Regularized Lift from Harmonic Weak Maass Forms

• The paper demonstrates that regularized lifts using the Millson theta kernel translate harmonic weak Maass forms into half-integral weight forms, encoding cycle integrals, CM traces, and central L-values.
• It details a precise construction and regularization method that handles divergent integrals by subtracting constant terms, with mapping properties established through the xi-diagram.
• This research establishes clear arithmetic identities among cycle integrals and links vanishing central L-values with holomorphic properties, thereby extending the theta lift framework.

Regularized lifts from harmonic weak Maass forms—most notably, those built on the Millson kernel—constitute a core mechanism for transferring analytic, geometric, and arithmetic information between spaces of modular forms of disparate weights and types. Such regularized lifts, generalizing the classical Shintani and Borcherds theta lifts, relate integral or negative weight harmonic weak Maass forms to half-integral weight harmonic Maass forms, and encode deep relationships among special values, traces, cycle integrals, and central values of $L$-functions. The canonical construction involves a delicate regularization of divergent integrals against vector-valued theta kernels, with output characterized by explicit formulas for Fourier coefficients in terms of traces of CM values and geodesic cycle integrals.

1. Definition of the Millson Theta Kernel and Regularized Integral

Let $V$ be a rational quadratic space of signature $(1,2)$, $L \subset V$ an even lattice, and $D$ the Grassmannian of positive lines (identifiable with $\mathbb H$). For $z = x+iy \in \mathbb H$ and $\tau = u+iv \in \mathbb H$, the Millson–Schwartz kernel $\psi_{M,k}(X, \tau, z)$ is defined as

$$\psi_{M,k}(X, \tau, z) = v^{k+1} p_z(X) Q_X(\bar z)^k \exp(-2\pi v R(X,z)) e^{2\pi i Q(X) \tau},$$

where

$$p_z(X) = \sqrt{2} (X, X_1(z)), \quad Q_X(z) = \sqrt{2N} y (X, X_2(z) + iX_3(z)), \quad R(X, z) = \frac{1}{2} p_z(X)^2 - (X,X).$$

The theta kernel is the vector-valued series

$$\Theta_{M,k}(\tau,z) = \sum_{h \in L'/L} \sum_{X \in h + L} \psi_{M,k}(X, \tau, z) e_h,$$

which in $\tau$ transforms with weight $1/2 - k$ for the Weil representation $\rho_L$, and in $z$ with weight $-2k$ under $\Gamma$.

For $F \in H_{-2k}^+(\Gamma)$, a harmonic weak Maass form of weight $-2k$, the naive integral

$$I_T(F, \tau) = \int_{M_T} F(z)\, \Theta_{M,k}(\tau, z)\, y^{-2k} d\mu(z)$$

diverges linearly in $T$ when $k > 0$; a careful regularization is thus required. Subtracting the constant terms of $F$ at each cusp, one obtains the limit

$$\Lambda_M(F, \tau) = \lim_{T \to \infty} \left(I_T(F, \tau) - C \cdot T \right),$$

defining the regularized Millson theta lift. For $k=0$, no subtraction is needed due to exponential decay.

(Alfes-Neumann et al., 2016)

2. Mapping Properties and the $\xi$-Diagram

The Millson lift induces

$$\Lambda_M: H_{-2k}^+(\Gamma) \to H_{1/2 - k}^+(\Gamma_0(4N), \rho_L),$$

so $\Lambda_M(F, \tau)$ is a harmonic weak Maass form in the Kohnen plus-space of weight $1/2 - k$ for $\rho_L$. If $F$ is weakly holomorphic, $\Lambda_M(F)$ is also weakly holomorphic.

The key mapping principle is the “$\xi$-diagram,” relating the Millson lift to the Shintani lift via the Bruinier–Funke $\xi$-operator:

$$\xi_{1/2-k, \tau}(\Lambda_M(F, \tau)) = - (2 \sqrt{N})^{-1} \Lambda_\mathrm{Sh}(\xi_{-2k}F, \tau).$$

Thus, the nonholomorphic part of the lift is governed by the image of $F$ under $\xi_{-2k}$ and its Shintani lift.

3. Fourier Expansion: Cycle Integrals and CM Traces

The Fourier expansion of $\Lambda_M(F, \tau)$ is explicitly given as: $$\Lambda_M(F, \tau)^+ = \sum_{m \gg -\infty} a_F^+(m)q^m, \quad \Lambda_M(F, \tau)^- = \sum_{m<0} a_F^-(m) \Gamma(1/2 - k, 4\pi|m|v) q^m.$$ For $k > 0$ and in the untwisted case $\Delta=1$,

$$a_F^+(m) = \frac{1}{2\sqrt{m}} \frac{1}{(4\pi\sqrt{m})^k} \left[t^+(R_{-2k}^k F; m) + (-1)^{k+1} t^-(R_{-2k}^k F; m)\right],\quad m>0,$$

where $t^+$ and $t^-$ are CM value traces for positive and negative definite forms, respectively, and $R_{-2k}^k$ is the iterated Maass raising operator. The nonholomorphic coefficients relate to geodesic cycle integrals of the shadow: $$a_F^-(m) = -\frac{1}{2(4\pi|m|)^{k + 1/2}} \overline{t(\xi_{-2k}F; m)}, \quad m < 0,$$ where $t(\xi F; m)$ is the sum of integrals over geodesic cycles associated to quadratic forms of discriminant $m$.

The constant term and certain “complementary traces” are described via Hurwitz zeta values and the constant terms of $F$ at the cusps (see Theorem 5.7 in (Alfes-Neumann et al., 2016)).

4. Identities between Cycle Integrals

A fundamental arithmetic identity emerges between cycle integrals of modular forms of varying weights: for $X \in V$ with $Q(X)=m<0$ and $c(X)$ the associated closed geodesic, repeated use of integration by parts and operator commutation establishes

$$C(F,j) := \int_{c(X)} R_{-2k}^{2j+1} F(z) Q_X(z)^k dz = \operatorname{const}_{j,k,|m|,N} \cdot \overline{\int_{c(X)} \xi_{-2k}F(z) Q_X(z)^k dz},$$

with

$$\operatorname{const}_{j,k,|m|,N} = (4|m|N)^{-(k-j)} j!(k-j)!(2k)! / (k!(2k-2j)!).$$

This identity demonstrates proportionality among various cycle integrals associated to iterated Maass-raising of $F$ and the shadow $\xi_{-2k}F$.

Such identities are critical in the interplay between cycle integrals of weakly holomorphic, harmonic, and cusp forms, underpinning the structure of special value formulas and arithmetic applications.

5. Connection to Central $L$-Values

Suppose $F \in H_{-2k}^+(\Gamma_0(N))$ is such that $G := \xi_{-2k}F$ is a normalized newform of weight $2k+2$. For a fundamental discriminant $\Delta < 0$ coprime to $N$, the twisted Millson theta lift $\Lambda_{M, \Delta}(F, \tau)$ is weakly holomorphic if and only if the central $L$-value $L(G, \chi_{\Delta}, k+1)$ vanishes. The connection is realized as

$$\xi(\Lambda_{M, \Delta}(F, \tau)) = \text{constant} \cdot \Lambda_{\mathrm{Sh}, \Delta}(G, \tau),$$

with the Shintani lift yielding the relevant half-integral weight modular form whose Waldspurger coefficients encode central $L$-values. Thus,

$$\Lambda_{M, \Delta}(F, \tau) \ \text{holomorphic} \iff L(G, \chi_{\Delta}, k+1)=0.$$

This criterion is fundamental for understanding the role of the Millson and Shintani lifts in the nonvanishing of critical $L$-values, providing a bridge to the arithmetic of modular forms.

6. Broader Context, Generalizations, and Related Lift Constructions

Regularized lifts from harmonic weak Maass forms generalize and unify a range of theta lifting constructions, encapsulating Borcherds’ singular theta lifts (Schwagenscheidt, 2018), the Bruinier–Funke–Imamoğlu regularizations (Bruinier et al., 2011), algebraic formulas for traces and partition values (Bruinier et al., 2011), and the Shintani fractional derivative lifts (Bringmann et al., 2014).

The structural regularization mechanisms—truncation, analytic continuation, constant-term extraction—are common features, with the choice of Schwartz kernel (Millson, Shintani, Kudla–Millson, Siegel) dictating the target weights and modular representation. Via these lifts, one obtains explicit generating series of CM value traces, cycle integrals, and arithmetic invariants, as well as explicit formulas for mock modular forms and harmonic preimages under the $\xi$ operator.

The automorphic and cohomological properties of these lifts are reflected in the identities among cycles and the adjointness or period relations in higher-rank settings (Branchereau, 28 Dec 2025). Furthermore, regularized lifts are crucial in constructing arithmetic theta series, Borcherds products, and in proving rational and Galois-theoretic results for coefficients of harmonic Maass forms associated to CM newforms (Ehlen et al., 2022).

7. Significance in the Theory of Modular Forms and Automorphic Functions

The regularized Millson lift from harmonic weak Maass forms is an essential tool for:

• Constructing half-integral weight harmonic Maass forms (and mock theta functions) with prescribed arithmetic or geometric properties.
• Analyzing the arithmetic of special cycles, CM values, and central $L$-values.
• Deriving explicit proportionality identities among period integrals, which feed into far-reaching results in the intersection theory of modular forms.
• Facilitating the transfer of analytic data—growth, eigenvalue, principal parts—between spaces of differing weights and representations.

It interlocks with the broader theta correspondence, the theory of Borcherds products, arithmetic intersection, and automorphic $L$-functions, furnishing both structural and computational insights into the arithmetic and geometry of modular forms. The Millson lift’s regularization technique exemplifies the power of analytic continuation and constant-term extraction in controlling divergent integrals arising in automorphic representation theory.

(Alfes-Neumann et al., 2016, Bruinier et al., 2011, Bruinier et al., 2011, Bringmann et al., 2014, Branchereau, 28 Dec 2025, Ehlen et al., 2022)