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

Technical Summary: Diagonal Cycles on Shtukas and the Adjoint L-Function

Problem Statement
The paper addresses the search for a Gross–Zagier-type identity over function fields for reductive groups of arbitrary type. While the classical Gross–Zagier formula relates the height of Heegner points to the central derivative of an $L$-function, and the arithmetic Gan–Gross–Prasad (GGP) conjectures generalize this to higher dimensions, existing frameworks have been largely restricted to groups of type A, B, or D, and to situations where the governing representation is the standard representation. Previous function field analogues, such as the work of Yun–Zhang and subsequent generalizations by the author and collaborators, were limited to "strongly tempered" spherical varieties, imposing strict constraints on the modification types of Shtukas.

This work investigates the group case $G = H \times H$ and $X = H$, where $H$ is a split almost simple reductive group. The central problem is to relate the self-intersection numbers of the diagonal cycle in the moduli space of $G$-Shtukas (with arbitrary modification types) to higher derivatives of adjoint $L$-functions, thereby extending the Gross–Zagier framework to groups of arbitrary type without the strongly tempered restriction.

Methodology
The proof strategy diverges from the relative trace formula methods used in prior arithmetic intersection studies. Instead, it employs a geometric approach rooted in the geometric Langlands program in positive characteristic and categorical trace formalisms. The methodology proceeds through the following key steps:

1. Geometric Period Integrals and Observables: The author defines a geometric period integral functor $\int_X$ on the category of sheaves on the moduli stack of $G$-bundles ($\text{Bun}_G$). For a geometrically strongly irreducible local system $\sigma$, the cohomology of this period integral is studied.
2. Clifford Algebra Action: A central technical innovation is the construction of a Clifford algebra action on the cohomology of the geometric period integral. This is achieved by establishing a commutator relation between "diagonal" and "adjoint" L-observables. This relation is derived via local computations in the Plancherel algebra using special cohomological correspondences.
3. Atiyah–Bott Type Formula: The paper proves an analogue of the Atiyah–Bott formula for these geometric period integrals, showing that the cohomology ring is a free (odd) polynomial ring generated by tautological classes. This structure allows the intersection pairing to be interpreted as a Frobenius-twisted trace of a specific operator, termed the "intersection observable."
4. Local-to-Global Correspondence: The intersection observable is decomposed into a "derivative part" and a "scalar part." Using the local Plancherel algebra, the author computes these parts explicitly. The derivative part is shown to correspond to the action of the adjoint representation, while the scalar part relates to the determinant line bundle.
5. Trace Formula: The self-intersection number of the diagonal cycle (twisted by determinant line bundles) is expressed as the trace of the intersection observable acting on the cohomology of the period integral. This trace is then computed using the spectral decomposition provided by the Clifford algebra action.

Key Contributions and Results

• Generalized Gross–Zagier Identity: The main result (Theorem 1.4) establishes an identity relating the intersection pairing of the diagonal cycle on the moduli of Shtukas for $H \times H$ to the $r$-th derivative of the adjoint $L$-function $L(s, \check{h}_{\sigma_H})$ at $s=0$. The formula is:
  $$\langle \Delta_{Sht, !}([Sht_{H, \leq \lambda_{H,I}}] \cup \prod c_1(L_{det, i})), \Delta_{Sht, !}[Sht_{H, \leq \lambda_{H,I}}] \rangle^* = \epsilon_{\lambda_I} \frac{L(0, \check{h}_{\sigma_H})}{\log q} \left( -\frac{d}{ds} \right)^r \left( q^{-(g-1)r(H)s} L(s, \check{h}_{\sigma_H}) \right) \Big|_{s=0}$$
  Here, $\epsilon_{\lambda}$ is a constant related to the Dynkin index of the representation, and $r(H)$ is the rank of $H$.
• Arbitrary Modification Types: Unlike previous works restricted to strongly tempered varieties, this result holds for essentially arbitrary representations $V \in \text{Rep}(\check{G})$ governing the modification types of the Shtukas.
• Nondegeneracy of Pairing: The paper proves (Theorem 1.2) that the intersection pairing on the $\sigma$-isotypic component of the intersection cohomology of Shtukas is nondegenerate for geometrically strongly irreducible local systems.
• Clifford Algebra Structure: The construction of the Clifford algebra action on the geometric period integral (Theorem 3.9) is a significant structural contribution, providing a new tool for analyzing intersection numbers in the geometric Langlands setting.
• Explicit Constants: The paper provides explicit formulas for the constants $\epsilon_\lambda$ (Dynkin indices) for various split simple groups and minuscule coweights, highlighting their role as multiplicity factors in the intersection formula.

Significance and Claims
The paper claims to provide the first instance of a Gross–Zagier-type formula for groups of arbitrary type over function fields. By removing the "strongly tempered" restriction, it significantly broadens the scope of the relative Langlands program's geometric realization.

The author suggests that this result points toward a parallel conjectural picture for arithmetic intersections on Shimura varieties. Specifically, the appearance of the constant $\epsilon_\mu$ (the Dynkin index of the Hodge cocharacter) as a multiplicity factor in the intersection formula is identified as a novel phenomenon not previously observed in the literature. The paper posits that Theorem 1.4 serves as a function field model for a generalization of the arithmetic GGP conjectures to arbitrary Shimura varieties, where the behavior of these constants plays a subtle and crucial role.

The work relies on the assumption that the local system $\sigma_H$ is automorphic (which is known for $GL_n$ and expected generally); if $\sigma_H$ is not automorphic, the intersection number vanishes. The results are formulated within the framework of $\infty$-categories and étale sheaves with $\mathbb{Q}_\ell$-coefficients, utilizing the geometric Langlands correspondence as developed in recent works by Gaitsgory, Lurie, and others.