TY - JOUR

T1 - The CM class number one problem for curves of genus 2

AU - Kılıçer, Pınar

AU - Streng, Marco

N1 - Funding Information:
This paper is based on Chapter 2 of the PhD thesis of the first author, completed at Universiteit Leiden and Université de Bordeaux, and supervised by Andreas Enge, Peter Stevenhagen, and the second author. The authors wish to thank Maarten Derickx and Florian Hess for the helpful discussions and the anonymous referee for various helpful suggestions.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2023/3

Y1 - 2023/3

N2 - Gauss’s class number one problem, solved by Heegner, Baker, and Stark, asked for all imaginary quadratic fields for which the ideal class group is trivial. An application of this solution gives all elliptic curves that can be defined over the rationals and have a large endomorphism ring (CM). Analogously, to get all CM curves of genus two defined over the smallest number fields, we need to find all quartic CM fields for which the CM class group (a quotient of the ideal class group) is trivial. We solve this CM class number one problem. We prove that the list given in Bouyer–Streng [LMS J Comput Math 18(1):507–538, 2015, Tables 1a, 1b, 2b, and 2c] of maximal CM curves of genus two defined over the reflex field is complete. We also prove that there are exactly 21 simple CM curves of genus two over C that can be defined over Q.

AB - Gauss’s class number one problem, solved by Heegner, Baker, and Stark, asked for all imaginary quadratic fields for which the ideal class group is trivial. An application of this solution gives all elliptic curves that can be defined over the rationals and have a large endomorphism ring (CM). Analogously, to get all CM curves of genus two defined over the smallest number fields, we need to find all quartic CM fields for which the CM class group (a quotient of the ideal class group) is trivial. We solve this CM class number one problem. We prove that the list given in Bouyer–Streng [LMS J Comput Math 18(1):507–538, 2015, Tables 1a, 1b, 2b, and 2c] of maximal CM curves of genus two defined over the reflex field is complete. We also prove that there are exactly 21 simple CM curves of genus two over C that can be defined over Q.

KW - Abelian varieties

KW - Algebraic curves

KW - Class number

KW - CM fields

KW - CM types

UR - http://www.scopus.com/inward/record.url?scp=85146455450&partnerID=8YFLogxK

U2 - 10.1007/s40993-022-00417-7

DO - 10.1007/s40993-022-00417-7

M3 - Article

AN - SCOPUS:85146455450

SN - 2363-9555

VL - 9

JO - Research in Number Theory

JF - Research in Number Theory

IS - 1

M1 - 15

ER -