论文标题
曲线上的动机本地系统和梅达的猜想
Motivic local systems on curves and Maeda's conjecture
论文作者
论文摘要
我们表明，只有有限的许多复杂属两条曲线和四个刺破球录入了两个几何来源的局部系统，而且每种系统都有有限的许多。这为Esnault和Kerz的猜想提供了进一步的反例：Landesman和Litt最近获得了非常一般曲线的反例。在第二部分中，我们证明了这一结果的类似物，即超过$ \ edline {\ mathbb {f}} _ p $，只有有限的许多属曲线有限地接纳了从固定的quaternionic shimura品种中撤回的两个无处不在的局部系统，相同，$ \ mathbb^po}。猜想的是，每个等级两个本地系统都会出现这种回调。这为MAEDA在功能字段上特征形式的Galois轨道上的猜想提供了结果。这些证据利用了地位者和利特的工作中的思想，例如等词，以及至关重要的描述了由于田和小而引起的戈伦 - 奥特层。
We show that only finitely many complex genus two curves and four punctured spheres admit rank two local systems of geometric origin, and moreover each carries finitely many. This gives further counterexamples to a conjecture of Esnault and Kerz: counterexamples over very general curves were recently obtained by Landesman and Litt. In the second part we prove an analogue of this result in positive characteristic, namely that over $\overline{\mathbb{F}}_p$, only finitely many genus two curves admit non-trivial rank two local systems pulled back from a fixed quaternionic Shimura variety, and the same for $\mathbb{P}^1$ minus four points; conjecturally, every rank two local system arises as such a pullback. This provides results towards Maeda's conjecture on Galois orbits of eigenforms over function fields. The proofs make use of ideas from the work of Landesman and Litt such as isomonodromy, as well as crucially the description of the Goren-Oort strata due to Tian and Xiao.
