Mathematical Research Letters

Volume 15, Issue 1, January 2008 pp. 121-127.

Principally polarizable isogeny classes of abelian surfaces over finite fields

Authors: Everett W. Howe (1), Daniel Maisner (2) Enric Nart (3), and Christophe Ritzenthaler (4)
Author institution: Center for Communications Research (1), Universidad Autónoma de la Ciudad de México and Indiana University (2), Universitat Autònoma de Barcelona (3), and Institut de Mathématiques de Luminy (4)

Summary: Let $\calA$ be an isogeny class of abelian surfaces over $\fq$ with Weil polynomial $x^4 + ax^3 + bx^2 + aqx + q^2$. We show that $\calA$ does not contain a surface that has a principal polarization if and only if $a^2 - b = q$ and $b < 0$ and all prime divisors of $b$ are congruent to $1$ modulo~$3$.

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