# Terminology about Abelian varieties over finite fields

Is there a standard meaning for ordinary and supersingular Abelian varieties over finite fields? If so, where can I find it (together with basic properties about them)?

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## 1 Answer

See http://en.wikipedia.org/wiki/Hasse-Witt_matrix#Abelian_varieties_and_their_p-rank . "Ordinary" is always defined by p-rank equal to the dimension (the maximum possible). The article gives one definition of "supersingular", but that usage may not be universal.

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The article you mentioned has a definition of supersingular but not of ordinary, right? – expmat May 11 '12 at 18:31
My answer has a definition of ordinary but not of supersingular, on the other hand. – Charles Matthews May 11 '12 at 19:07