Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

2 · Going further

The central problem is to enhance a cuspidal excursion parameter occurring at a specified full finite level for a split connected semisimple group over a global function field. Assuming the finite-level Ramanujan–Arthur decomposition stated in its Theorem 1.1, Manuscript 1 constructs a global Arthur enhancement. The announced mechanism uses a single algebraic ${\mathrm{SL}}_2$ and a map from the Weil group into its centralizer: their diagonal specialization recovers the original parameter on the entire Weil group, including inertia. The abstract gives no further argument details and asserts neither ellipticity, Arthur-packet classification, nor a multiplicity formula.

Manuscript 3 supplies the decomposition strand: over global function fields, cuspidal automorphic functions for split semisimple groups are decomposed over $\mathbb{Q}$ and ${\overline{\mathbb{Q}}}_{\ell }$ into subspaces indexed by nilpotent orbits of the dual group. This holds at every full finite level, with arbitrary divisor multiplicities; the same orbit governs each summand at every unramified place and every complex embedding. At trivial level, the text presents this as resolving Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin. For split connected adjoint absolutely simple groups, one generic unramified local component suffices to obtain temperedness at every unramified place. Manuscript 4 extends the conclusion to every place for globally generic cuspidal representations in the exceptional-group setting, without restrictions on characteristic or ramification depth.

On the geometric side, Manuscript 5 shows the ${\overline{\mathbb{Q}}}_{\ell }$-linear restricted geometric Langlands equivalence in its stated regimes for connected reductive groups and smooth projective connected curves, with $\ell \ne p$. Over ${\overline{\mathbb{F}}}_q$, four characteristic conditions are required; over an arbitrary algebraically closed field of positive characteristic, $p$ must additionally be very good for the group and satisfy $p\nmid |W_G|$. The text presents this as resolving Gaitsgory–Raskin’s full-support conjecture in these regimes. Under the characteristic hypotheses of the restricted theory, Manuscript 2 shows that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over $\mathbb{Q}$, including the noncuspidal part and every closed invariant nilpotent support; this addresses the Gaitsgory–Lafforgue–Raskin rationality conjecture in that setting.

Manuscripts 6–8 construct nonzero Hecke eigensheaves on curves of genus at least two, with Borel level at marked points. Manuscript 6 concerns one point, a simple simply connected group subject to four characteristic hypotheses, and a dense geometric local system with tame regular-unipotent monodromy. Manuscript 7 treats ${\mathrm{SL}}_n$ for $p>n$ with at least two points: the nilpotent logarithms of monodromy may have any Jordan type, including zero. These constructions have parabolic nilpotent singular support and preserve tensor and fusion compatibilities. Manuscript 8 adds, for ${\mathrm{SL}}_n$ with one marked point and $p>n$, a Weil structure and Frobenius compatibility of the full multi-leg eigenstructure, after a finite extension of the constant field, for a geometrically dense arithmetic parameter with tame regular-unipotent monodromy.

Useful notions

• Curves over finite fields and function fields
• Reductive groups and the dual group
• Automorphic functions and local components
• Ramification, inertia, and the Weil group
• Local systems and sheaves

What was known before / what is new

Limits and assumptions

The main enhancement assumes the finite-level Ramanujan–Arthur decomposition; it supplies neither ellipticity, Arthur-packet classification, nor a multiplicity formula. The four characteristic conditions for the geometric strand are mentioned but not detailed in the supplied excerpt; over an arbitrary algebraically closed field, the additional conditions that p be very good and $p\nmid |W_G|$ are also required. Temperedness at every place is announced for globally generic cuspidal representations of split connected adjoint exceptional groups in the stated scope; the broader result concerns only unramified places and requires a generic unramified component. The sheaf constructions have their own restrictions on genus, group, marked points, and monodromy. The source does not specify what is known outside these settings. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Arthur enhancement

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In the main manuscript, additional data involving an algebraic ${\mathrm{SL}}_2$ and a commuting Weil-group map whose diagonal specialization returns the original parameter.

Inertia

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The part of the Weil group carrying local ramification information.

Temperedness

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The property of local components that serves here as the expected conclusion of the Ramanujan conjecture.

Filtration

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An organization of a space by a family of nested subspaces.

Hecke eigensheaf

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A geometric object equipped with isomorphisms expressing compatibility with Hecke operations and a prescribed local system.

Keywords : Restricted geometric Langlands · Arthur enhancements · Ramanujan

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family