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

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arxiv 2308.10979 v3 pith:JJWJYREY submitted 2023-08-21 math.NT math.AG

Tony Feng , Zhiwei Yun , Wei Zhang This is my paper · ORCID
classification math.NTmath.AG
keywords modularityalgebraicderivedfourierhigherseriesthetaclassical
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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 $\ell$-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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Modularity of Higher Theta Series III: Proof of the Modularity Conjecture

    math.NT 2026-08 conditional novelty 8.0 of 10

    The higher theta series on Hermitian shtukas are shown to be modular, independent of the chosen Lagrangian, with a stronger supermodularity result for general linear groups.

  2. Diagonal cycles on Shtukas and the adjoint $L$-function

    math.NT 2026-07 conditional novelty 7.0 of 10

    For split almost simple groups over function fields, self-intersections of diagonal cycles on shtuka moduli, with determinant line-bundle insertions, equal higher derivatives of adjoint L-functions.

  3. On the relative Langlands duality for $\operatorname{Sp}_{2n} \backslash \operatorname{GL}_{2n+1}$ (with an appendix by Zeyu Wang)

    math.RT 2025-04 conditional novelty 6.0 of 10

    The symplectic period of certain Eisenstein series on GL(2n+1) matches the L-function ratio predicted by relative Langlands duality, confirming the dual pair Sp(2n)\GL(2n+1) and GL(n)xGL(n+1)\GL(2n+1) in the tested cases.

  4. Introduction to Shtukas and their moduli

    math.NT 2024-11 unverdicted

    A survey of Shtukas and their moduli stacks, written as lecture notes, covering definitions, geometry, cohomological structures, and applications.

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