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On a theorem of Tate

100%
Bogomolov F., Tschinkel Y.
Open Mathematics
|
2008
|
tom 6
|
nr 3
343-350
EN
We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.
2
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The groups of points on abelian varieties over finite fields

76%
Rybakov S.
Open Mathematics
|
2010
|
tom 8
|
nr 2
282-288
EN
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).
3
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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

46%
Bandman T., Garion S., Kunyavskiĭ B.
Open Mathematics
|
2014
|
tom 12
|
nr 2
175-211
EN
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
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