arXiv subjects

Xuanyou Li

Publications and source records attributed to Xuanyou Li.

4 recordsLinked to original sources

arXiv2026

Arithmetic hypergeometric $\mathcal {D}$-modules and exponential sums on reductive groups

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric $\mathcal D$-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic $\mathcal D$-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric $\mathcal {D}$-module defines an $F$-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.

math.AG↗

arXiv2026

Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups

We define the hypergeometric exponential sum associated to a family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the hypergeometric exponential sum. Motivated by the definition of the hypergeometric sheaf, we introduce the hypergeometric $\mathcal D$-module, prove it is holonomic and estimate its rank. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we show how the hypergeometric $\mathcal D$-module controls the general behavior of the hypergeometric sheaf. We apply our results to the estimation of the hypergeometric exponential sum.

math.AG↗

arXiv2026

Twisted Higgs bundles and coendoscopy

This short note is devoted to the study of $G$-Higgs bundles twisted by a central gerbe. These objects arise naturally in the decomposition of the inertia stacks of $G$-Higgs bundles in terms of coendoscopic data. We establish that stabilised point-counts and cohomology are insensitive to the central twist. Along the way we show an analogue of Ngô's product formula for twisted Hitchin fibres.

math.AG↗

arXiv2025

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

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.

math.AG↗