---
title: Adjoint Kudla Lift and Lambda-adic Interpolation
url: https://www.emergentmind.com/topics/adjoint-kudla-lift
type: topic
---

# Adjoint Kudla Lift and Lambda-adic Interpolation

The $\Lambda$-adic Kudla lift is a canonical $p$-adic analytic family of automorphic lifts from Hida families of ordinary Picard modular forms to families of higher-dimensional Picard modular forms, interpolating the (classical and $p$-modified) Kudla theta lift across weights and characters. The construction, centered on the arithmetic geometry of unitary Shimura varieties and their associated Iwasawa-theoretic cohomology, provides a powerful extension of the classical integral theta correspondences to the realm of $p$-adic analytic families and enables deep links to $p$-adic $L$-functions, special cycles, and arithmetic geometry on Picard modular surfaces. The $\Lambda$-adic Kudla lift constitutes a key structural component in the nascent $p$-adic Kudla program and underpins recent advances on the $p$-adic interpolation of special cycle generating series, including the adjoint Kudla (Cogdell) lift.

## 1. Classical Kudla Lift for Picard Modular Forms

The classical Kudla lift is an automorphic theta lift associated with the reductive dual pair $(\text{GU}(2), \text{GU}(3))$ over an imaginary quadratic field $K/\mathbb{Q}$ of discriminant $D$ [2410.19992]. Consider the Hermitian space $V = K^3$ with signature $(2,1)$, the similitude unitary group $G = \text{GU}(V)$, and its subgroup $U = \ker \mu = \text{SU}(V)$. Picard modular forms of weight $k$ and level $U_f \subset G(\mathbb{A}_f)$ are scalar-valued functions transforming by the usual automorphy factor.

Given a modular form $f \in M_{k-1}(\Gamma_1(D), \chi_{K/\mathbb{Q}})$ and an unramified Hecke character $\mathfrak{E}$, Kudla’s kernel $\Theta^\mathfrak{E}(T, g)$ forms an adelic theta series on $\mathcal{H} \times G(\mathbb{A})$. The Kudla lift is defined by integrating $f$ against this kernel,
$$
L_{k,\mathfrak{E}}(f)(g) = \int_{\Gamma_1(D) \backslash \mathcal{H}} f(T)\,\Theta^\mathfrak{E}(T, g)\, y^{k-3}\, dx\, dy,
$$
yielding a Picard modular form in $M_k(G, U_f, \mathfrak{E})$. The Fourier–Jacobi expansion along the Siegel parabolic reveals the image as a sum over theta-type functions, indexed by fractional ideals and integer variables, with explicit coefficients provided by Finis’s formula in terms of special values of $f$ [F99].

## 2. Formulation of the $\Lambda$-adic Kudla Lift

The $\Lambda$-adic Kudla lift interpolates the $p$-modified Kudla lift in ordinary $p$-adic families, formalized in the Iwasawa-algebraic context. Let $\Gamma = 1 + p\mathbb{Z}_p$, $\Lambda = \mathbb{Z}_p[[\Gamma]]$ the cyclotomic Iwasawa algebra, and $\mathcal{W} = \mathrm{Hom}_{\mathrm{cont}}(\Gamma, \mathbb{C}_p^\times)$ the $\Lambda$-adic weight space. Hida families $\mathfrak{F}$ of ordinary Picard modular forms are formal $q$-expansions in $\Lambda[[q]]$ such that each specialization at an arithmetic point yields a classical $p$-stabilized newform.

The $\Lambda$-adic Kudla lift $\mathcal{L}_\Lambda(\mathfrak{F})$ is given by a formal $q$-expansion
$$
\mathcal{L}_\Lambda(\mathfrak{F})(q) = \sum_{n \geq 1}[\, \xi_{n, \infty}^{\mathrm{n.o.}},\, \mathfrak{F}\,]_\Lambda \, q^n \in \Lambda[[q]],
$$
where the coefficients are obtained by a $\Lambda$-valued (Poincaré) pairing of big Iwasawa-theoretic cycle classes $\xi_{n, \infty}^{\mathrm{n.o.}}$ with the modular form family $\mathfrak{F}$ [2410.19992, 2601.10077]. For each arithmetic specialization corresponding to $(k, \varepsilon)$, the $q$-expansion specializes to the $p$-modified (ordinary) Kudla lift in weight $k$ and Nebentypus $\varepsilon$,
$$
\nu_P(\mathcal{L}_\Lambda(\mathfrak{F}))(q) = \mathcal{L}_k(\mathfrak{F}(P))(q).
$$
This family is uniquely characterized by its interpolation property, Hecke equivariance, and integrality of coefficients.

## 3. Structural Properties and $p$-adic Analyticity

The main structural attributes of the $\Lambda$-adic Kudla lift are as follows:

- **Hecke-Equivariance:** For all primes $\ell \nmid pD$, the lift commutes with Hecke operators: $\mathcal{L}_\Lambda(T_\ell\mathfrak{F}) = T_\ell\mathcal{L}_\Lambda(\mathfrak{F})$.
- **Integrality:** The coefficients $A_n(\mathfrak{F})$ lie in $\Lambda$, reflecting the integrality of the input cycles and modular forms.
- **Control Theorems:** Specialization at arithmetic points realizes the classical (or $p$-modified) Kudla lift.
- **Finite Generation:** The formal $q$-expansion coefficients are assembled from explicit theta-type series whose coefficients admit $p$-adic analytic variation in the Iwasawa algebra [2410.19992].
- **$p$-adic Cohomology:** The construction uses families of ordinary cycle classes in the étale Iwasawa cohomology of towers of Picard varieties, with U$_p$-compatibility ensuring analytic continuation over the weight space [2601.10077].

The Finis formula gives explicit Fourier–Jacobi coefficients as sums over cosets, theta polynomials, and $p$-adic modular data, which package into finite free $\Lambda$-modules parameterizing $\Lambda$-adic theta bundles.

## 4. Construction Methodology

The construction synthesizes analytic, automorphic, and arithmetic tools in several key steps [2410.19992, 2601.10077]:

1. **Big Special Cycle Classes:** For each $n\geq 1$, a compatible system of étale (Iwasawa) cohomology classes $\xi_{n, \infty}^{\mathrm{n.o.}}$ is constructed, satisfying $U_p$-recurrence.
2. **Ordinary Projectors:** Use of Hida’s ordinary projector $e_{\mathrm{ord}}$ restricts to 'ordinary' parts, yielding modules of finite rank over $\Lambda$.
3. **Pairing with Modular Forms:** Poincaré duality enables pairings $[\,\cdot,\cdot\,]$ between cycle classes and $\Lambda$-adic modular forms, defining $q$-expansion coefficients.
4. **Hecke and Weight Interpolation:** The formal $q$-expansion so obtained is shown to interpolate the Kudla lift at all algebraic specializations and is compatible with Hecke actions outside $p$.
5. **Fourier–Jacobi Expansion:** Explicit formulas for the period polynomials and theta-type coefficients enable the analytic variation across the family ($\mathcal{L}_\Lambda(\mathfrak{F}) \in \Lambda[[q]]$).

## 5. Extension to Special Cycles, Adjoint Lifts, and Applications

The $\Lambda$-adic Kudla lift plays a vital role in the $p$-adic interpolation of generating series of special cycles. This includes the adjoint (Kudla–Millson) and Cogdell lifts, where intersection numbers of special cycles on Picard surfaces are $q$-expansion coefficients of modular forms [2601.10077]. Loeffler’s norm-relation techniques yield Iwasawa-theoretic classes of special cycles, compatible with the big cycles of the Kudla lift, facilitating $p$-adic analytic families whose specialization recovers the classical cycle-generating modular forms.

Furthermore, the $\Lambda$-adic Kudla lift is expected to enable:

- Construction of $p$-adic $L$-functions via triple product and pullback formulas.
- Interpolation of geometric invariants and regulators associated to CM cycles in Iwasawa theory.
- Linkage with overconvergent and small-slope families beyond the ordinary setting, providing a pathway to extensions in the context of $p$-adic Hodge theory.
- Development of multivariable $p$-adic theta lifts for higher rank unitary groups.

## 6. Context within the $p$-adic Kudla Program and $\Lambda$-adic Generalizations

The $\Lambda$-adic Kudla lift is situated at the center of the emerging $p$-adic Kudla program, which seeks to generalize classical theta correspondences, generating series of cycles, and automorphic periods to the $p$-adic analytic and Iwasawa-theoretic setting [2404.12936]. A complementary approach constructs $\Lambda$-adic variants of the Shintani and Borcherds lifts for rigid analytic cocycles, indicating that the full suite of classical arithmetic theta lifting machinery, including the Eichler–Shimura–Shintani–Kudla formalism, admits robust $p$-adic and $\Lambda$-adic analogues.

A plausible implication is that the ordinary and overconvergent $\Lambda$-adic Kudla lifts will continue to enable new $p$-adic and geometric results on the arithmetic of Picard modular forms, CM cycles, and modularity of generating series for higher-dimensional cycles.

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**References**: arXiv:2410.19992, arXiv:2601.10077, arXiv:2404.12936.

Source: https://www.emergentmind.com/topics/adjoint-kudla-lift