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Principally polarizable isogeny classes of Abelian surfaces over finite fields. (English) Zbl 1145.11045
An isogeny class of abelian varieties defined over a field \(k\) is said to be principally polarizable if it contains a variety that admits a principal polarization defined over \(k\). This paper considers the case of abelian surfaces defined over a finite field. By the Honda-Tate theorem, two abelian varieties defined over a finite field are isogenous to one another if, and only if, they share the same Weil polynomial (the Weil polynomial is the characteristic polynomial of the Frobenius endomorphism). Now the Weil polynomial of an abelian surface \(A\) over a finite field \(\mathbb F_{q}\) (where \(q\) is a power of a prime \(p\)) has the form
\[ x^{4}+ax^{3}+bx^{2}+aqx+q^{2}\in\mathbb Z[x]. \] Thus, the isogeny class of \(A\) may be denoted by \(\mathcal A_{(a,b)}\). The main theorem of the paper is the following simple criterion: \(\mathcal A_{(a,b)}\) is principally polarizable except when the following three conditions hold (i) \(a^{2}-b=q\), (ii) \(b<0\), and (iii) all prime divisors of \(b\) are congruent to 1 modulo 3.
The authors intend to use the above criterion in a forthcoming paper which will deal with the problem of determining which isogeny classes of abelian surfaces contain Jacobians.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G25 Varieties over finite and local fields
14G15 Finite ground fields in algebraic geometry
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