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Modularity of higher theta series II: Chow group of the generic fiber
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abstract
Higher theta series on moduli spaces of Hermitian shtukas were constructed by Feng--Yun--Zhang and conjectured to be modular, parallel to classical conjectures in the Kudla program. In this paper we prove the modularity of higher theta series after restriction to the generic locus. The proof is an upgrade, using motivic homotopy theory, of earlier work of Feng--Yun--Zhang which established generic modularity of $\ell$-adic realizations. In the process, we develop some general tools of broader utility. One such is the "motivic sheaf-cycle correspondence", a categorical trace formalism for extracting computations in the Chow group from computations in Voevodsky's derived category of motives. Another new tool is the "derived homogeneous Fourier transform", which we use to implement a form of Fourier analysis for motives.
Forward citations
Cited by 2 Pith papers
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Modularity of Higher Theta Series III: Proof of the Modularity Conjecture
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.
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Diagonal cycles on Shtukas and the adjoint $L$-function
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.
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