Isogeny classes of abelian varieties with no principal polarizations

Abstract

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\ell$ of $k$ of odd prime degree $p$, and an elliptic curve $E$ over $k$ that has no complex multiplication over $k$ and that has no $k$-defined $p$-isogenies to another elliptic curve, we construct a simple $(p-1)$-dimensional abelian variety $X$ over $k$ such that every polarization of every abelian variety isogenous to $X$ has degree divisible by $p^2$. We note that for every odd prime $p$ and every number field $k$, there exist $\ell$ and $E$ as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field $k$ have positive characteristic and that there be a Galois extension of $k$ with a certain non-abelian Galois group. Note: Theorem 3.2 in this paper is incorrect. The current version of the paper includes comments explaining the mistake.

This version highlights an error in the paper, and includes marginal notes pointing out the incorrect result and the error in the proof

References in corpus (3)

• Kernels of polarizations of abelian varieties over finite fields
• Modular representations arising from self-dual $\ell$-adic representations of finite groups
• Polarizations on abelian varieties

Cited by in corpus (4)

• Rank growth of elliptic curves over $N$-th root extensions
• Quadratic twists of abelian varieties and disparity in Selmer ranks
• Arithmetic local constants for abelian varieties with extra endomorphisms
• On the integral Hodge conjecture for real abelian threefolds