Work

Modularity of higher theta series I: cohomology of the generic fiber

Year: 2026

Type: article

Abstract: In a previous paper we constructed higher theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the ℓ -adic realization of higher theta series in cohomology. The proof debuts a new type of Fourier transform, occurring on the Borel–Moore homology of moduli spaces for shtuka-type objects, that we call the arithmetic Fourier transform . Another novelty in the argument is a sheaf-cycle correspondence extending the classical sheaf-function correspondence, which facilitates the deployment of sheaf-theoretic methods to analyze algebraic cycles. Although the modularity property is a statement within classical algebraic geometry, the proof relies on derived algebraic geometry, especially a nascent theory of derived Fourier analysis on derived vector bundles, which we develop.

Source: Publications mathématiques de l IHÉS

DOI: 10.5802/pmihes.27

Authors: Tony Feng, Zhiwei Yun, Wei Zhang

Institutions: University of California, Berkeley, Massachusetts Institute of Technology

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,590

Funders: National Science Foundation, Deutsche Forschungsgemeinschaft, Hausdorff Center for Mathematics, Division of Mathematical Sciences