The CM class number one problem for curves of genus 2

2015/11/16 by Pınar Kılıçer, Kilicer, Pinar, Marco Streng +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1511.04869

openalex publication_date 2015/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus 2 defined over the reflex field. We solve the problem by showing that the list given in Bouyer and Streng [LMS J. Comput. Math., 18(1):507538, 2015] is complete.

Related

• Families of genus two curves with many elliptic subcovers
• The class of the d-elliptic locus in genus 2
• Curves of genus two with maps of every degree to a fixed elliptic curve
• A bound on the primes of bad reduction for CM curves of genus 3
• A CRT algorithm for constructing genus 2 curves over finite fields