arXiv math/99021641999-02-26math.NT

阿贝尔簇的极化与惯性群的self-dual ell-adic表示

Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups

Alice Silverberg, Yuri G. Zarhin

摘要：

GL_d（Q_{\ell}）的每个有限子群都共轭于GL_d（Z_{\ell}）的一个子群.然而，如果我们用辛群代替一般线性群，这就不成立了。我们称G是惯性型群，如果G是有限群，且G的正规Sylow-p-子群具有循环商.证明了若\ell >d+1，且G是Sp_{2d}（Q_{\ell}）的惯性型子群，则G在GL_{2d}（Q_{\ell}）中共轭到\Sp_{2d}（Z_{\ell}）的子群.尽管G不能在\GL_{2d}（Q_ \ell）中共轭到\Sp_{2d}（Z_ \ell）的一个子群，但我们证明了只要\ell >3，它仍然可以嵌入到\Sp_{2d}（F_ \ell）中，使得特征多项式保持不变（mod \ell）.后一个结果适用于任意有限群，不一定是惯性型的，也适用于对称形式，而不仅仅是交替形式。我们给出的例子表明，边界是尖锐的。我们应用这些结果来构建，为每一个奇素数\ell，issues类的阿贝尔簇所有的极化度可分的\ell。本文是ANT-0151的修订版，标题为“有限群的自对偶ell-adic表示”。

英文摘要：

It is well-known that every finite subgroup of GL_d(Q_{\ell}) is conjugate to a subgroup of GL_d(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 Sp_{2d}(Q_{\ell}) of inertia type, then G is conjugate in GL_{2d}(Q_{\ell}) to a subgroup of \Sp_{2d}(Z_{\ell}). Despite the fact that G can fail to be conjugate in \GL_{2d}(Q_\ell) to a subgroup of \Sp_{2d}(Z_\ell), we prove that it can nevertheless be embedded in \Sp_{2d}(F_\ell) in such a way that the characteristic polynomials are preserved (mod \ell), as long as \ell>3. The latter result holds for arbitrary finite groups, not necessarily of inertia type, and holds also for symmetric forms, not just alternating forms. We give examples which show that the bounds are 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. This paper is a revised version of ANT-0151, titled `Self-dual ell-adic representations of finite groups'.

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