The groups of points on abelian varieties over finite fields

Sergey Rybakov

Open Mathematics (2010)

  • Volume: 8, Issue: 2, page 282-288
  • ISSN: 2391-5455

Abstract

top
Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).

How to cite

top

Sergey Rybakov. "The groups of points on abelian varieties over finite fields." Open Mathematics 8.2 (2010): 282-288. <http://eudml.org/doc/268981>.

@article{SergeyRybakov2010,
abstract = {Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).},
author = {Sergey Rybakov},
journal = {Open Mathematics},
keywords = {Abelian variety; The group of rational points; Finite field; Newton polygon; Hodge polygon; abelian variety; Weil polynomial},
language = {eng},
number = {2},
pages = {282-288},
title = {The groups of points on abelian varieties over finite fields},
url = {http://eudml.org/doc/268981},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Sergey Rybakov
TI - The groups of points on abelian varieties over finite fields
JO - Open Mathematics
PY - 2010
VL - 8
IS - 2
SP - 282
EP - 288
AB - Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).
LA - eng
KW - Abelian variety; The group of rational points; Finite field; Newton polygon; Hodge polygon; abelian variety; Weil polynomial
UR - http://eudml.org/doc/268981
ER -

References

top
  1. [1] Berthelot P., Ogus A., Notes on crystalline cohomology, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1978 Zbl0383.14010
  2. [2] Demazure M., Lectures on p-divisible groups, Lecture notes in mathematics 302, Springer-Verlag, Berlin, 1972 
  3. [3] Kedlaya K., p-adic differential equations, Cambridge University Press, to appear, web draft available at http://wwwmath.mit.edu/?kedlaya/papers/pde.pdf 
  4. [4] Rück H.-G., A note on elliptic curves over finite fields. Math. Comp. 1987, 49(179), 301–304 http://dx.doi.org/10.2307/2008268 Zbl0628.14019
  5. [5] Schoof R., Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 1987, 46(2), 183–211 http://dx.doi.org/10.1016/0097-3165(87)90003-3 
  6. [6] Tsfasman M.A., The group of points of an elliptic curve over a finite field, Theory of numbers and its applications, Tbilisi, 1985, 286–287 
  7. [7] Tsfasman M., Vladut S., Nogin D., Algebraic geometric codes: basic notions, Mathematical Surveys and Monographs, 139, American Mathematical Society, Providence, RI, 2007 Zbl1127.94001
  8. [8] Voloch J.F., A note on elliptic curves over finite fields, Bull. Soc. Math. France, 1988, 116(4), 455–458 Zbl0688.14015
  9. [9] Waterhouse W., Abelian varieties over finite fields, Ann. scient. Éc. Norm. Sup., 1969, 4 serie 2, 521–560 Zbl0188.53001
  10. [10] Waterhouse W., Milne J., Abelian varieties over finite fields, Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969, 53–64 Zbl0216.33102
  11. [11] Xing Ch., The structure of the rational point groups of simple abelian varieties of dimension two over finite fields, Arch. Math., 1994, 63, 427–430 http://dx.doi.org/10.1007/BF01196672 Zbl0813.14015
  12. [12] Xing Ch., On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl., 1996, 2(4), 407–421 http://dx.doi.org/10.1006/ffta.1996.0024 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.