On two geometric theta lifts

@article{Bruinier2002OnTG,
  title={On two geometric theta lifts},
  author={Jan H. Bruinier and Jens Funke},
  journal={Duke Mathematical Journal},
  year={2002},
  volume={125},
  pages={45-90}
}
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result between Borcherds's singular theta lift and the Kudla-Millson lift. We extend this result to arbitrary signature by introducing a new singular theta lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character, in the sense of Cheeger and Simons, for the cycles… 
The Shimura-Shintani correspondence via singular theta lifts and currents
Abstract. We describe the construction and properties of a singular theta lift for the orthogonal group SO(2, 1). We obtain locally harmonic Maass forms in the sense of [6] with singular sets along
Regularized theta lifts for orthogonal groups over totally real fields
Abstract We define a regularized theta lift from SL2 to orthogonal groups over totally real fields. It takes harmonic ‘Whittaker forms’ to automorphic Green functions and weakly holomorphic Whittaker
CM values of automorphic Green functions on orthogonal groups over totally real fields
Generalizing work of Gross--Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing
Regularized theta liftings and periods of modular functions
In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of
Lifting cusp forms to Maass forms with an application to partitions
TLDR
This construction answers a question of Dyson by providing the general framework “explaining” Ramanujan's mock theta functions by showing that the number of partitions of a positive integer n is the “trace” of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi–Yau threefold.
Deformations of Theta Integrals and A Conjecture of Gross-Zagier
In this paper, we complete the proof of Gross-Zagier’s conjecture concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an
LIFTING ELLIPTIC CUSP FORMS TO MAASS FORMS WITH AN APPLICATION TO PARTITIONS
Abstract. For 2 < k ∈ 1 2 Z, we define lifts of cuspidal Poincaré series in Sk(Γ0(N)) to weight 2 − k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general
Regularized theta lifts and (1,1)-currents on GSpin Shimura varieties
We introduce a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of
A Singular Theta Lift and the Shimura Correspondence
Modular forms play a central and critical role in the study of modern number theory. These remarkable and beautiful functions have led to many spectacular results including, most famously, the proof
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
...
...

References

SHOWING 1-10 OF 56 REFERENCES
Correspondence of modular forms to cycles associated to Sp(p,q)
In their study of Hubert modular surfaces, Hirzebruch and Zagier [13] have discovered a striking connection between geometry and number theory. It was established that intersection numbers of cycles
Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables
Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq)
Tubes, Cohomology with Growth Conditions and an Application to the Theta Correspondence
In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point
Heegner Divisors and Nonholomorphic Modular Forms
We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a
Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors
Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The
Borcherds products and arithmetic intersection theory on Hilbert modular surfaces
We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coecients of an elliptic modular form of weight two.
Integrals of automorphic Green's functions associated to Heegner divisors
In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group $\Orth(2,p)$ and for the integrals of certain automorphic Green's
Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms
The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the
Automorphic forms with singularities on Grassmannians
We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to
Derivatives of Eisenstein series and Faltings heights
We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an
...
...