Details
| Original language | English |
|---|---|
| Pages (from-to) | 436-470 |
| Number of pages | 35 |
| Journal | Journal of number theory |
| Volume | 242 |
| Early online date | 25 May 2022 |
| Publication status | Published - 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).
Keywords
- Class field theory, Complex multiplication, Fields of definition, K3 surfaces
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Journal of number theory, Vol. 242, 01.2023, p. 436-470.
Research output: Contribution to journal › Article › Research › 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 -