Arithmetic Chow Groups Research Articles

A genus two arithmetic Siegel-Weil formula on 𝑋₀(𝑁)

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X 0 ( N ) X_0(N) , for N > 3 N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q q -coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.

Modularity of generating series of divisors on unitary Shimura varieties

We form generating series of special divisors, valued in the Chow group and in the arithmetic Chow group, on the compactified integral model of a Shimura variety associated to a unitary group of signature (n-1,1), and prove their modularity. The main ingredient of the proof is the calculation of the vertical components appearing in the divisor of a Borcherds product on the integral model.

The Maillot–Rössler current and the polylogarithm on abelian schemes

We give a conceptual proof of the fact that the realisation of the degree zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut-K\"ohler higher analytic torsion form of the Poincar\'e bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincar\'e bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.

Higher arithmetic intersection theory

We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over an arithmetic field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a heigh pairing between two higher algebraic cycles of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of higher arithmetic intersection pairing in dimension zero.

On two arithmetic theta lifts

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of$U(n,1)$Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology

In 2014, Kings and Rössler showed that the realization of the degree zero part of the abelian polylogarithm in analytic Deligne cohomology can be described in terms of a class of currents which was previously defined by Maillot and Rössler and strongly related to the Bismut–Köhler higher torsion form of the Poincaré bundle. In this paper we show that, if the base of the abelian scheme is proper, Kings and Rössler’s result can be refined to hold already in Deligne–Beilinson cohomology. More precisely, by means of Burgos’ theory of arithmetic Chow groups, we prove that the class of currents defined by Maillot and Rössler has a representative with logarithmic singularities at the boundary and therefore defines an element in Deligne–Beilinson cohomology. This element coincides with the realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology.

Arakelov motivic cohomology II

We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from BGL \operatorname {BGL} to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic K K -theory and arithmetic Chow groups. For example, this implies a decomposition of higher arithmetic K K -groups in its Adams eigenspaces. Finally, we give a conceptual explanation of the height pairing: it is the natural pairing of motivic homology and Arakelov motivic cohomology.

Complex multiplication cycles and Kudla-Rapoport divisors, II

This paper is about the arithmetic of {\it Kudla-Rapoport divisors} on Shimura varieties of type ${\rm GU}(n-1,1)$. In the first part of the paper we construct a toroidal compactification of N.~Kr\"amer'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-K\"uhn. 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 {\it 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.

Heights of Kudla–Rapoport divisors and derivatives of $$L$$ L -functions

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f) and the theta function of a positive definite hermitian lattice of rank n-1. When specialized to the case n=2, this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature (1,1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

Unitary cycles on Shimura curves and the Shimura lift II

AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series \({\widehat{\phi}_{\mathcal C}(\tau)}\) defined using arithmetic 0-cycles on the moduli space \({\mathcal C}\) of elliptic curves with CM by the ring of integers \({O_{\kappa}}\) of an imaginary quadratic field. The second such series \({\widehat{\phi}_{\mathcal M}(\tau)}\) has weight 3/2 and takes values in the arithmetic Chow group \({\widehat{{\rm CH}}^1(\mathcal{M})}\) of the arithmetic surface associated to an indefinite quaternion algebra \({B/\mathbb{Q}}\). For an embedding \({O_\kappa \rightarrow O_B}\), a maximal order in B, and a two sided O B -ideal Λ, there is a morphism \({j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}\) and a pullback \({j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}\). Our main result is an expression for the pullback \({j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}\) as a linear combination of products of \({\widehat{\phi}_{\mathcal C}(\tau)}\)’s and classical weight \({\frac{1}{2}}\) theta series.

Higher arithmetic Chow groups

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. For projective varieties the degree zero group agrees with the arithmetic Chow groups defined by Gillet and Soulé, and in general with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.

Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space $${L^2_1}$$ . We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross–Zagier formula and a calculation due independently to Bost and Kühn we compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the Néron-Tate height pairing.

On Goncharov’s Regulator and Higher Arithmetic Chow Groups

In this paper, we show that the regulator defined by Goncharov in [10] from higher algebraic Chow groups to Deligne–Beilinson cohomology agrees with Beilinson’s regulator. We give a direct comparison of Goncharov’s regulator to the construction given by Burgos Gil and Feliu in [5]. As a consequence, we show that the higher arithmetic Chow groups defined by Goncharov agree, for all projective arithmetic varieties over an arithmetic field, with the ones defined by Burgos Gil and Feliu.

Semipurity of tempered deligne cohomology

In this paper we define the formal and tempered Deligne cohomology groups, that are obtained by applying the Deligne complex functor to the complexes of formal differential forms and tempered currents respectively. We then prove the existence of a duality between them, a vanishing theorem for the former and a semipurity property for the latter. The motivation of these results comes from the study of covariant arithmetic Chow groups. The semi-purity property of tempered Deligne cohomology implies, in particular, that several definitions of covariant arithmetic Chow groups agree for projective arithmetic varieties.

COHOMOLOGICAL ARITHMETIC CHOW RINGS

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

Arithmetic characteristic classes of automorphic vector bundles

We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values in an arithmetic Chow ring constructed by means of differential forms with certain log-log type singularities. We first study the cohomological properties of log-log differential forms, prove a Poincaré lemma for them and construct the corresponding arithmetic Chow groups. Then, we introduce the notion of log-singular hermitian vector bundles, which is a variant of the good hermitian vector bundles introduced by Mumford, and we develop the theory of arithmetic characteristic classes. Finally we prove that the hermitian metrics of automorphic vector bundles considered by Mumford are not only good but also log-singular. The theory presented here provides the theoretical background which is required in the formulation of the conjectures of Maillot-Roessler in the semi-abelian case and which is needed to extend Kudla's program about arithmetic intersections on Shimura varieties to the non-compact case.