Jens Funke

On two geometric theta lifts

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 Borcherdss 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 under consideration.

Traces of CM values of modular functions

Abstract Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL2(ℤ) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagiers Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

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 certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

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 case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.

The geometric theta correspondence for Hilbert modular surfaces

We give a new proof and an extension of the celebrated theorem of Hirzebruch and Zagier [17] that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles in (certain) Hilbert modular surfaces is a classical modular form of weight 2. In our approach we replace Hirzebuch’s smooth complex analytic compactification of the Hilbert modular surface with the (real) Borel-Serre compactification. The various algebro-geometric quantities that occur in [17] are replaced by topological quantities associated to 4-manifolds with boundary. In particular, the “boundary contribution” in [17] is replaced by sums of linking numbers of circles (the boundaries of the cycles) in 3-manifolds of type Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.

On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift.

We consider the Kudla-Millson lift from elliptic modular forms of weight (p + q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p,q). We study the L 2 -norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L 2 -norm of the lift, which often implies that the lift is in- jective. For O(p,2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift.

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.