How is the co-rank 1 arithmetic Siegel-Weil formula reduced to local geometric quantities?

What the co-rank 1 formula actually encodes

The arithmetic Siegel-Weil formula in co-rank 1 states that degrees of Kudla-Rapoport arithmetic special 1-cycles are encoded in the first derivatives of Fourier coefficients of unitary Eisenstein series. In other words, a geometric intersection count on one side is matched to an analytic derivative on the other. The setting is a unitary Shimura variety of arbitrarily large even arithmetic dimension, and the cycles live on the exotic smooth Rapoport-Smithling-Zhang integral model of odd relative dimension n-1, which is empty when n is congruent to 0 modulo 4.

The special cycles themselves are concrete. The moduli stack carries a family of Hermitian lattices obtained by taking O_F-linear homomorphisms from one abelian scheme to another, and for a Hermitian matrix T the cycle Z(T) is the substack of tuples whose Gram matrix equals T. Over the generic fiber the morphism Z(T) has relative dimension n-1-rank(T), and Z(T) is empty when T is not positive semi-definite. Co-rank 1 means the rank is one below the ambient dimension, which is why the formula concerns special 1-cycles.

Why global intersection numbers must be split

A global arithmetic intersection number is a single real quantity, but it is built from contributions at every place of the base field. The paper's main objective is to explain how to reduce these global intersection numbers to quantities appearing on the geometric sides of the local main theorems in the companion papers. That reduction is the bridge: the companion papers supply local statements, and this paper shows that the global number is exactly what those local statements add up to.

The decomposition is written out explicitly for arithmetic degrees without boundary contributions. For m at most n, the degree of the cycle paired with a power of the first arithmetic Chern class of a metrized tautological bundle splits into three pieces: an Archimedean integral of a Green current against the Chern class, a horizontal contribution given by the degree of the tautological bundle restricted to the flat part of Z(T), and a sum over primes of local vertical degrees multiplied by log p. This split is conditional on convergence of the Archimedean integral, and it is stated for the case where Z(T) is proper, since the moduli stack is not proper in general and the cycle may need modification on a compactification.

Archimedean and vertical parts are handled separately

The Archimedean piece is not a straightforward integral. The Green current g_{T,y} depends on an auxiliary positive definite matrix y, as is typical in Kudla's program, and the paper constructs the horizontal class using currents that satisfy a modified current equation: the combination of the ddbar-type expression with a delta current supported on the complex fiber of Z(T), wedged with a Chern class, is required to be represented by a smooth form. The paper follows the approach of Garcia and Sankaran, who built such currents using the Mathai-Quillen theory of superconnections, and applies it to the flat part of Z(T).

The non-Archimedean piece is handled by uniformization. The paper studies formal completions of the integral model, defines framing objects and a framed stack, and obtains a uniformization of the local special cycles. This lets the local vertical intersection numbers be computed on the framed side. The paper also notes that the Archimedean and horizontal parts might be treated by the Garcia-Sankaran idea while the vertical part might be treated by derived or K-theoretic methods, and that it seems unknown whether the two approaches can be combined, since there appears to be no derived arithmetic intersection theory at present.

Local horizontal numbers are sums over quasi-canonical cycles

For the horizontal side, the paper defines a local horizontal intersection number for a local Hermitian matrix T' of rank n-1, with m equal to n-1 or n. The definition sums over lattices between the span of a chosen vector and its dual, subject to a length condition, and each term is twice the degree of a quasi-canonical lifting cycle times a local change of tautological height. The extra factor of 2 accounts for the fact that the ring of integers tensored with the p-adic integers has degree 2 over the p-adic integers.

Two features make this definition usable. First, it does not depend on the choice of the vector x_p with Gram matrix T', by the action of the unitary group on the relevant space and Witt's theorem. Second, the formula for the degree of the quasi-canonical lifting cycle shows that each local horizontal intersection number is an integer. The paper also defines a local degree for the horizontal cycle itself, again independent of the choice of vector, and sets both quantities to zero when no such vector exists.

Height changes along isogenies supply the local terms

The local terms are not assumed; they are computed. The paper studies how two heights, the Faltings height and a tautological height, change along isogenies of p-divisible groups. For a specific O_F-linear isogeny built by tensoring an isogeny with the ring of integers, the paper computes the change in the Faltings height and shows it equals minus twice the change in the tautological height, giving a formula in terms of the degree p^s of the isogeny and a constant that depends on whether p is inert, ramified, or split in the imaginary quadratic field.

The computation goes through the Serre tensor construction. The paper identifies the induced map on Lie algebras with an explicit matrix whose determinant generates the different ideal, so the length of the cokernel is expressed in terms of the p-adic valuation of the discriminant. It then shows that the norm of the isogeny on the relevant line bundle is 1, and that the cokernel on the dual tautological bundles vanishes, so the tautological height change is zero in that case. These are the ingredients that let the local horizontal numbers be evaluated rather than merely defined.

Cycle classes are proposed beyond nonsingular matrices

The paper also proposes a new candidate definition of arithmetic cycle classes for possibly singular matrices of arbitrary co-rank. The class is written as a sum of a horizontal part and a sum over primes of vertical parts. The horizontal part is meant to capture both the flat part of Z(T) and a Green current on the Archimedean fiber, while each vertical part is an element in a graded piece of a K-zero group with support on the reduction of Z(T) at that prime.

This construction is presented as a proposal, and the paper is explicit about what it depends on. Because the moduli stack is not proper in general, the class should likely be modified on a suitable compactification, though the authors expect the construction to apply in already-compact situations such as the Rapoport-Smithling-Zhang setup for CM extensions of totally real fields other than the rationals. It may also be necessary to modify the Green currents differently than in the Garcia-Sankaran definition. The paper suggests treating the Archimedean and horizontal parts by the Garcia-Sankaran idea and the vertical part by derived or K-theoretic methods, while noting that combining the two approaches remains open.

What the reduction does not cover

The results are stated under specific hypotheses. In the introduction the imaginary quadratic field has odd discriminant, the Hermitian space has signature (n-1,1) and contains a full-rank self-dual lattice, and the prime 2 is assumed split in the ring of integers. The main interest is the exotic smooth integral model of odd relative dimension n-1, which is empty when n is congruent to 0 modulo 4. The paper notes that versions of the main global results hold for more general models, including odd arithmetic dimension n, at the price of discarding finitely many primes, particularly ramified primes for odd n.

There are also local restrictions. In the section on local horizontal intersection numbers, the paper requires p not equal to 2 when p is inert, because that assumption was needed for the discussion of quasi-canonical lifting cycles. The arithmetic degree decomposition is conditional on convergence of the Archimedean integral and is stated for proper Z(T). These conditions matter because they determine which primes and which dimensions the local-to-global reduction currently reaches.

Where the reduction changes the picture

The most consequential point is the direction of the argument. The paper does not prove the global formula directly; it shows that the global arithmetic intersection number is a sum of local geometric quantities that the companion papers already control. That means the practical test of the program is local: whether the Archimedean integral converges, whether the horizontal local numbers can be summed over lattices with the stated length condition, and whether the vertical local numbers can be computed on the framed stack.

For a reader deciding how much the paper delivers, the key distinction is between the reduction and the construction. The reduction from global intersection numbers to local geometric quantities is carried out, with the explicit three-part decomposition and the local horizontal definition. The arithmetic special cycle classes for singular matrices of arbitrary co-rank remain a proposal, with open questions about compactification and about combining the Archimedean and vertical approaches. The paper's contribution is therefore strongest where the local main theorems already exist and the cycle is proper; it is most tentative exactly where singular matrices and non-proper models enter.

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