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Polarizations on abelian varieties and self-dual \(l\)-adic representations of inertia groups. (English) Zbl 1015.14021

Summary: It is well known that every finite subgroup of \(\mathrm{GL}_d (\mathbb Q_{\ell})\) is conjugate to a subgroup of \(\mathrm{GL}_d(\mathbb Z_{\ell})\). However, this does not remain true if we replace general linear groups by symplectic groups. We say that \(G\) is a group of inertia type if \(G\) is a finite group which has a normal Sylow-\(p\)-subgroup with cyclic quotient. We show that if \(\ell>d +1\), and \(G\) is a subgroup of \(\mathrm{Sp}_{2d} (\mathbb Q_{\ell})\) of inertia type, then \(G\) is conjugate in \(\mathrm{GL}_{2d} (\mathbb Q_{\ell})\) to a subgroup of \(\mathrm{Sp}_{2d} (\mathbb Z_{\ell})\). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime \(\ell\), isogeny classes of abelian varieties all of whose polarizations have degree divisible by \(\ell^2\). We prove similar results for Euler characteristic of invertible sheaves on abelian varieties over fields of positive characteristic.

MSC:

14K02 Isogeny
20C11 \(p\)-adic representations of finite groups
11G10 Abelian varieties of dimension \(> 1\)
11S23 Integral representations
20D25 Special subgroups (Frattini, Fitting, etc.)
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