Details
| Original language | English |
|---|---|
| Article number | 107772 |
| Journal | Advances in mathematics |
| Volume | 385 |
| Early online date | 10 May 2021 |
| Publication status | Published - 16 Jul 2021 |
| Externally published | Yes |
Abstract
We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains Br(X‾)GK for any K3 surface X/K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.
Keywords
- Brauer groups, Complex multiplication, K3 surfaces
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Advances in mathematics, Vol. 385, 107772, 16.07.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Complex multiplication and Brauer groups of K3 surfaces
AU - Valloni, Domenico
N1 - Publisher Copyright: © 2021
PY - 2021/7/16
Y1 - 2021/7/16
N2 - We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains Br(X‾)GK for any K3 surface X/K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.
AB - We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains Br(X‾)GK for any K3 surface X/K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.
KW - Brauer groups
KW - Complex multiplication
KW - K3 surfaces
UR - http://www.scopus.com/inward/record.url?scp=85105528632&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2021.107772
DO - 10.1016/j.aim.2021.107772
M3 - Article
AN - SCOPUS:85105528632
VL - 385
JO - Advances in mathematics
JF - Advances in mathematics
SN - 0001-8708
M1 - 107772
ER -