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Principally polarizable isogeny classes of abelian surfaces over finite fields

Everett W. Howe·Daniel Maisner·Enric Nart·Christophe Ritzenthaler
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Abstract

Let A be an isogeny class of abelian surfaces over F_q with Weil polynomial x^4 + ax^3 + bx^2 + aqx + q^2. We show that A 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. We use this result in a forthcoming paper in which we determine which isogeny classes of abelian surfaces over finite fields contain Jacobians.

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Everett W. Howe, Daniel Maisner, Enric Nart, Christophe Ritzenthaler. 2006-02-28. Principally polarizable isogeny classes of abelian surfaces over finite fields. https://arxiv.org/abs/math/0602650

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