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Arithmetic hypergeometric π’Ÿ-modules and exponential sums on reductive groups

  • Lei Fu
  • Xuanyou Li
  • Chenhan Liu
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Abstract

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 π’Ÿ-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic π’Ÿ-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. We prove that over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric π’Ÿ-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026arithmetic,
  title = {{Arithmetic hypergeometric $\mathcal D$-modules and exponential sums on reductive groups}},
  author = {Lei Fu and Xuanyou Li and Chenhan Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.00470},
  eprint = {2608.00470},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.00470},
}
RIS
TY  - UNPB
TI  - Arithmetic hypergeometric π’Ÿ-modules and exponential sums on reductive groups
AU  - Lei Fu
AU  - Xuanyou Li
AU  - Chenhan Liu
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2608.00470
UR  - https://arxiv.org/abs/2608.00470
AB  - 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 π’Ÿ-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic π’Ÿ-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. We prove that over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric π’Ÿ-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.
ER  -