Polarizations on abelian varieties

Abstract

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]². We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]².