Dwork Motive、Monodromy与Potential Automorphy
Dwork Motives, Monodromy and Potential Automorphy
AI总结：
本文研究了作为Dwork族上同调直和项的motive族，通过计算找到几何monodromy稠密、实现多种unipotent算子且Hodge数不超过1的族，并借此构造potential automorphic表示及给出Matsumoto定理的新证明。
摘要：
本文研究了某些motive族，它们作为Dwork族上同调的直和项出现。我们通过计算找到了具有以下三个性质的有趣族示例。首先，它们的几何monodromy群在$\operatorname{SL}_n$中是Zariski稠密的。其次，它们在$t = \infty$处作为monodromy算子实现了许多不同的unipotent算子。第三，它们的所有Hodge数均$\leq 1$。这对Galois表示具有推论。即，如果幂零算子$N$在我们某个族中作为$t = \infty$处的monodromy出现，我们便可构造出potential automorphic表示，其在固定素数$p$处的$\ell$-adic monodromy由$N$给出。作为另一应用，我们获得了Matsumoto近期local-global compatibility定理某些情形的新证明。
英文摘要：
In this paper we study certain families of motives, which arise as direct summands of the cohomology of the Dwork family. We computationally find examples of interesting families with the following three properties. Firstly, their geometric monodromy group is Zariski dense in $\operatorname{SL}_n$. Secondly, they realise many different unipotent operators as the monodromy operator at $t = \infty$. Thirdly, all their Hodge numbers are $\leq 1$. This has consequences for Galois representations. Namely, if a nilpotent operator $N$ appears as the monodromy at $t = \infty$ in one of our families, we can construct potentially automorphic representations with $\ell$-adic monodromy given by $N$ at a fixed prime $p$. As another application, we obtain a new proof of some cases of the recent local-global compatibility theorem of Matsumoto.
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