Dwork Motives, Monodromy and Potential Automorphy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917734061703168 |
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| author | A'Campo, Lambert |
| author_facet | A'Campo, Lambert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dwork Motives, Monodromy and Potential Automorphy A'Campo, Lambert Number Theory Algebraic Geometry 11R39, 14D05 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. |
| title | Dwork Motives, Monodromy and Potential Automorphy |
| topic | Number Theory Algebraic Geometry 11R39, 14D05 |
| url | https://arxiv.org/abs/2407.16481 |