Abelian surfaces with prescribed groups

Title:
Abelian surfaces with prescribed groups
Authors:
David, Chantal; Garton, Derek; Scherr, Zachary; Shankar, Arul; Smith, Ethan; Thompson, Lola
Abstract:
Let A be an abelian surface over Fq, the field of q elements. The rational points on A/Fq form an abelian group A(Fq)≃Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z do not occur if n1 is very large with respect to n2,n3,n4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).
Citation:
David, C., D. Garton, Z. Scherr, A. Shankar, E. Smith, and L. Thompson. 2014. "Abelian surfaces with prescribed groups." Bulletin of the London Mathematical Society 46: 779-792. 
Publisher:
London Mathematical Society
DATE ISSUED:
2014
Department:
Mathematics
Type:
Article
PUBLISHED VERSION:
10.1112/blms/bdu033
PERMANENT LINK:
http://hdl.handle.net/11282/566913

Full metadata record

DC FieldValue Language
dc.contributor.authorDavid, Chantalen
dc.contributor.authorGarton, Dereken
dc.contributor.authorScherr, Zacharyen
dc.contributor.authorShankar, Arulen
dc.contributor.authorSmith, Ethanen
dc.contributor.authorThompson, Lolaen
dc.date.accessioned2015-08-13T10:36:48Zen
dc.date.available2015-08-13T10:36:48Zen
dc.date.issued2014en
dc.identifier.citationDavid, C., D. Garton, Z. Scherr, A. Shankar, E. Smith, and L. Thompson. 2014. "Abelian surfaces with prescribed groups." Bulletin of the London Mathematical Society 46: 779-792. en
dc.identifier.issn0024-6093en
dc.identifier.urihttp://hdl.handle.net/11282/566913en
dc.description.abstractLet A be an abelian surface over Fq, the field of q elements. The rational points on A/Fq form an abelian group A(Fq)≃Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z do not occur if n1 is very large with respect to n2,n3,n4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).en
dc.language.isoen_USen
dc.publisherLondon Mathematical Societyen
dc.identifier.doi10.1112/blms/bdu033en
dc.subject.departmentMathematicsen
dc.titleAbelian surfaces with prescribed groupsen
dc.typeArticleen
dc.identifier.journalBulletin of the London Mathematical Societyen
dc.identifier.volume46en
dc.identifier.issue4en
dc.identifier.startpage779en
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