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arXiv · 2610.05781

Automorphic commutator relations in relative geometric Langlands

Shurui Liu
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Abstract

We prove the automorphic commutator relation conjectured by Liu and Wang arXiv:2504.00275, which is the geometric input for the automorphic Clifford relations in their framework relating higher period integrals to higher derivatives of $L$-functions over function fields. The relation was previously established only under additional cohomological vanishing assumptions and for curves of genus different from one; we remove these assumptions. The main new ingredient is a "universal-local" stage between the local and the semi-local settings: we introduce the classifying stack of formal multidisks with two ordered marked sections, and we construct universal-local analogues of the Hecke stacks, unit objects, Hecke actions and special cohomological correspondences. Fusion and purity on this stack yield a universal-local commutator identity, which is proved using only the local assumptions on the Plancherel algebra that enter the conjecture. Specialization to a curve and the six-functor formalism then give the conjecture for every smooth projective curve.

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Shurui Liu. 2026-10-05. Automorphic commutator relations in relative geometric Langlands. https://arxiv.org/abs/2610.05781

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