Λ-adic Kudla Lift Overview

• Λ-adic Kudla lift is a p-adic analytic construction that interpolates classical Kudla theta lifts to generate families of Picard modular forms.
• It employs a p-modification and Hida theory to overcome non-ordinarity and ensure p-adic interpolation of eigenvalues.
• Its construction interweaves Iwasawa cohomology with explicit Fourier-Jacobi expansions, advancing p-adic L-function theory and theta correspondence.

The $\Lambda$-adic Kudla lift is a $p$-adic analytic construction that interpolates the classical Kudla theta lift in families of automorphic forms, specifically for Picard modular forms associated with unitary groups over imaginary quadratic fields. It synthesizes techniques from Iwasawa theory, Hida theory, and the theory of special cycles on Shimura varieties, enabling the construction of canonical $p$-adic analytic families (“Hida families”) of Picard modular forms whose $q$-expansions interpolate those of $p$-modified Kudla lifts at classical or “arithmetic” weights. The $\Lambda$-adic Kudla lift has played a central role in recent advances toward $p$-adic and $\Lambda$-adic theta correspondences and in the construction of $p$-adic families of special cycles and $L$-functions.

1. Classical Kudla Lift and Automorphic Framework

The classical Kudla lift is a theta correspondence between automorphic representations of the unitary similitude groups $GU(2)$ and $GU(3)$ (for a fixed imaginary quadratic field $K/\mathbb{Q}$). On the geometric side, these groups encode the moduli of abelian varieties with $O_K$-multiplication, and Picard modular forms are identified as functions on the associated Picard modular surfaces $$X= GU(2,1)(\mathbb{Q}) \backslash [\mathcal{H}_2 \times GU(2,1)(\mathbb{A}_f)/U_f]$$.

Given an elliptic modular form $f \in M_{k-1}(\Gamma_1(D),\chi_{K/\mathbb{Q}})$ of even weight $k-1\geq 1$ (with $D$ the discriminant of $K$ and $\chi_{K/\mathbb{Q}}$ the associated quadratic character), and a Hecke character $\mathcal{E}$ of $K$, the Kudla theta kernel produces a theta series $\Theta^\mathcal{E}(T,g)$, which is a rapidly decaying holomorphic form of weight $k-1$ when held at fixed local components. The integrals

$$L_{k, \mathcal{E}}(f)(g) := \int_{\Gamma_1(D) \backslash \mathcal{H}} f(T)\,\Theta^\mathcal{E}(T, g)\,y^{k-3}\,dx\,dy$$

define the Kudla lift, yielding a weight $k$ Picard modular form.

This construction admits an explicit Fourier-Jacobi expansion, with the $r$-th Fourier-Jacobi coefficient valued in finite-dimensional spaces of theta functions on the moduli.

2. Fourier-Jacobi Expansions and Finis's Formula

Explicit arithmetic formulas for the classical Kudla lift were first given by Finis, who computed the Fourier–Jacobi coefficients as explicit sums indexed by ideals and coset representatives in $\Gamma_1(D)\backslash SL_2(\mathbb{Z})$. For $f\in S_{k-1}(\Gamma_1(D),\chi_{K/\mathbb{Q}})$, the expansion reads: $$C_k(f)(z,w) = \sum_{a\in \mathrm{Cl}(K)} \sum_{n\geq 0} g_{n/N(a),a}(w)\,q^n,$$ with

$$g_{n/N(a), a}(w) = |\mathcal{O}_K / N(a)|^{-k/2} \sum_{b\in \mathrm{Cl}(K)} \mathcal{E}(b)^{-1} N(b)^{k/2} \sum_{\gamma \in \Gamma_1(D)\backslash SL_2(\mathbb{Z}),\,\det\gamma=n} f|_{k-1}\gamma(\tfrac{w+\tau_{a,b}}{N(a)}).$$

Each $g_{n/N(a), a}(w)$ is a holomorphic theta function in $w$ of weight $k$, and the arithmeticity of the $q$-expansion follows by standard $q$-expansion principles.

3. $p$-Modification and Obstacles to $p$-Adic Interpolation

A principal obstruction to $p$-adic interpolation is the non-ordinarity of the classical Kudla lift at $p$: the $p$-th Hecke eigenvalue is always divisible by $p$. This is overcome by constructing a $p$-modified Kudla lift, $L^{(p)}_{k, \mathcal{E}}(f)$, using a $p$-modified theta kernel $\Theta^{(p)}$. The $p$-modified lift matches the classical Kudla lift at all Hecke operators away from $p$, but has its $p$-eigenspace controlled so that the $p$-eigenvalue is a $p$-adic unit. This $p$-modification ensures compatibility with ordinary $p$-adic families and is crucial for defining a $\Lambda$-adic interpolation.

4. $\Lambda$-Adic Weight Space, Hida Families, and the Main Interpolation Theorem

The $\Lambda$-adic setting is governed by the Iwasawa algebra $\Lambda = \mathbb{Z}_p[[\Gamma]]$ for $\Gamma = 1 + p\mathbb{Z}_p$, and its analytic weight space $$\mathcal{X}(\Lambda_1) = \operatorname{Hom}_{\mathrm{cont}}(\mathbb{Z}_p^\times, \mathbb{C}_p^\times)$$ encapsulates all possible twists by finite-order and $p$-adic analytic characters.

A Hida family is a formal $q$-expansion

$$\mathbf{f}(q) = \sum_{n=1}^\infty a_n(\mathbf{f})\,q^n \in \Lambda[[q]]$$

such that, for every arithmetic character $\chi$ of signature $(k,\varepsilon)$, the specialization $$f_{k,\varepsilon}(q) = \sum a_n(\mathbf{f})(\chi) q^n$$ is a $p$-stabilized newform of weight $k$.

The principal result is the existence and uniqueness of a $\Lambda$-adic Kudla lift: $$\mathcal{L}_{\omega}(\mathbf{f}) = \sum_{a,n} \mathbf{g}_{n/N(a), a}(W)\,q^n \in \Lambda [[q]] \otimes \bigoplus_a \Theta_{*, a}^\Lambda,$$ where each $\mathbf{g}_{n/N(a), a}(W)$ belongs to a finite free $\Lambda$-module of $\Lambda$-adic theta functions. At each arithmetic specialization, the $\Lambda$-adic Kudla lift recovers the $p$-modified Kudla lift (up to a $p$-adic period), thus providing a rigid-analytic, $p$-adic family of Picard modular forms interpolating all $L^{(p)}_{k, \mathcal{E}_k}(f_k)$ (Iudica, 2024, Iudica, 15 Jan 2026).

5. Cohomological Construction, Ordinary Projectors, and Key Formalism

The $\Lambda$-adic Kudla lift is constructed via pairing $\Lambda$-adic Hida families of Picard modular forms against "big" classes in the Iwasawa cohomology of Shimura varieties. The main ingredients are:

• Construction of Iwasawa classes $$\xi_{n, \infty}^{\mathrm{n.o.}} \in e' H^2_{\mathrm{ét}, \mathrm{Iw}}(Q_G^0, \mathbb{Z}_p(1))_\theta$$ compatible by the norm relations and $U_p$-operator,
• Use of the Hida ordinary projector $e_{\mathrm{ord}}$ to project to the ordinary part and control interpolation,
• Pairing against a Hida family $\mathfrak{F}$ to produce $q$-expansions in $\Lambda[[q]]$:

$$\mathcal{L}_\Lambda(\mathfrak{F})(q) = \sum_{n\geq 1} [\xi_{n, \infty}^{\mathrm{n.o.}}, \mathfrak{F}]_\Lambda\,q^n.$$

The interpolation property guarantees that for each arithmetic point $P$,

$$\nu_P(\mathcal{L}_\Lambda(\mathfrak{F}))(q) = \mathcal{L}_k(\mathfrak{F}(P))(q)$$

where $\mathcal{L}_k$ is the classical Kudla lift in weight $k$. Hecke equivariance for all primes $\ell \nmid pD$ holds by Eichler–Shimura theory and the equivaraince properties of cohomology and the theta kernel (Iudica, 2024, Iudica, 15 Jan 2026).

6. Structural Properties and Applications

The table below outlines principal structural features:

Property	Statement	Source
Hecke Equivariance	$$\mathcal{L}_\Lambda(T_\ell\mathfrak{F}) = T_\ell\mathcal{L}_\Lambda(\mathfrak{F})$$	(Iudica, 2024)
Integrality	Coefficients $A_n(\mathfrak{F}) \in \Lambda$	(Iudica, 2024)
Interpolation	Specialization at each $P$ recovers the classical Kudla lift	(Iudica, 2024)

Applications include:

• Construction of $p$-adic analytic cohomology classes of special cycles, used in $p$-adic interpolation of the adjoint (Cogdell) lift (Iudica, 15 Jan 2026).
• Foundation for $p$-adic $L$-functions of several variables via pullback formulas and seesaw identities; expected to generalize the Greenberg–Stevens $L$-function for $\mathrm{GL}_2$ (Iudica, 2024).
• Structural tools for exploring $p$-adic analogues of Kudla–Millson theory—interpolation of intersection numbers of special cycles and their relations to regulators of CM-cycles (Iudica, 2024).

7. Extensions, Open Problems, and Context in the $p$-Adic Kudla Program

The $\Lambda$-adic Kudla lift realizes a $p$-adic analytic family of automorphic theta lifts and serves as a model for anticipated higher-dimensional $\Lambda$-adic correspondences. Key open directions include:

• Extension of the construction to overconvergent families of small slope beyond ordinary forms,
• Classicality results for the $\Lambda$-adic lift under slope conditions (analogous to Coleman's classicality in the overconvergent setting),
• Relating $\Lambda$-adic interpolation of the adjoint Kudla–Millson lift to $p$-adic regulators,
• Multivariable $p$-adic theta-lifts for unitary and other reductive groups.

This suggests that the full scope of the $p$-adic Kudla program encompasses a robust framework for lifting $p$-adic (and $\Lambda$-adic) automorphic forms and cycles, mirroring the entire structure of the theta correspondence and its arithmetic avatars in the $p$-adic setting (Negrini, 2024, Iudica, 2024, Iudica, 15 Jan 2026).