Mathematical Research Letters

Volume 15 (2008)

Number 1

Principally polarizable isogeny classes of abelian surfaces over finite fields

Pages: 121 – 127



Everett W. Howe (Center for Communications Research)

Daniel Maisner (Universidad Autónoma de la Ciudad de México and Indiana University)

Enric Nart (Universitat Autònoma de Barcelona)

Christophe Ritzenthaler (Institut de Mathématiques de Luminy)


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~$3$.

Published 1 January 2008