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 -