$\infty$-Categorical Generalized Langlands Correspondence II: Langlands Program Formalism

TL;DR

This paper extends the Langlands program by introducing new functorial perturbations and constructions in various settings, broadening its scope and formalism across different mathematical frameworks and characteristics.

Contribution

It develops generalized functorial perturbations of the Langlands program, incorporating mixed-parity, Robba-Frobenius sheafification, and arithmetic D-module approaches, extending the formalism after Fargues-Scholze.

Findings

Constructed Langlands correspondence from smooth representations to Weil side.

Introduced new functorial perturbations applicable in all characteristics.

Extended the formalism of the Langlands program with novel geometric and arithmetic methods.

Abstract

We extend the Langlands program in various subprograms with certain different generalizations: (1) Mixed-parity functorial perturbation of the usual Langlands program after Fargues-Scholze in all characteristics; (2) Robba-Frobenius sheafified functorial perturbation of the usual Langlands program after Fargues-Scholze and Kedlaya-Liu in all characteristics; (3) Arithmetic $D$-module theoretic functorial perturbation of the usual Langlands program after Fargues-Scholze and Abe-Kedlaya-Xu. In certain localized setting we construct Langlands correspondence from smooth representation side to Weil side through geometrized generalized Bernstein center.