Abelian n -Division Fields of Elliptic Curves and Brauer Groups of Product Kummer & Abelian Surfaces
| dc.citation.articleNumber | e26 | |
| dc.citation.journalTitle | Forum of Mathematics, Sigma | |
| dc.citation.volumeNumber | 5 | |
| dc.contributor.author | Várilly-Alvarado, Anthony | |
| dc.contributor.author | Viray, Bianca | |
| dc.date.accessioned | 2018-11-09T15:00:00Z | |
| dc.date.available | 2018-11-09T15:00:00Z | |
| dc.date.issued | 2017 | |
| dc.description.abstract | Let Y be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of Q. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes BrY/Br1Y is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic 0, we prove that the existence of a strong uniform bound on the size of the odd torsion of BrY/Br1Y is equivalent to the existence of a strong uniform bound on integers n for which there exist non-CM elliptic curves with abelian n-division fields. Using the same methods we show that, for a fixed prime ℓ, a number field k of fixed degree r, and a fixed discriminant of the geometric Néron–Severi lattice, #(BrY/Br1Y)[ℓ∞] is bounded by a constant that depends only on ℓ, r, and the discriminant. | |
| dc.identifier.citation | Várilly-Alvarado, Anthony and Viray, Bianca. "Abelian n -Division Fields of Elliptic Curves and Brauer Groups of Product Kummer & Abelian Surfaces." <i>Forum of Mathematics, Sigma,</i> 5, (2017) Cambridge University Press: https://doi.org/10.1017/fms.2017.16. | |
| dc.identifier.digital | division-fields | |
| dc.identifier.doi | https://doi.org/10.1017/fms.2017.16 | |
| dc.identifier.uri | https://hdl.handle.net/1911/103304 | |
| dc.language.iso | eng | |
| dc.publisher | Cambridge University Press | |
| dc.rights | This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. | |
| dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | |
| dc.title | Abelian n -Division Fields of Elliptic Curves and Brauer Groups of Product Kummer & Abelian Surfaces | |
| dc.type | Journal article | |
| dc.type.dcmi | Text | |
| dc.type.publication | publisher version |
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