Vol. 4, No. 1, 2020

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Algorithms to enumerate superspecial Howe curves of genus 4

Momonari Kudo, Shushi Harashita and Everett W. Howe

Vol. 4 (2020), No. 1, 301–316
Abstract

A Howe curve is a curve of genus 4 obtained as the fiber product of two genus-1 double covers of P1. We present a simple algorithm for testing isomorphism of Howe curves, and we propose two main algorithms for finding and enumerating superspecial Howe curves: One involves solving multivariate systems coming from Cartier–Manin matrices, while the other uses Richelot isogenies of curves of genus 2. Comparing the two algorithms by implementation and by complexity analyses, we conclude that the latter enumerates superspecial Howe curves more efficiently. Using these algorithms, we show that there exist superspecial curves of genus 4 in characteristic p for every prime p with 7 < p < 20000.

Keywords
algebraic curves, superspeciality
Mathematical Subject Classification
Primary: 11G20
Secondary: 14G15, 14H45
Supplementary material

Algorithms for enumerating superspecial Howe curves

Milestones
Received: 25 February 2020
Revised: 1 August 2020
Accepted: 31 August 2020
Published: 29 December 2020
Authors
Momonari Kudo
Department of Mathematical Informatics, Graduate School of Information Science and Technology
The University of Tokyo
Bunkyo-ku
Tokyo
Japan
Shushi Harashita
Graduate School of Environment and Information Sciences
Yokohama National University
Hodogaya-ku
Yokohama
Japan
Everett W. Howe
San Diego, CA
United States