论文标题
关于阿贝利亚表面的局部全球原理
On the local-global principle for isogenies of abelian surfaces
论文作者
论文摘要
令$ \ ell $为素数。我们对$ \ operatorName {sp} _4（\ Mathbb {f} _ \ ell）$和$ \ propatatorName {gsp} _4（\ Mathbb {f} _ \ ell）$进行分类。一个$ \ mathbb {f} _ \ ell $ - vector的尺寸1。我们使用此分类来证明，在许多情况下，阿伯利亚表面之间的局部global原理$ \ ell $在许多情况下都存在，尤其是每当亚伯层表面的范围很大的是$ $ $ $ \ eys $。最后，我们证明存在一些任意大的Primes $ \ ell $，其中某些Abelian Surface $ a/\ mathbb {q} $失败了local-lobal原理，用于$ \ ell $的同学。
Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.
