$p$-adic hypergeometric $\mathscr{D}^{\dagger}(\infty)$-module and exponential sums on reductive groups

Xuanyou Li; Chenhan Liu

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arXiv · 2512.11302

$p$-adic hypergeometric $\mathscr{D}^{\dagger}(\infty)$-module and exponential sums on reductive groups

Xuanyou Li·Chenhan Liu

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Abstract

We study the $p$-adic analogue of the $\ell$-adic hypergeometric sheaves for reductive groups, called the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules. They are overholonomic objects in the derived category of arithmetic $\mathscr{D}$-modules with Frobenius structures. Over the non-degenerate locus, the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules define $F$-isocrystals overconvergent along the complement of the non-degenerate locus. As an application, we use the theory of $L$-functions of overholonomic arithmetic $\mathscr{D}$-modules to study hypergeometric exponential sums on reductive groups.

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BibTeX RIS

Xuanyou Li, Chenhan Liu. 2025-12-12. $p$-adic hypergeometric $\mathscr{D}^{\dagger}(\infty)$-module and exponential sums on reductive groups. https://arxiv.org/abs/2512.11302

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