Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 436-470 |
| Seitenumfang | 35 |
| Fachzeitschrift | Journal of number theory |
| Jahrgang | 242 |
| Frühes Online-Datum | 25 Mai 2022 |
| Publikationsstatus | Veröffentlicht - Jan. 2023 |
Abstract
Let X/C be a K3 surface with complex multiplication by the ring of integers of a CM field E. We show that X can always be defined over an Abelian extension K/E explicitly determined by the discriminant form of the lattice NS(X). We then construct a model of X over K via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for X in terms of the class number of E and the discriminant of NS(X).
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
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in: Journal of number theory, Jahrgang 242, 01.2023, S. 436-470.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Fields of definition of K3 surfaces with complex multiplication
AU - Valloni, Domenico
N1 - Funding Information: This work was written while the author was a PhD student at Imperial College London. The research was funded by an EPSRC studentship (project reference EP/N509486/1).
PY - 2023/1
Y1 - 2023/1
N2 - Let X/C be a K3 surface with complex multiplication by the ring of integers of a CM field E. We show that X can always be defined over an Abelian extension K/E explicitly determined by the discriminant form of the lattice NS(X). We then construct a model of X over K via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for X in terms of the class number of E and the discriminant of NS(X).
AB - Let X/C be a K3 surface with complex multiplication by the ring of integers of a CM field E. We show that X can always be defined over an Abelian extension K/E explicitly determined by the discriminant form of the lattice NS(X). We then construct a model of X over K via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for X in terms of the class number of E and the discriminant of NS(X).
KW - Class field theory
KW - Complex multiplication
KW - Fields of definition
KW - K3 surfaces
UR - http://www.scopus.com/inward/record.url?scp=85132307174&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1907.01336
DO - 10.48550/arXiv.1907.01336
M3 - Article
AN - SCOPUS:85132307174
VL - 242
SP - 436
EP - 470
JO - Journal of number theory
JF - Journal of number theory
SN - 0022-314X
ER -