Abelian surfaces with prescribed groups
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).
Repository 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
Publication Date
1-1-2014
Publication Title
Bulletin of the London Mathematical Society
Department
Mathematics
Document Type
Article
DOI
https://dx.doi.org/10.1112/blms/bdu033
Language
English
Format
text