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Abelian surfaces over finite fields with prescribed groups
Author(s) -
David Chantal,
Garton Derek,
Scherr Zachary,
Shankar Arul,
Smith Ethan,
Thompson Lola
Publication year - 2014
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdu033
Subject(s) - abelian group , mathematics , combinatorics , prime power , congruence relation , prime (order theory) , field (mathematics) , characterization (materials science) , discrete mathematics , pure mathematics , physics , optics
Let A be an abelian surface over F q , the field of q elements. The rational points on A / F qform an abelian group A ( F q ) ≃ Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z . 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 / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z do not occur if n 1 is very large with respect ton 2 , n 3 , n 4(Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).