The -numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6
Volume 216 / 2024
Acta Arithmetica 216 (2024), 227-248
MSC: Primary 14H45; Secondary 14H50, 33C05
DOI: 10.4064/aa231014-27-5
Published online: 17 October 2024
Abstract
We study non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6. Over an algebraically closed field K of characteristic \neq 2,3, such curves are written as plane quartics C_r: x^3 z + y^4 + r y^2 z^2 + z^4 = 0 with one parameter r. As the first main theorem, we show that r\neq 0,\pm 2 and give a necessary and sufficient condition on r and r’ for C_r \cong C_{r’}. By describing the Hasse–Witt matrix of C_r in terms of a certain Gauss hypergeometric series, we obtain the second main theorem, where we determine the possible a-numbers of C_r, and give the exact number of isomorphism classes over K of such curves attaining the possible maximal a-number.
