is maximal or minimal over Fp2 . Moreover, in this case we will provide a simple criterion in terms of r, s, t, p that tells whether C is maximal (resp. minimal) over Fp2 . (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper studies Ciani curves in characteristic p >= 3 and their properties. By determining whether a Ciani curve is superspecial, it can be determined whether it is a maximal or minimal curve over Fp2.