Quasi-canonical AFL and Arithmetic Transfer conjectures at parahoric levels

Technical Summary: Quasi-Canonical AFL and Arithmetic Transfer Conjectures at Parahoric Levels

Problem Statement
The paper addresses the formulation and verification of Arithmetic Transfer (AT) conjectures, which are variants of the Arithmetic Fundamental Lemma (AFL) conjecture in the presence of ramification. The classical AFL conjecture, established by Zhang, Mihatsch–Zhang, and Z. Zhang, relates the derivative of an orbital integral of the unit element in a spherical Hecke algebra to an arithmetic intersection number on a Rapoport–Zink (RZ) formal moduli space of $p$-divisible groups. This classical setting requires an unramified quadratic extension $F/F_0$, a special vector of unit length, and a hyperspecial level structure.

The authors investigate the scenario where the "unramifiedness" hypotheses are relaxed, specifically by considering non-hyperspecial parahoric level structures. This introduces ramification into the problem. A primary obstacle in formulating AT conjectures in this context is the geometric side: the ambient RZ spaces must be regular to define Serre intersection multiplicities. When the level structure is non-hyperspecial, the natural product spaces often fail to be regular, necessitating the construction of new integral models and the resolution of singularities.

Methodology
The paper is divided into two independent parts, employing a blend of arithmetic geometry, deformation theory, and harmonic analysis.

1. Geometric Construction and Analysis (Part 2):

   • The authors focus on the space $\widetilde{\mathcal{N}}_n^{[1]}$, an integral model of a member of the RZ tower corresponding to a non-parahoric subgroup in a non-quasisplit unitary group. This space arises as a fiber product linking $\mathcal{N}_n^{[1]}$ (level $r=1$) and $\mathcal{N}_{n+1}^{[0]}$ (hyperspecial level).
   • Using Dieudonné theory and Grothendieck–Messing deformation theory, the authors analyze the $\bar{k}$-points and the local structure of $\widetilde{\mathcal{N}}_n^{[1]}$ and the special divisor $Z(u)$ (associated with a special vector of valuation 1).
   • They prove that $\widetilde{\mathcal{N}}_n^{[1]}$ is regular of dimension $n$. The morphism $\pi_1: \widetilde{\mathcal{N}}_n^{[1]} \to \mathcal{N}_n^{[1]}$ is shown to be finite flat of degree $q+1$, étale away from a "balloon locus" and totally ramified along it. The morphism $\pi_2: \widetilde{\mathcal{N}}_n^{[1]} \to Z(u)$ is a blow-up along a zero-dimensional subscheme.
   • The special fiber of $Z(u)$ is shown to be regular outside a zero-dimensional closed subset, with the singularities being of a specific type (quasi-canonical).

2. Arithmetic Transfer Conjectures and Proofs (Part 1):

   • The authors define arithmetic intersection numbers on the regular spaces constructed above, specifically on products like $\widetilde{\mathcal{N}}_n^{[r]} \times \mathcal{N}_{n+1}^{[0]}$ and their "graph versions" $\widetilde{\mathcal{N}}_n^{[r]} \times \mathcal{N}_{n+1}$.
   • They formulate AT conjectures relating these intersection numbers to the derivatives of orbital integrals of specific test functions. These test functions are constructed using base change homomorphisms between spherical Hecke algebras and atomic Hecke functions.
   • Graph Version: The authors prove a "graph version" of the AT conjecture. By relating the intersection number on the graph of the morphism $\pi_2$ to the intersection number on the special divisor $Z(u)$, they reduce the problem to the known AFL (for even $r$) and the newly established "quasi-canonical AFL" (for odd $r$).
   • Quasi-Canonical AFL: A central result is the proof of the quasi-canonical AFL, which handles the case where the special vector has valuation 1 (level 1). This involves establishing a transfer theorem for specific non-unit elements in the Hecke algebra and proving the corresponding identity for the intersection number on the quasi-canonical divisor $Z(u)$.
   • Reduction to Known Cases: For cases where $r$ is even, the authors show that the AT conjecture follows from the AFL for the whole Hecke algebra (Conjecture 9.2.3 in [14]), provided the necessary transfer functions exist. For specific small cases (e.g., $n=r=1$), they verify the conjectures explicitly.

Key Contributions and Results

• Structure of $\widetilde{\mathcal{N}}_n^{[1]}$: The paper provides a complete structural description of the mysterious space $\widetilde{\mathcal{N}}_n^{[1]}$, proving its regularity and detailing the geometry of the morphisms $\pi_1$ and $\pi_2$. This confirms conjectures by Kudla and Rapoport regarding the geometry of almost self-dual RZ towers.
• Quasi-Canonical AFL: The authors establish the Quasi-Canonical AFL (Theorem 7.2.1), which relates the intersection number on the quasi-canonical divisor $Z(u)$ (valuation 1) to the derivative of orbital integrals of a specific transfer function. This fills a gap in the AFL theory for ramified settings.
• Graph Version of AT Conjectures: The paper proves the graph version of the AT conjecture for type $(r, 0)$ (Theorem 1.0.2). This result is "reasonably complete," covering cases where $r$ is even (via reduction to AFL) and $r$ is odd (via reduction to the quasi-canonical AFL).
• Explicit AT Conjectures: The authors formulate explicit AT conjectures for various types $(r, 0)$ and $(0, r)$, identifying candidate test functions via base change homomorphisms.
  • For even $r$, the conjecture is linked to the AFL for the full Hecke algebra.
  • For odd $r=n$, a specific conjecture is formulated and proven for $n=1$.
  • For other odd $r$, the existence of a transfer function is postulated, but explicit candidates are not provided in all cases.
• Verification in Simple Cases: The paper proves the conjectures in specific simple cases, notably $n=1$ and $r=1$, and demonstrates how the general conjectures follow from the AFL for the whole Hecke algebra in the even $r$ case.

Significance and Scope
The paper claims to have exhausted the possible cases for AT conjectures under natural constraints: the ambient space must be a product of RZ spaces of maximal parahoric levels and be regular, and the cycles must be pull-backs of exceptional special divisors (Z-divisors or Y-divisors).

The significance lies in extending the AFL framework beyond the unramified, hyperspecial setting. By proving the regularity of the necessary integral models and establishing the quasi-canonical AFL, the authors provide the geometric foundation required to formulate and partially prove AT conjectures in the presence of parahoric level structures. The results verify the hypotheses of the graph version of the AT conjectures in a particular case and offer a roadmap for future work on the remaining cases where explicit test functions are not yet known. The authors explicitly note that while they have formulated conjectures for all remaining cases, they do not have explicit candidates for the test functions in the general odd $r$ case (beyond $r=n=1$) and leave this for future investigation.