Arithmetic volume of Shtukas and Langlands duality

Abstract: We extend the work of Feng--Yun--Zhang relating the arithmetic volume of Shtukas with derivatives of zeta functions by allowing arbitrary coweights for split semisimple algebraic groups. As in their original work, the formula involves some numbers called eigenweights. We obtain uniform formulas for the eigenweights in terms of the Langlands dual group, marking the first structural role for the dual group in such formulas governing derivatives of L-functions.

• The paper introduces a uniform arithmetic volume formula for Shtukas using general dominant coweights and explicitly links eigenweights with the Langlands dual group.
• It employs relative Hecke operators and intersection theory to compute volume invariants across classical and exceptional groups.
• The framework advances the computation of higher derivatives of L-functions and offers structural insights into Gross–Zagier-type phenomena.

Arithmetic Volumes of Shtukas and Langlands Duality

Introduction and Context

The paper "Arithmetic volume of Shtukas and Langlands duality" (2604.03763) presents a conceptual and technical advance in the study of arithmetic volumes in the moduli theory of Shtukas, especially their relation to derivatives of $L$-functions and explicit structural connections with the Langlands dual group. Moduli spaces of Shtukas play a role analogous to Shimura varieties in the function field setting, and the arithmetic intersection theory on these stacks provides deep input into understanding the Langlands correspondence, special cycles, and Gross–Zagier-type phenomena.

The authors extend the results of Feng–Yun–Zhang (Feng et al., 26 Jan 2026) who related arithmetic volumes, for the case of minuscule modifications, with higher derivatives of zeta functions of curves. The current work achieves three major improvements: generalization to arbitrary coweights (hence arbitrary modification types), a uniform and conceptually satisfying description of the so-called "eigenweights" in terms of the Langlands dual group $\hat G$, and explicit elementary computations and parameterizations for these invariants.

Setup: Shtuka Moduli, Hecke Correspondences, and Volumes

Let $C$ be a smooth, proper, connected curve over $\mathbb{F}_q$, $G$ a split semisimple group over $\mathbb{F}_q$, and $\operatorname{Bun}_G$ the moduli stack of $G$-bundles on $C$. Dominant coweights $\lambda\in X_*(T)_+$ (where $\hat G$0 is a maximal torus in $\hat G$1) index modification types; via Hecke stacks and their iterations, one builds the moduli of Shtukas $\hat G$2 with multi-leg modifications of type $\hat G$3.

The paper refines the definition of arithmetic volume for these moduli—even in the absence of smoothness and properness—by leveraging cohomological correspondences, Hecke algebra actions, and the Grothendieck–Lefschetz trace formula. The central numerical invariant is

$\hat G$4

for a cohomology class $\hat G$5 coming from an appropriate "volume datum," abstracting all previous ad hoc definitions and extending them via intersection cohomology.

Main Results: Explicit Formulae and the Role of the Langlands Dual Group

The key advances are:

Formula Structure for Arbitrary Coweights

The arithmetic volume formula for the general setup ($\hat G$6, arbitrary dominant coweights) is given as an iterated differential operator (formed from "relative Hecke operators" parameterized by the data) applied to a zeta-type $\hat G$7-function of the curve:

$\hat G$8

where the $\hat G$9 are specific differential operators encoding both a numerical term and a linear combination of partial derivatives weighted by the eigenweights

$C$0

and

$C$1

with $C$2 the fundamental degrees of $C$3, $C$4 the eigenweights, and $C$5 an additional explicit constant.

Structural Identification of Eigenweights

A novel contribution is the description of the eigenweights $C$6 in terms of the Langlands dual group $C$7. They are realized as explicit traces of invariants on the "Gross motive" for $C$8, filtered and graded via the geometry of local models, and ultimately expressed in terms of the Langlands dual adjoint representation and the Kostant slice. This links the structure constants appearing in the volume formula to spectral data of $C$9, thus making the dual group directly and explicitly responsible for the arithmetic of Shtuka intersection volumes.

Uniformity and Explicit Computations

The general formula is shown to yield uniform and computationally effective descriptions of eigenweights and constants for all classical groups and, for many cases, for exceptional groups. The authors present explicit recursions and combinatorial formulas for types $\mathbb{F}_q$0, $\mathbb{F}_q$1, $\mathbb{F}_q$2, $\mathbb{F}_q$3, and some exceptional types ($\mathbb{F}_q$4, $\mathbb{F}_q$5, $\mathbb{F}_q$6, $\mathbb{F}_q$7, $\mathbb{F}_q$8), with all the relevant invariants ($\mathbb{F}_q$9, $G$0, and degrees) computed.

Methods and Technical Framework

Compared to the previous literature (Feng et al., 26 Jan 2026), which relied on the geometry of Vinberg semigroups and wonderful compactifications (effectively restricting attention to minuscule coweight cases), this paper develops an approach based on the careful study of "relative Hecke operators"—first locally, in terms of convolution and the Satake category, and then globally, via a local–global compatibility theorem. This methodology allows for recursion in coweights, functoriality across group homomorphisms, and histrionic connections with the Satake isomorphism and the categorical structure on affine Grassmannians ([BF], [yun2011integral]).

Fundamental to the technical argument is the derivation of explicit commutator formulas between Hecke correspondences and Frobenius, reducing the global computation to "eigenweight" data on the dual group. The identification of duality actions, filtration compatibilities, and the universal commutator formula are all integral.

Implications and Connections

These results provide the first structural link between Langlands duality and the explicit arithmetic formulae governing derivatives of $G$1-functions in the context of Shtukas, giving robust evidence to the general philosophy that "the Langlands dual group governs the intersection theory of special cycles" (in the function field field).

Practically, this enables concrete computations of intersection numbers, arithmetic volumes, and traces of period operators for arbitrary modification types, which is essential for formulating and testing geometric versions of Gross–Zagier-type formulas and arithmetic Siegel–Weil theorems in higher rank and general type. Conceptually, the explicit appearance of $G$2 in functional invariants renders the "relative Langlands duality" paradigm truly operative in this domain (Ben-Zvi et al., 2024).

Moreover, the explicit computations of eigenweights as ratios of invariants on $G$3, and the ability to express these in elementary terms for classical types, makes the theory ready for applications in both theoretical exploration and computational experimentation.

Future Directions

The methods and results lay the foundation for several possible advances:

• Extension from split semisimple to general reductive groups, potentially incorporating twisted and non-quasi-split cases.
• Potential refinement of the local–global matching and "eigenweight" formalism to special cycles and arithmetic intersection theory for Shimura varieties in the number field case, bridging the currently wider gap between the function field and number field analogues.
• Application to conjectures on arithmetic Siegel–Weil and higher derivatives, critical values, and refined Birch–Swinnerton-Dyer–type formulae for $G$4-functions.
• Deeper exploration of the relation with the Plancherel algebra, the Poisson structure on the dual group, and new perspectives on categorical traces and their arithmetic significance (see [BZSV], (Liu et al., 31 Mar 2025)).

Conclusion

This paper provides a uniform, explicit, and deeply conceptual framework for the computation of arithmetic volumes of Shtukas for arbitrary modification types, connecting the output directly to canonical data on the Langlands dual group. It elevates the role of the dual group from a philosophical backdrop to an effective computational and structural tool in arithmetic intersection theory for moduli of Shtukas. The techniques introduced are poised to inform ongoing efforts in higher automorphic theory, arithmetic geometry over function fields, and beyond.

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References

• "Arithmetic volumes of moduli stacks of Shtukas" (Feng et al., 26 Jan 2026)
• "Eigenweights for arithmetic Hirzebruch Proportionality" (Feng, 30 Jan 2026)
• "Relative Langlands duality" (Ben-Zvi et al., 2024)
• "Integral homology of loop groups via Langlands dual groups" [yun2011integral]
• [BF], [MV] as cited in the original manuscript
• See full references within (2604.03763) for additional attributions.