Regularized Theta Lifts Research Articles

A classification of harmonic weak Maaß forms of half-integral weight

We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by Schulze-Pillot and parallels recent work by Bringmann–Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. Harish-Chandra modules in both integral and half-integral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to an initial understanding of a theta correspondence for extensions of Harish-Chandra modules.

Modularity of special cycles on unitary Shimura varieties over CM-fields

We study the modularity of the generating series of special cycles on unitary Shimura varieties over CM-fields of degree $2d$ associated with a Hermitian form in $n+1$ variables whose signature is $(n,1)$ at $e$ real places and $(n+1,0)$ at the remaining $d-e$ real places for $1\leq e <d$. For $e=1$, Liu proved the modularity and Xia showed the absolute convergence of the generating series. On the other hand, Bruinier constructed regularized theta lifts on orthogonal groups over totally real fields and proved the modularity of special divisors on orthogonal Shimura varieties. By using Bruinier's result, we work on the problem for $e=1$ and give an another proof of Liu's proof. For $e>1$, we prove that the generating series of special cycles of codimension $er$ in the Chow group is a Hermitian modular form of weight $n+1$ and genus $r$, assuming the Beilinson-Bloch conjecture with respect to orthogonal Shimura varieties. Our result is a generalization of $\textit{Kudla's modularity conjecture}$, solved by Liu unconditionally when $e=1$.

Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for Γ0(N) as regularized theta lifts of a single type of Poincaré series, due to Selberg. Using this realization and properties of the Poincaré series we derive the meromorphic continuation and Kronecker limit formulas for the above Eisenstein series. The corresponding Kronecker limit functions are then given by the logarithm of the absolute value of the Borcherds product associated to a special value of the underlying Poincaré series.

Cycle integrals of meromorphic modular forms and coefficients of harmonic Maass forms

In this paper, we investigate traces of cycle integrals of certain meromorphic modular forms. By relating them to regularized theta lifts we provide explicit formulae for them in terms of coefficients of harmonic Maass forms.

Theta lifts for Lorentzian lattices and coefficients of mock theta functions

We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews’ spt-function, and Ramanujan’s mock theta functions.

Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)

These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh. In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SLN (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GLN, GL1). This suggests looking to reductive dual pairs (GLN, GLk) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms. In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.

CM values of Green functions associated to special cycles on Shimura varieties with applications to Siegel 3-fold X2(2)

We generalize the definition of CM cycles beyond the ‘small’ and ‘big’ CM ones studied by various authors, such as in [BY09] and [BKY12] and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of regularized theta lifts. Finally, as an application to Siegel 3-fold X2(2), we can compute special values of theta functions and Rosenhain λ-invariants at a CM cycle, which is useful for genus two curve cryptography as in [CDSLY14].

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.

Errata for Stephan Ehlen, “CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1,” Duke Math. J., Volume 166, Number 13 (2017), 2447–2519

Errata for Stephan Ehlen, “CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1,” Duke Math. J., Volume 166, Number 13 (2017), 2447–2519

CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maa{\ss} forms of weight one. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight one studied by Kudla, Rapoport and Yang. The results of the paper are much improved in comparison to the 2012 preprint arxiv:1208.2386 which contained partial results in the same direction. Moreover, they are also an improvement of the main result of the authors thesis (CM values of regularized theta lifts, TU Darmstadt, 2013).

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 $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we prove that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In the second part to this paper, we will show how to evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts.

Regularized pairings of meromorphic modular forms and theta lifts

We show that the meromorphic modular forms recently considered by Bringmann and Kane can be obtained as images of regularized theta lifts of Poincaré series under weight raising operators. We use this fact in order to simplify the evaluation of the regularized pairing, also defined by Bringmann and Kane, of these functions with other meromorphic modular forms.

Periods of the j -function along infinite geodesics and mock modular forms

Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight 1 2 to traces of values of the modular j-function at imaginary quadratic points. A real quadratic analogue was recently studied by Duke, Imamoḡlu, and Tóth. They showed that the coefficients of certain weight 1 2 mock modular forms f D = ∑ d > 0 a ( d , D ) q d , D > 0 are given in terms of traces of cycle integrals of the j-function. Their result applies to those coefficients a ( d , D ) for which d D is not a square. Recently, Bruinier, Funke, and Imamoḡlu employed a regularized theta lift to show that the coefficients a ( d , D ) for square d D are traces of regularized integrals of the j-function. In the present paper, we provide an alternate approach to this problem. We introduce functions j m , Q (for Q a quadratic form) which are related to the j-function and show, by modifying the method of Duke, Imamoḡlu, and Tóth, that the coefficients for which d D is a square are traces of cycle integrals of the functions j m , Q .

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.

Regularized theta lifts and Niebur-type Poincaré series on n -dimensional hyperbolic space

We study theta liftings from H n to H 2 and vice versa. The theta-lift is realized via an integral transform with a Siegel theta series as kernel function. Since this Siegel theta series fails to be square integrable it has to be regularized. The regularization is obtained by applying a suitable differential operator built from the Laplacian. The Siegel theta series is seen to be related to an automorphic Selberg kernel function on H n and therefore the Selberg transform applies to compute the Fourier coefficients of the lift. This gives formulas of Katok–Sarnak type for Fourier coefficients of positive as well as for negative index, involving geodesic cycles. As an application we can lift Poincaré series of Niebur type. The lifted series are again Poincaré series built with Whittaker functions.

Regularized theta lifts for orthogonal groups over totally real fields

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 forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results, we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.

Special Values of Green Functions at Big CM Points

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin-Selberg type convolution.

Borcherds forms and generalizations of singular moduli

We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature (n,2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can occur in this factorization. One remarkable phenomenon we observe is that the regularized theta lift of a weakly holomorphic modular form is always finite.