Adjoint Kudla Lift and Lambda-adic Interpolation

• Adjoint Kudla Lift is a p-adic analytic extension of the classical theta correspondence, interpolating automorphic forms in Lambda-adic families.
• It utilizes Hecke-equivariant methods and integrality properties to connect Hida families of ordinary Picard modular forms with special cycle generating series.
• Its construction leverages étale Iwasawa cohomology, ordinary projectors, and explicit Fourier–Jacobi expansions to bridge arithmetic geometry with p-adic L-functions.

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 field 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 $D$ (Iudica, 2024). 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 $\mathfrak{F}$ (Iudica, 2024, Iudica, 15 Jan 2026). 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 (Iudica, 2024).
• $p$-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 (Iudica, 15 Jan 2026).

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 (Iudica, 2024, Iudica, 15 Jan 2026):

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 (Iudica, 15 Jan 2026). 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 (Negrini, 2024). 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: (Iudica, 2024, Iudica, 15 Jan 2026, Negrini, 2024).