Complex multiplication cycles and Kudla-Rapoport divisors

Howard, Benjamin

doi:10.4007/annals.2012.176.2.9

Cited by 30 publications

(81 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2) The manuscript for our paper has been in circulation for several years and, in the intervening time, results concerning our special cycles in the case of M.n 1; 1/ have been obtained by Ben Howard [15,16], and by Bruinier, Howard and Yang [5]. In particular, a very complete treatment of proper integral models of compactifications is contained in [16].…”

Section: Introductionmentioning

confidence: 95%

Special cycles on unitary Shimura varieties II: Global theory

Kudla

Rapoport

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

145

View full text Add to dashboard Cite

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature .n 1; 1/. We define arithmetic cycles on these models and study their intersection behavior. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s D 0 of a certain incoherent Eisenstein series for the group U.n; n/. This is done by relating the arithmetic cycles to their formal counterpart from part I [33] via nonarchimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of part I and a counting argument.

show abstract

Section: Introductionmentioning

confidence: 95%

Special cycles on unitary Shimura varieties II: Global theory

Kudla

Rapoport

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

145

View full text Add to dashboard Cite

show abstract

“…First we expose some basic properties of the ideal J Φ . Following [10], let Lie Φ be defined by the exactness of the sequence of O K ⊗ O L modules…”

Section: Application To Moduli Spaces Of Abelian Schemesmentioning

confidence: 99%

Serre’s tensor construction and moduli of abelian schemes

Amir-Khosravi

2017

manuscripta math.

View full text Add to dashboard Cite

Given a polarized abelian scheme with action by a ring, and a projective finitely presented module over that ring, Serre's tensor construction produces a new abelian scheme. We show that to equip these abelian schemes with polarizations it's enough to equip the projective modules with non-degenerate positive-definite hermitian forms. As an application, we relate certain moduli spaces of principally polarized abelian schemes with action by the ring of integers of a CM field. More specifically, we consider integral models of zero-dimensional Shimura varieties associated to compact unitary groups. We show that all abelian schemes in such moduli spaces are,étale locally over their base schemes, Serre constructions of CM abelian schemes with positive-definite hermitian modules. We also describe the morphisms between such objects in terms of morphisms between the constituent data, and formulate these relations as an isomorphism of algebraic stacks. IntroductionLet R be a ring, possibly non-commutative, and free of finite rank over . Let (A, ι) be an abelian scheme A over a base S, with an injective ring homomorphism ι : R ֒→ End S (A) giving an R-action on A. Take M to be a projective finitely presented right R-module. Serre's tensor construction associates to this data a new abelian scheme M ⊗ R A over S, which is characterized by its functor of points Sch /S → Ab, T → M ⊗ R A(T ) (Definition 1). The map A → M ⊗ R A is functorial in A and M , and preserves many desirable properties of A. This suggests the possibility of using it to relate families of abelian schemes. In order to do this, we first need to equip M ⊗ R A with extra structures, in particular a polarization [20,6] that is compatible with the R-action in the following sense.Assume R is equipped with a positive involution r → r * (Definition 4). Then the pair (A, ι) has a dual (A ∨ , ι ∨ ), where A ∨ is the dual abelian scheme of A, and ι ∨ (r) = ι(r * ) ∨ , for r ∈ R. A polarization λ :The dual module M ∨ = Hom R (M, R) has a natural right R-module structure, with r ∈ R acting on f ∈ M ∨ by (f · r)(m) = r * f (m). Then R-linear maps h : M → M ∨ may be identified with sesquilinear forms H : M × M → R via H(m, m ′ ) = h(m)(m ′ ). Such a map h is called hermitian if H(m, m ′ ) = H(m ′ , m) * , and non-degenerate if it's an isomorphism. Since M É ≃ R n É , we may identify h with an element of H n (R É ), the set of n × n hermitian matrices with entries in R É . Then H n (R É ) ⊗ Ê is a formally real Jordan algebra (Definition 8) over Ê. We say h is positive-definite if its image under H n (R É ) ⊂ H n (R É ) ⊗ Ê is positive (Definition 15). This notion does not depend on the choice of isomorphism M É ≃ R n É (Lemma 16).Theorem A. Suppose (A, ι, λ) consists of an abelian scheme A/S, an R-action ι : R ֒→ End(A), and R-linear polarization λ : A → A ∨ . Let h : M → M ∨ be R-linear. The map h ⊗ λ : M ⊗ R A → M ∨ ⊗ R A ∨ is a polarization on M ⊗ R A if and only if h is a positive definite R-valued hermitian form.The above is Theorem 17 in the main text. That the abel...

show abstract

“…The main results of [14] consist of calculations of the intersection multiplicity of naive versions of X Φ and Z(m) on the (non-compact, non-regular, and non-flat) Shimura variety M (1,0) 1) . By reducing to the calculations of [14], we are able to prove in Section 4.2 a precise formula for [ Z(m, v) : X Φ ], and show that this value is related to the Fourier coefficients of Eisenstein series.…”

Section: Intersections With CM Cycles Having Constructed Arithmetic mentioning

confidence: 99%

“…This implies, by dimension considerations, that Z(m) contains an entire irreducible component of M. As M is flat over O k , it follows that Z(m) /C contains an irreducible component of M /C . But Z(m) /C is a divisor on M /C , as one can check using the explicit complex uniformization of [20] or [14].…”

Section: This Gives a Decompositionmentioning

confidence: 99%

Complex multiplication cycles and Kudla-Rapoport divisors, II

Howard¹

2015

ajm

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is about the arithmetic of Kudla-Rapoport divisors on Shimura varieties of type GU(n − 1, 1). In the first part of the paper we construct a toroidal compactification of N. Krämer's integral model of the Shimura variety. This extends work of K.-W. Lan, who constructed a compactification at unramified primes.In the second, and main, part of the paper we use ideas of Kudla to construct Green functions for the Kudla-Rapoport divisors on the open Shimura variety, and analyze the behavior of these functions near the boundary of the compactification. The Green functions turn out to have logarithmic singularities along certain components of the boundary, up to log-log error terms. Thus, by adding a prescribed linear combination of boundary components to a Kudla-Rapoport divisor one obtains a class in the arithmetic Chow group of Burgos-Kramer-Kühn.In the third and final part of the paper we compute the arithmetic intersection of each of these divisors with a cycle of complex multiplication points. The computation is quickly reduced to the calculations of the author's earlier work Complex multiplication cycles and Kudla-Rapoport divisors. The arithmetic intersection multiplicities are shown to appear as Fourier coefficients of the diagonal restriction of the central derivative of a Hilbert modular Eisenstein series.

show abstract

Complex multiplication cycles and Kudla-Rapoport divisors

Cited by 30 publications

References 36 publications

Special cycles on unitary Shimura varieties II: Global theory

Special cycles on unitary Shimura varieties II: Global theory

Serre’s tensor construction and moduli of abelian schemes

Complex multiplication cycles and Kudla-Rapoport divisors, II

Contact Info

Product

Resources

About