Dwork Motives, Monodromy and Potential Automorphy

TL;DR

This paper investigates motives from the Dwork family, demonstrating their monodromy properties and implications for automorphic representations, providing new insights into Galois representations and local-global compatibility.

Contribution

It introduces new families of motives with dense monodromy, unipotent operators, and bounded Hodge numbers, linking these to potential automorphy and Galois representations.

Findings

Monodromy groups are Zariski dense in SL_n

Existence of unipotent monodromy operators at infinity

Hodge numbers are all less than or equal to 1

Abstract

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.