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Group Structures of Elementary Supersingular Abelian Varieties over Finite Fields

✍ Scribed by Hui June Zhu

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
176 KB
Volume
81
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let A be a supersingular abelian variety over a finite field k which is k-isogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e. We show that the group of k-rational points A(k) on A is isomorphic to (ZÂg(1) Z) e unless A's simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (ZÂg(1) Z) a _ (ZÂ( g(1)Â2) Z_ZÂ2Z) b for some non-negative integers a, b with a+b=e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k)$(ZÂg(1) Z) e .

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