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# A Λ-ADIC KUDLA LIFT

Francesco Iudica

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Francesco Iudica. A Λ-ADIC KUDLA LIFT. 2024. [https://hal.science/hal-04754606v1](https://hal.science/hal-04754606v1)

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Preprint submitted on 26 Oct 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# A $\Lambda$-ADIC KUDLA LIFT

FRANCESCO IUDICA

**ABSTRACT.** The Kudla lift studied in this article is a classical version for Picard modular forms of the automorphic theta lift between $\mathrm{GU}(2)$ and $\mathrm{GU}(3)$. We construct an explicit $p$-adic analytic family of Picard modular forms varying with respect to the weight and level, which interpolates a so-called $p$-modification of the lift at arithmetic weights, by exploiting a formula of Finis for the Fourier-Jacobi coefficients of a lifted form.

## CONTENTS

<table><tr><td>0. Introduction</td><td>1</td></tr><tr><td>1. Notation</td><td>3</td></tr><tr><td>1.1. Conventions on Hecke characters</td><td>4</td></tr><tr><td>2. The unitary Shimura variety of signature (2, 1)</td><td>5</td></tr><tr><td>2.1. PEL type moduli spaces</td><td>7</td></tr><tr><td>3. Picard modular forms</td><td>8</td></tr><tr><td>3.1. The Fourier-Jacobi expansion</td><td>9</td></tr><tr><td>3.2. Arithmetic theta functions</td><td>10</td></tr><tr><td>4. The $p$-modified Kudla lift</td><td>10</td></tr><tr><td>4.1. The level one Kudla lift</td><td>10</td></tr><tr><td>4.2. The $p$-modification</td><td>13</td></tr><tr><td>5. The $p$-adic interpolation of the Kudla lift</td><td>21</td></tr><tr><td>5.1. On $\Lambda$-adic elliptic cusp forms</td><td>22</td></tr><tr><td>5.2. On $\Lambda$-adic Picard modular forms</td><td>24</td></tr><tr><td>5.3. The $p$-adic lift</td><td>26</td></tr><tr><td>5.4. The $\Lambda$-adic lift</td><td>27</td></tr><tr><td>Acknowledgements</td><td>29</td></tr><tr><td>References</td><td>29</td></tr></table>

## 0. INTRODUCTION

In his seminal paper [K79], Kudla defined a lift $\mathcal{L}$ from elliptic modular forms to holomorphic modular forms for unitary groups of signature $(m, 1)$. He proved that this lifting is in fact arithmetic, in the sense that if $f$ is a modular form on the upper half-plane with respect to a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and whose Fourier coefficients are in $\mathbb{Q}^{ab}$, then there exists a complex period $\Omega$ independent of $f$ such that $\Omega^{-1}\mathcal{L}(f)$ is arithmetic in the sense of Shimura [S78, p. 580]. Other authors, such as Gelbart-Rogawski [GeR91], and Murase-Sugano [MS07], have studied the theta lift for the dual reductive pair

$$(\mathrm{GU}(1, 1), \mathrm{GU}(2, 1))$$
from an automorphic point of view. This theta lift is an extension of the Kudla lift for $m = 2$ to the space of automorphic forms for GU(1, 1).

As Kudla suggests, [K79, p. 2], and as proved by Tong and Wang in [TW83], the holomorphic lift is, in a suitable sense, the adjoint of the geometric theta lift of Kudla and Millson [KM90, p. 127]. The Kudla-Millson lift, roughly speaking, takes special cycles on the Picard modular surface and maps them to elliptic modular forms, whose coefficients are intersection multiplicities of special cycles. Indeed, it was proved in loc. cit., and independently in [TW83], that $\mathcal{L}$ yields the type (1, 1)-forms which represent the cohomology classes of the composite cycles defined in [Cog85, p. 121], in the following sense. For $\mathcal{L}^\dagger$ the adjoint lift to $\mathcal{L}$ with respect to the Petersson inner product $\langle, \rangle$, we have

**Theorem 1** ([TW83, p. 497]). Let $f \in S_3(\Gamma_1(D), \omega_{K/\mathbb{Q}})$ and consider an homology class $C \in H_{1,1}(\mathrm{Sh}(\mathrm{GU}(2,1)))^{\mathrm{res}}, \mathbb{C})$ which is orthogonal to the first Chern class in $\mathrm{Sh}(\mathrm{GU}(2,1))$. Then we have:

$$ \mathcal{L}_4(f) = \frac{i}{2} \langle f, \Theta(\tau, z) \rangle, \quad \mathcal{L}_4^\dagger(C) = \frac{i}{2} \int_C \Theta(\tau, z), $$

where

$$ \Theta(\tau, z) = \sum_{n>0} [C_n^c]^\vee q^n $$

is the generating series of the Poincaré dual classes of the special cycles defined by Cogdell.

In his unpublished Ph.D. thesis [F99], Finis further studied the arithmeticity of the classical level 1 Kudla lift

$$ \mathcal{L}_k : M_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}}) \to \mathcal{M}(k, L, \chi) \\ f \mapsto \mathcal{L}_k(f) $$

for $k \ge 6$ an even integer, obtaining in particular an explicit formula for the Fourier-Jacobi coefficients of the lifted form $\mathcal{L}_k(f)$ in terms of $f$. In this article we address a problem first proposed by Finis in his thesis, [F99, p. 6]: for $p \ge 5$ a prime which splits in a fixed imaginary quadratic number field $K/\mathbb{Q}$, we construct $p$-adic analytic families of Picard modular forms varying in weight and level, which interpolate the classical Kudla lifts of Hida families of elliptic cusp forms with Nebentypus. In order to do so, we first recall the definition of the level 1 Kudla lift. As we shall see, the Fourier-Jacobi coefficients obtained by Finis, which are so-called generalized theta functions in the terminology of Shimura [S78], do not vary $p$-adically. This leads to our first result, namely the construction of a suitable modification of the above lift, which we call the $p$-modified lift:

$$ \mathcal{L}_k^p : S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}}) \to \mathcal{S}(k, \Gamma, \chi), $$

for a certain congruence subgroup $\Gamma \subset \mathrm{GU}(2,1)(\mathbb{Q})$. This procedure can be thought of as a sort of $p$-stabilization of the level 1 Kudla lift. Indeed, the effect of this modification is that of adding a factor $p$ in the level, and of removing any $p$-power from the coefficients. However, we note here that we have not been able to compare this procedure with the usual $p$-stabilization of eigenforms. Having removed the obstruction to the interpolation of the level 1
Kudla lift, we get the following result (see Theorem 17 in the text for a more general version):

**Main Theorem.** Given a Hida family $\mathfrak{F}(\kappa)$ with ordinary specializations

$$ \mathfrak{F}(k) = f_k \in S_k^{\mathrm{ord}}(\Gamma_1(pD), \omega_{K/\mathbb{Q}}) $$

for $k \in \mathbb{Z}$, there exists a $\Lambda$-adic Fourier-Jacobi expansion defining a $\Lambda$-adic Picard modular form $\mathcal{L}_{\omega^j}(\mathfrak{F})(\kappa)$, for $\omega$ the Teichmüller character, such that

$$ \Omega_0^{-k} \mathcal{L}_{\omega^j}(\mathfrak{F})(k) = \mathcal{L}_k^p(f_{k-1}), $$

where $\Omega_0$ is a period only depending on $K$.

It is noteworthy that both the level 1 and the $p$-modified lifts are never ordinary in the sense of Hida. Indeed, for each rational prime $q$, the eigenvalue attached to $q$ of any eigenform obtained by this lift will not be invertible modulo $q$.

As an application of these results, we mention a possible interpolation of the aforementioned adjoint lift. Indeed, applying the results of Loeffler [L19] to our setting, we obtain, in a suitable sense, $p$-adic families of the special cycles defined in [Cog85]. It makes thus sense to ask whether the $p$-adic variation of these cycles is compatible with the interpolation of their image under the adjoint Kudla lift, which is an elliptic cusp form in a Hida family. This will be pursued in future work.

The paper is organized as follows: in §1-2 we introduce the necessary notations and recall the definition of Picard modular surfaces. In §3 we define Picard modular forms following the formulation of Finis, and we introduce the notion of arithmetic theta functions. The description of the theta lifts, of level 1 and $p$ respectively, occupies §4. The main result of the paper, Theorem 17, is in the last section §5.

# 1. NOTATION

Let $K$ be an imaginary quadratic number field and fix an inclusion $\sigma : K \hookrightarrow \mathbb{C}$, and let $\bar{\sigma} : K \hookrightarrow \mathbb{C}$ be its complex conjugate. We use the following notation:

* $d$ is the squarefree integer such that $K = \mathbb{Q}(\sqrt{d})$, $d < 0$,

* $D$ is the discriminant of $K$, that is

$$ D = \begin{cases} d & d \equiv 1 \mod 4 \\ 4d & d \equiv 2,3 \mod 4, \end{cases} $$

* $\delta = \sqrt{D}$ is the square root of the discriminant with positive imaginary part,

* $O_K = \mathbb{Z} + \mathbb{Z}\tau_K$, so that

$$ \tau_K = \begin{cases} \frac{1+\sqrt{d}}{2} & \text{if } D \text{ is odd,} \\ \sqrt{d} & \text{otherwise.} \end{cases} $$

Throughout the article, we fix an odd prime $p$ split in $K$, so that $pO_K = \mathfrak{p}\bar{\mathfrak{p}}$. From now on, we fix $\iota_\infty : \mathbb{Q} \hookrightarrow \mathbb{C}$ an embedding of an algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ such that $\iota_\infty|_K = \sigma$, and we identify the image of $\iota_\infty$ with the image under a fixed embedding $\iota_p : \mathbb{Q} \hookrightarrow \mathbb{Q}_p$ once and for all.
Let $\mathbb{A}_{\mathbb{Q}}$ denote the ring of adeles of $\mathbb{Q}$ and $\mathbb{A}_{\mathbb{Q},f}$ the ring of finite adeles. The ideal class group of $K$ is denoted by $\text{Cl}_K$. We write $w_K$ for the number of units in $K$. Write $\mathbb{A}_K = \mathbb{A}_{\mathbb{Q}} \otimes_{\mathbb{Q}} K$ and $\mathbb{A}_{K,f} = \mathbb{A}_{\mathbb{Q},f} \otimes_{\mathbb{Q}} K$. For any finite set $S$ of places of $\mathbb{Q}$, write $\mathbb{A}_{\mathbb{Q}}^S$ for the adeles away from $S$. We will pass freely from finite adeles to fractional ideals of $K$, and thus we will use the same notation for both objects.

**1.1. Conventions on Hecke characters.** We give here a description of the conventions used throughout the article for Hecke characters associated to the imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$. If $f$ is a non-zero ideal of $\mathcal{O}_K$ and $I_K^{S(f)}$ is the group of fractional ideals of $K$ coprime to $f$, a classical Hecke character is a group homomorphism $\varphi : I_K^{S(f)} \to \mathbb{C}^\times$ satisfying

$$ \varphi(\alpha \mathcal{O}_K) = \varphi_{\text{fin}}(\alpha) \alpha^a \bar{\alpha}^b $$

for all $\alpha \in \mathcal{O}_K$ that are coprime to $f$, where $\varphi_{\text{fin}} : (\mathcal{O}_K/f)^\times \to \mathbb{C}^\times$ is a character, called the finite part of $\varphi$, and $a, b$ are complex numbers. We call the pair $(a, b)$ the infinity type of $\varphi$, and $f$ the conductor of $\varphi$. In the case an algebraic character has infinity type of the form $(-k/2, k/2)$ for some integer $k$, we just say that it has weight $k$. We say that a character is unramified if it has trivial conductor, i.e. $f = \mathcal{O}_K$. With this convention, the norm character defined by $N(\mathfrak{q}) = |\mathcal{O}_K/\mathfrak{q}|$ on prime ideals is unramified, has trivial finite part, and infinity type (1, 1). It is sometimes useful to view $\varphi$ as being defined on all fractional ideals of $\mathcal{O}_K$, setting $\varphi(\mathfrak{a}) = 0$ if $\mathfrak{a}$ is not coprime to $f$. Recall that classical Hecke characters are in bijection with adelic Hecke characters, which are defined as continuous homomorphisms $\mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times$. A straightforward way to describe this bijection is by letting

$$ \text{ide} : \mathbb{A}_{K,f}^\times \to I_K \\ \alpha \mapsto \prod \mathfrak{q}^{v_{\mathfrak{q}}(\alpha_{\mathfrak{q}})} $$

the map that associates to a finite idele $\alpha = (\alpha_v)$, for $v$ the places of $K$, its corresponding fractional ideal; then a classical Hecke character $\varphi : I_K^{S(f)} \to \mathbb{C}^\times$ corresponds to a continuous character $\varphi_{\mathbb{C}} : \mathbb{A}_K^\times/K^\times \to \mathbb{C}^\times$ such that $\varphi_{\mathbb{C}}(\alpha) = \varphi(\text{ide}(\alpha))$ for every $\alpha \in \mathbb{A}_{K,f}^\times$ which is trivial at the places of $S(f)$. Under this correspondence, we find that if $\varphi$ has infinity type $(a, b)$ then the local factor $\varphi_{\mathbb{C},\infty}$ at the infinity place is given by $\varphi_{\mathbb{C},\infty}(z) = z^{-a} \bar{z}^{-b}$. Thus our convention for the infinity type for an adelic Hecke character is that it corresponds to the negative of the exponents of $z$ and $\bar{z}$ in the local component at the infinity place. In particular, the adelic absolute value $|| \cdot ||_{\mathbb{A}_K}$ is a character of infinity type $(-1, -1)$, and it corresponds to the inverse of the norm character.

A classical or adelic Hecke character of $K$ is called algebraic if its infinity type $(a, b)$ consists of integers. Algebraic adelic Hecke characters are in bijection with algebraic $p$-adic Hecke characters, which we briefly describe. A $p$-adic Hecke character is a continuous homomorphism $\psi : \mathbb{A}_K^\times/K^\times \to \overline{\mathbb{Q}}_p^\times$, where the domain is endowed with the idelic topology, and $\psi$ is algebraic of integral weights $(a, b)$ if its local factors $\psi_p$ and $\psi_{\bar{p}}$ on $K_p \simeq \mathbb{Q}_p^\times$ and $K_{\bar{p}} \simeq \overline{\mathbb{Q}}_p^\times$ are given by $\psi_p(x) = x^{-a}$ and $\psi_{\bar{p}}(x) = x^{-b}$ on some neighborhoods of the identity in these multiplicative groups of local fields. Then, an algebraic adelic Hecke
character $\varphi_{\mathbb{C}}$ of infinity type $(a, b)$ corresponds to a $p$-adic Hecke character $\varphi_{\mathbb{Q}_p}$ of weights $(a, b)$ by the formula

$$ \varphi_{\mathbb{Q}_p}(\alpha) = (\iota_p \circ \iota_{\infty}^{-1})(\varphi_{\mathbb{C}}(\alpha)\alpha_{\infty}^a \bar{\alpha}_{\infty}^b)\alpha_{\bar{p}}^{-a}\alpha_{\bar{p}}^{-b} $$

for any idele $\alpha = (\alpha_v)$. It is easy to see that this defines a continuous character $A_K^\times \to \bar{\mathbb{Q}}_p^\times$, and it is trivial on $K^\times$ because if $\alpha \in K^\times$ is treated as a principal idele, then $\iota_\infty^{-1}(\alpha_\infty) = \iota_p^{-1}(\alpha_p)$ and $\iota_\infty^{-1}(\bar{\alpha}_\infty) = \iota_p^{-1}(\alpha_{\bar{p}})$, because of our convention on embeddings.

## 2. THE UNITARY SHIMURA VARIETY OF SIGNATURE (2, 1)

Endow the $K$-vector space $V = K^3$ with the hermitian pairing

$$ (u, v) = \bar{u}^t J v, $$

for

$$ J = \begin{pmatrix} & \delta^{-1} \\ -\delta^{-1} & 1 \end{pmatrix}. $$

We identify $V_{\mathbb{C}} = V \otimes_K \mathbb{C}$ with $\mathbb{C}^3$, letting $K$ act on it via $\sigma$. The space $V_{\mathbb{C}}$ is hermitian of signature (2, 1).

**Definition 1.** Let $G := \text{GU}(V, (,))$ be the general unitary group of $V$. Given a $\mathbb{Q}$-algebra $A$, define the $A$-points of the $\mathbb{Q}$-group scheme $G$ as,

$$ G(A) := \{ (g, \mu(g)) \in \text{GL}_3(A \otimes_{\mathbb{Q}} K) \times A^\times \mid (gu, gv) = \mu(g)(u, v) \ \forall u, v \in V \otimes_{\mathbb{Q}} A \}. $$

The map $\mu : G \to \mathbb{G}_m$ is a character of algebraic groups over $\mathbb{Q}$, and is called the similitude factor. Note that we can restate the condition satisfied by $g \in G$ as $t \bar{g} J g = \mu(g) J$, and taking determinants we see that

$$ \mu(g)^3 = \det \bar{g} \det g = |\det g|^2. $$

We define $U := \ker \mu$. As $\mu$ is uniquely determined by $g$, we will often identify the couple $(g, \mu(g))$ with $g$. We let $B$ denote the Borel subgroup of all upper triangular matrices in $G$, which is also the stabilizer of the point $(1 : 0 : 0)$ in $\mathbb{P}^2(K)$. Its unipotent radical is denoted by $N$. We write elements of $N$ as $[v, u]$ with $v \in \text{Res}_{K/\mathbb{Q}}\mathbb{G}_a$ and $u \in \mathbb{G}_a$, where

$$ [v, u] = \begin{pmatrix} 1 & \delta \bar{v} & u + \delta v \bar{v}/2 \\ 0 & 1 & v \\ 0 & 0 & 1 \end{pmatrix}. $$

The center of $N$ is the group of all elements $[0, u]$ and is therefore isomorphic to $\mathbb{G}_a$.

The group $G_\infty = G(\mathbb{R})$ acts on $\mathbb{P}_{\mathbb{C}}^2 = \mathbb{P}(V_{\mathbb{C}})$ by projective linear transformations and preserves the open subdomain $\mathcal{X}$ of negative definite lines, which is biholomorphic to the open unit ball in $\mathbb{C}^2$. Every negative definite line is represented by a unique vector $(z, w, 1)^t$ and such a vector represents a negative line if and only if

$$ \eta(z, w) := (z - \bar{z})/\delta - w\bar{w} > 0. $$
For a point $\mathfrak{z} = (z, w) \in \mathfrak{X}$, we denote $\mathfrak{z} = (z, w, 1)^t$ the corresponding negative vector in $V$. The upper half-plane $\mathbb{H}$ embeds in $\mathfrak{X}$ as the set of points with $w = 0$. Fix the point $\mathfrak{z}_0 = (\delta/2, 0) \in \mathfrak{X}$, and let $K_\infty$ be its stabilizer in $G_\infty$. The latter is compact modulo center: $K_\infty/Z(G_\infty) \cong K_\infty \cap U(\mathbb{R})$ is compact and isomorphic to $U(2)(\mathbb{R}) \times U(1)(\mathbb{R})$. Since $G_\infty$ acts transitively on $\mathfrak{X}$, we may identify the latter with

$$G_\infty/K_\infty \cong U(2, 1)(\mathbb{R})/(U(2)(\mathbb{R}) \times U(1)(\mathbb{R})).$$

The pair $(G, \mathfrak{X})$ is a Shimura datum in the sense of Deligne, with reflex field $K$. Let $L$ be a full $O_K$-lattice in $V$, such that $L$ is included in

$$L^\vee := \{v \in V \mid \text{tr}((u, v)) \in \mathbb{Z}, \forall u \in L\},$$

the dual of $L$. Let $G(L)_f \subset G(\mathbb{A}_f)$ the subgroup stabilizing $\hat{L} := L \otimes \hat{\mathbb{Z}}$. An adelic level subgroup is an open compact subgroup $U_f \subset G(L)_f$. We will suppose that $U_f = U^p U_p$ with $U^p \subset G(\mathbb{A}^{p,\infty})$ and $U_p \subset G(\mathbb{Z}_p)$. In the following, we will only be interested in the case where $U_p$ is hyperspecial, i.e. $U_p = G(\mathbb{Z}_p)$, or parahoric. More precisely, fix the canonical basis $\{e_1, e_2, e_3\}$ for $\mathbb{F}_p^3$. Call $P_1 \subset GL_3(\mathbb{Z}_p)$ (resp. $P_2 \subset GL_3(\mathbb{Z}_p)$), the inverse images via the reduction map $GL_3(\mathbb{Z}_p) \to GL_3(\mathbb{F}_p)$ of the subgroups of $GL_3(\mathbb{F}_p)$ that stabilize the line $\langle e_1 \rangle$ (resp. the plane $\langle e_1, e_2 \rangle$). Since $p$ splits in $K$, under the isomorphism $G(\mathbb{Z}_p) \cong \mathbb{G}_m(\mathbb{Z}_p) \times GL_3(\mathbb{Z}_p)$, the parahoric level structure that we consider corresponds to the case $U_p = \mathbb{G}_m(\mathbb{Z}_p) \times P_2$.

For each choice of adelic level subgroup $U_f \subset G(L)_f$, define $\Gamma_{U_f} := U_f \cap G(\mathbb{Q})$, the associated classical level subgroup. Call the conductor of $U_f$, or of $\Gamma_{U_f}$ indistinctly, the squarefree integer which is the product of the primes $\ell$ at which $U_\ell \subsetneq G(\mathbb{Z}_\ell)$. The double coset space

$$S_{U_f}(\mathbb{C}) := G(\mathbb{Q})\backslash G(\mathbb{A}_{\mathbb{Q}})/U_f K_\infty = G(\mathbb{Q})\backslash (\mathfrak{X} \times G(\mathbb{A}_{\mathbb{Q},f}))/U_f,$$

is in fact the set of complex points of a quasi-projective variety $S_{U_f}/\mathbb{C}$ of dimension two, the Shimura variety associated to the reductive group $G$, of level $U_f$. The character $\nu : \det/\mu : G \to T$, where $T$ is the torus $\text{Res}_{K/\mathbb{Q}}\mathbb{G}_m$, induces a map

$$\nu : S_{U_f}(\mathbb{C}) \to T(\mathbb{A}_{\mathbb{Q}})/T(\mathbb{Q})\nu(Z(\mathbb{R})U_f)$$

from $S_{U_f}(\mathbb{C})$ to a finite group, whose fibers are the connected components of $S_{U_f}(\mathbb{C})$. In particular, for $U_f = G(L)_f$, the Shimura surface $S$ of level 1 has $h_K$ connected components, cf. [dSG16, p. 13]. The principal congruence subgroup $U_f(N)$ of $G(L)_f$ is the normal subgroup obtained as the kernel of the reduction modulo $N$ map on $G(L)_f$, i.e. the subgroup operating trivially on $\hat{L}/N\hat{L}$.

The Shimura varieties obtained this way allow an interpretation as moduli spaces for polarized abelian varieties with additional endomorphism and level structures. To sketch this interpretation, let $V = K^3$, $L = \mathcal{O}_K^3$, $J$, be as above. given a point $\mathfrak{z} \in \mathfrak{X}$, we obtain a splitting $V_{\mathbb{C}} = V_+ \otimes V_-$ as a sum of a positive two-dimensional space $V_+$ and a negative one-dimensional space $V_-$. This splitting defines a canonical complex structure $j_{\mathfrak{z}}$ on $V_{\mathbb{C}}$: for $v = v_+ + v_- \in V_+ \oplus V_-$ and $z \in \mathbb{C}$, set

$$j_{\mathfrak{z}}(z)(v_+ + v_-) := zv_+ + \bar{z}v_-.$$
Additionally, the alternating $\mathbb{R}$-linear form $E = \mathrm{Tr}_{\mathbb{C}/\mathbb{R}}\delta(\cdot, \cdot)$ has the properties $E(j_3(i)x, j_3(i)y) = E(x, y)$ and $E(j_3(i)x, x) < 0$ for $x \in V_{\mathbb{C}} \setminus \{0\}$. If therefore $\Lambda \subset V$ is on $O_K$-lattice such that the form $E$ is integral on $\Lambda$, we obtain an abelian threefold $V_{\mathbb{C}}/\Lambda$, with complex structure $j_3$ and polarization given by $E$, together with an $O_K$-multiplication, such that the Rosati involution induces the non-trivial automorphism of $O_K$.

We now associate to a pair $(\zeta, g_f) \in \mathfrak{X} \times G(\mathbb{A}_{\mathbb{Q}_f})$ the lattice $\Lambda = g_f L$ and change the hermitian form by the factor $|\mu(g_f)|_{\mathbb{A}_{\mathbb{Q}}}$ . Then, our construction gives a polarized abelian threefold $A$ with additional $O_K$-structure. Multipli- cation of $g_f$ by an element of $G(L)_f$ on the right does not change anything, and multiplication of $(\zeta, g_f)$ by an element of $G(\mathbb{Q})$ from the left induces an isomorphism of the two structures. Consequently, we have an interpretation, over $\mathbb{C}$, of the above surface of level 1 as a moduli space. To get a correspond- ing interpretation for level $N$ Picard variety, we must add a level structure, i.e. a trivialization of the $N$-torsion of $A$, cf. [dSG16, p. 11].

2.1. PEL type moduli spaces. Since $S_{\mathcal{U}_f}(\mathbb{C})$ parametrizes abelian varieties with additional structure over $\mathbb{C}$, it is natural to give it an arithmetic structure by showing the representability of the corresponding moduli functor problem over a number ring.

Fix a prime ideal over $p$, say $\mathfrak{p}$. For each level subgroup $\mathcal{U}_f = \mathcal{U}^p \mathcal{U}_p \subset G(\mathbb{A}_{\mathbb{Q},f})$, we define the following moduli problem that sends each connected scheme $T$ over $\mathrm{Spec}(\mathcal{O}_{K,\mathfrak{p}})$ to the set of tuples $(A, \lambda, \eta_{\mathcal{U}_f}, i)$ up to isomorphism, where:

(i) $A \to T$ is an abelian scheme of relative dimension 3.

(ii) $i : \mathcal{O}_K \to \mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$ is an injective morphism of $\mathbb{Z}_{(p)}$-algebras.

(iii) $\lambda : A \to A^\vee$ is a principal polarization such that the Rosati involution induces complex conjugation on $i(\mathcal{O}_K)$

(iv) $\eta_{\mathcal{U}_f}$ is a $\mathcal{U}_f$-level structure.

We explain what the level structure $\eta_{\mathcal{U}_f}$ means. For an abelian variety $A$ of dimension 3 over $T$ satisfying the first three conditions above, and a geometric point $x \in T$, we can look at the trivialization of the Tate module

$$
H_1(A_x, \mathbb{A}_{\mathbb{Q},f}^p) \cong V \otimes \mathbb{A}_{\mathbb{Q},f}^p.
$$

This is an isomorphism of symplectic modules, with the symplectic structure
on the left hand side given by the Weil pairing and on the right hand side by
$E$. The group $G(\mathbb{A}_{\mathbb{Q},f}^p)$ acts on $V \otimes \mathbb{A}_{\mathbb{Q},f}^p$ and so does $\mathcal{U}^p \subset G(\mathbb{A}_{\mathbb{Q},f}^p)$. We are
interested in two cases:

(i) If $\mathcal{U}_p = G(\mathbb{Z}_p)$, then $\eta_{\mathcal{U}_f}$ means a choice of a $\mathcal{U}^p$-orbit of the above isomorphism.

(ii) If $\mathcal{U}_p = \mathbb{Z}_p^\times \times P_2$, then $\eta_{\mathcal{U}_f}$ means a choice of a $\mathcal{U}^p$-orbit of the above isomorphism, plus a choice of a subgroup $H$ of $A[\mathfrak{p}]$ of order $p^2$.

The following is well-known and can be found in [Lan12, I.4.1.11]:

**Theorem 2.** If the level $\mathcal{U}^p$ is neat, the above moduli problem is represented by a smooth, quasi-projective scheme $S_{\mathcal{U}_f}$ over $\mathcal{O}_{K,\mathfrak{p}}$.

Above $S_{\mathcal{U}_f}$ there is a universal abelian scheme $\mathcal{A}$ of dimension 3 with an $\mathcal{O}_K$-action. We also have the relative differential sheaf $\omega_{\mathcal{A}} := e^*\Omega_{\mathcal{A}/S_{\mathcal{U}_f}}^1$, where
$e: S_{U_f} \to \mathcal{A}$ is the identity section. This is a locally free sheaf of rank 3 which inherits the $\mathcal{O}_K$-action of $\mathcal{A}$. We have a decomposition $\omega_{\mathcal{A}} = \omega_{\mathfrak{p}} \oplus \omega_{\bar{\mathfrak{p}}}$. In addition, by our assumption on the signature of $V$, the type of $\omega_{\mathcal{A}}$ as an $\mathcal{O}_{S_{U_f}}$-module is also $(2,1)$. Thus $\mathrm{rank}_{\mathcal{O}_{S_{U_f}}}(\omega_{\mathfrak{p}}) = 2$ and $\mathrm{rank}_{\mathcal{O}_{S_{U_f}}}(\omega_{\bar{\mathfrak{p}}}) = 1$ on all connected components of $S_{U_f}$.

### 3. PICARD MODULAR FORMS

Recall that $G = \mathrm{GU}(2,1)$, $G_\infty = G(\mathbb{R})$ and that we have fixed a point $\mathfrak{z}_0 \in \mathcal{X}$. In order to define scalar-valued modular forms for $G$, first of all we note that function

$$j(g, \mathfrak{z}) = g_{31}z + g_{32}w + g_{33}, \quad g = (g_{ij}) \in G_\infty, \mathfrak{z} \in \mathcal{X}$$

is a holomorphic factor of automorphy for $G$, cf. [dSG16, p. 22]. We give the classical analytic definition of Picard modular forms.

**Definition 2 (Picard modular form).** (i) A scalar-valued modular form $F$ for $G$ of weight $k \in \mathbb{Z}$ and level $U_f \subset G(L)_f$, is a complex valued function on the double coset space

$$G(\mathbb{Q}) \backslash G(\mathbb{A}_{\mathbb{Q}})/U_f \to \mathbb{C},$$

which satisfies the following: for each $g_f \in G(\mathbb{A}_f)$ the function

$$F_{g_f}(g_\infty) := j(g_\infty, \mathfrak{z}_0)^k \mu(g_\infty)^{-k/2} F(g_\infty g_f)$$

of $g_\infty \in G_\infty$ depends only on $\mathfrak{z} = g_\infty \mathfrak{z}_0$ and is holomorphic in $\mathfrak{z}$. We denote by $\mathcal{M}(k, \Gamma_{U_f})$ the space of all scalar-valued modular forms of weight $k$ and level $\Gamma_{U_f}$.

(ii) For $\chi$ a unitary Hecke character of $K$, let $\mathcal{M}(k, \Gamma, \chi)$ be the space of functions $F \in \mathcal{M}(k, \Gamma)$ with central character $\chi$, i.e.

$$F(zg) = \chi(z)F(g) \quad \forall z \in Z(\mathbb{Q}) \backslash Z(\mathbb{A}_{\mathbb{Q}}), g \in G(\mathbb{A}_{\mathbb{Q}}),$$

for $Z$ the center of $G$.

(iii) A modular form $F \in \mathcal{M}(k, \Gamma)$ is called a cusp form if for every $g_f \in G(\mathbb{A}_{\mathbb{Q},f})$, the function $F_{g_f}$ goes to zero for $\mathfrak{z} = (z, w)$, $z \to i\infty$. The subspace of cusp forms is denoted $\mathcal{S}(k, \Gamma)$. For $\Gamma = \Gamma(L)$ the stabilizer of the lattice $L$, we will simply write $\mathcal{M}(k, L)$ and $\mathcal{S}(k, L)$.

*Remark 1.* For the space $\mathcal{M}(k, \Gamma, \chi)$ to be non-zero, $\chi$ needs to be unramified and have infinity type $(-k/2, k/2)$.

We give two other equivalent definitions of Picard modular forms: one of a geometric nature, and the other algebraic. Let $R$ be an $\mathcal{O}_K[1/2D]$-algebra. A geometric modular form of weight $k \ge 0$ and neat level $U_f$ defined over $R$ is an element of the finite $R$-module

$$\mathcal{M}^{\mathrm{geom}}(k, U_f, R) := H^0(S_{U_f, R}, \omega_{\bar{\mathfrak{p}}}^k).$$

A cusp form is an element of the space

$$\mathcal{S}^{\mathrm{geom}}(k, U_f, R) := H^0(S_{U_f, R}, \omega_{\bar{\mathfrak{p}}}^k \otimes \mathcal{O}(C)^\vee),$$

where $C$ is the boundary divisor. An algebraic Picard modular form à la Katz of weight $k$ and level $U_f$ defined over $R$, is a rule $f$ which assigns to every
R-scheme $T$, and every $\underline{A} = (A, \lambda, \eta_{\underline{A}}; i) \in S_{\underline{A},R}(T)$, together with a nowhere vanishing section $\omega \in H^0(T, \omega_{\underline{A}/T}(\bar{\sigma}))$, an element $f(\underline{A}, \omega) \in H^0(T, \mathcal{O}_T)$ satisfying

• $f(\underline{A}, \lambda\omega) = \lambda^{-k}f(A, \omega)$ for every $\lambda \in H^0(T, \mathcal{O}_T)^*$;

• The rule $f$ is compatible with base change $T'/T$.

We denote by $\mathcal{M}^{\text{alg}}(k, \mathcal{U}_f, R)$ the set of all algebraic Picard modular forms defined over $R$, of weight $k$, level $\mathcal{U}_f$.

**Theorem 3.** We have a bijection between $\mathcal{M}^{\text{geom}}(k, \mathcal{U}_f, \mathbb{C})$, $\mathcal{M}^{\text{alg}}(k, \mathcal{U}_f, \mathbb{C})$ and $\mathcal{M}(k, \mathcal{U}_f, \mathbb{C})$.

*Proof.* See [dSG16, p. 25]. □

**3.1. The Fourier-Jacobi expansion.** In this section, we follow closely Finis's presentation. Let $\lambda$ be the additive character of $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ normalized by $\lambda(x_\infty) = e^{2\pi i x_\infty}$. The Fourier-Jacobi coefficients $F_r(g), r \in \mathbb{Q}$, of a form $F \in \mathcal{M}(k, \Gamma, \chi)$ are given by integration against the center of $N$:

$$F_r(g) = \int_{\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}} F((0, u)g)\lambda(-ru)du, \quad F(g) = \sum_{r \in \mathbb{Q}} F_r(g).$$

Following [S78, p. 570], we may define the space of generalized theta functions $T_{r,a,k}$, for $b$ a fractional ideal of $\mathcal{O}_K$ and an element $r \in \mathbb{Q}^{\ge 0}$, such that $r\text{N}(b)$ is integral, as the space of holomorphic functions $\vartheta: \mathbb{C} \to \mathbb{C}$ satisfying the functional equation

$$\vartheta(w + s) = \psi(s)e^{-2\pi i r \delta \bar{s}(w+s/2)}\vartheta(w), \quad s \in \mathfrak{b},$$

where $\psi(s) := (-1)^{rD|s|^2}$. The classical Fourier-Jacobi expansion may be extracted out of the $F_r(g)$ as follows: for $w_\infty \in \mathbb{C}$ and $a \in \mathbb{A}_K^\times$, define

$$g_{r,a}(w_\infty) = a_\infty^{-k} F_r((w_\infty, 0)a)e^{-\pi i r \delta(|a_\infty|^2 + |w_\infty|^2)}.$$

This depends only $w_\infty$ and on $a_f \in \mathbb{A}_{K,f}^\times$, the finite part of $a$, and is holomorphic in $w_\infty$. Then, for $\mathfrak{a}$ the fractional ideal of $K$ such that $\text{ide}(\mathfrak{a}) = a_f$, $g_{r,a} \in T_{r,a,k}$ whenever $r\text{N}(\mathfrak{a})$ is integral, and vanishes otherwise. Moreover, replacing $a$ by $\lambda a$ with $\lambda \in K^\times$, we have the relation

$$g_{r/N(\lambda), \lambda a}(\lambda w) = \lambda^{-k} g_{r,a}(w).$$

We will therefore write $g_{r,a} := g_{r,a}$. Fourier-Jacobi coefficients are therefore parametrized by pairs $(\mathfrak{a}, d)$ consisting of a fractional ideal $\mathfrak{a}$ of $\mathcal{O}_K$, and a non-negative integer $d = r\text{N}(\mathfrak{a})$. We may define the space $\mathcal{T}_{d,k}$ of Fourier-Jacobi coefficients of degree $d$ as the space of all vectors $(t_\mathfrak{a}) \in \prod_{\mathfrak{a} \in I_K} T_{d/N(\mathfrak{a}), \mathfrak{a},k}$ such that

$$t_{\lambda \mathfrak{a}}(\lambda w) = \lambda^{-k} t_{\mathfrak{a}}(w).$$

Obviously, we always have that $g_d := (g_{d/N(\mathfrak{a}),\mathfrak{a}}) \in \mathcal{T}_{d,k}$. After choosing a system of representatives $\mathcal{A}$ for the ideal classes of $K$ we get an isomorphism

$$\mathcal{T}_{d,k} \cong \bigoplus_{\mathfrak{a} \in \mathcal{A}} T_{d/N(\mathfrak{a}), \mathfrak{a},k}^1,$$

where $T_{r,a,k}^1 \subset T_{r,a,k}$ denotes the subspace of theta functions $\vartheta$ invariant under the action of the roots of unity. The coefficient $g_{0,a}$ is the constant term of $F$ at the cusp corresponding to $\mathfrak{a}$. It is an element of the one-dimensional
space $T_{0,a,k}$, which consists of constant theta functions. Consequently, $T_{0,k}$ has dimension $h_K = |\mathrm{Cl}_K|$.

**3.2. Arithmetic theta functions.** The notion of an arithmetic automorphic form for $G$ has been introduced by Shimura in [S78, p. 580]. An element in $\mathcal{M}(k, \Gamma)$ is said to be arithmetic if its Fourier-Jacobi coefficients are arithmetic theta functions:

**Definition 3.** (i) A theta function $g_d \in \mathcal{T}_{d,k}$ is called arithmetic if
$$e^{-\pi d\sqrt{D}|w|^2} g_d(w) \in \iota_\infty(\mathcal{O}_{K^{ab}}), \forall w \in K.$$

Denote by $\mathcal{T}_{d,k}^{ar} \subset \mathcal{T}_{d,k}$ the subspace of arithmetic theta functions.

(ii) Denote by $\mathcal{M}^{ar}(k, \Gamma)$ (resp. $\mathcal{S}^{ar}(k, \Gamma)$) the space of arithmetic Picard modular forms (resp. cusp forms).

The notions introduced in Shimura's are in analogy to the case of elliptic modular forms, where the $q$-expansion principle holds: the $q$-expansion homomorphism is injective, and if the $q$-expansion of a modular form $f$ is defined over a ring $S$, then the modular form is already defined over $S$. In the $\mathrm{GU}(2, 1)$ case, the existence of a good compactified moduli scheme (away from primes dividing $D$) is known from work of Larsen [Lar88],[Lar92], see also [Bel02]. This, together with the discussion in section 3, shows that we can read the ring of definition of a Picard modular form from its $q$-expansion, which justifies the above definitions.

**4. THE p-MODIFIED KUDLA LIFT**

In this section, we first give an account of the level one Kudla lift, and then introduce the $p$-modified lift, which is a modification of the particular Kudla lift considered by Finis, in order to make this compatible with the desired $p$-adic interpolation.

**4.1. The level one Kudla lift.** Throughout this subsection, let $L := \mathcal{O}_K^3$ be the standard lattice. For $k$ a positive even integer, we consider the theta lift

$$\mathcal{L} : S_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}}) \to \mathcal{S}(k, L, \chi),$$

where we recall that

$$\Gamma_1(D) = \left\{ \gamma \in \mathrm{SL}_2(\mathbb{Z}) \mid \gamma \equiv \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \mod D \right\},$$

and $\omega_{K/\mathbb{Q}}$ is the non-trivial quadratic character associated to $K/\mathbb{Q}$. We also take $\chi$ to be an unramified unitary Hecke character with infinity type $\chi_\infty(z) = (z/|z|)^{-k}$; we will say $\chi$ has weight $k$. Additionally, let $\varepsilon$ be an unramified Hecke character of weight 0, i.e. a character of $\mathrm{Cl}_K$. We define the theta kernel

$$\theta(\tau, \mathfrak{z}) = \theta_{k, \varepsilon, \chi}(\tau, \mathfrak{z}) : \mathbb{H} \times G(\mathbb{A}_{\mathbb{Q}}) \to \mathbb{C}$$

by

$$\overline{\theta(\tau, g)} = \sum_{\mathfrak{b} \in \mathrm{Cl}_K} \varepsilon(\det g) |N(\mathfrak{b})\mu(g_f)|_{\mathbb{A}_{\mathbb{Q}}}^{k/2} (\varepsilon^3 \chi^{-1})(\mathfrak{b}) \overline{\theta_{\mathfrak{b}}(\tau, g)}$$

with

$$\overline{\theta_{\mathfrak{b}}(\tau, g)} = 2^{k-1} D^{-k/2} y^2 \eta(\mathfrak{z}_0)^{-k} \mu(g_\infty)^{-k/2} \sum_{X \in \mathfrak{b}g_f L} \overline{f_{\tau, \mathfrak{z}}^{\mathfrak{b}g_f}(X)}.$$
where

$$ \overline{f_{\tau, \mathfrak{g}_f}^{\mathfrak{b}g_f}(X)} = (X, g_{\infty, \mathfrak{g}_0})^k e^{2\pi i |N(\mathfrak{b})\mu(g_f)|_{\mathcal{A}_{\mathbb{Q}}}(2iy|(X, g_{\infty, \mathfrak{g}_0})|^2(\mu(g_\infty)\eta(\mathfrak{g}_0)))^{-1} + \bar{\tau}(X, X))} $$

Here $\tau = x + iy \in \mathbb{H}$, and recall that $\mathfrak{g}_0 := (\delta/2, 0) \in \mathfrak{X}$. The following proposition is due to Kudla:

**Theorem 4.** [K79, p. 13]

(i) For a fixed $g_0 \in G(\mathbb{A}_{\mathbb{Q}})$ the function $\theta(\tau, g_0) : \mathbb{H} \to \mathbb{C}$ is automorphic, i.e. for $\gamma \in \Gamma_1(D)$, we have:

$$ \theta(\gamma\tau, g_0) = \omega_{K/\mathbb{Q}}(\gamma)j(\gamma, \tau)^{k-1}\theta(\tau, g_0). $$

Moreover, it is rapidly decreasing in $\tau$.

(ii) For a fixed $\tau_0 \in h$, $\theta(\tau_0, g) : G(\mathbb{Q})\backslash G(\mathbb{A}_{\mathbb{Q}})/G(L)_f \to \mathbb{C}$ is a smooth function.

(iii) Define the lifting $\mathcal{L} := \mathcal{L}_{k,\varepsilon,\chi}$ by

$$ \mathcal{L}(f)(g) = \int_{\Gamma_1(D)\backslash\mathbb{H}} f(\tau) \overline{\theta(\tau, g)} y^{k-3} dxdy $$

for $f \in M_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}})$. Then $\mathcal{L}(f)$ is an element of $\mathcal{M}(k, L, \chi)$. If $f$ is cuspidal, $\mathcal{L}(f)$ is also cuspidal.

We now define the intrinsic theta functions associated to $K$, in the sense of Shimura [S76, §4]. For a fractional ideal $\mathfrak{a}$ of $K$ set

$$ \vartheta_{\mathfrak{a}}(w, \tau) = \sum_{a \in \mathfrak{a}} e^{2\pi i (N(a)^{-1}N(a)\tau + aw)}, $$

for $w \in \mathbb{C}$ and $\tau \in \mathbb{H}$. This defines a holomorphic function

$$ \vartheta_{\mathfrak{a}} : \mathbb{C} \times \mathbb{H} \to \mathbb{C}. $$

**Proposition 1.** The function $\vartheta_{\mathfrak{a}}$ satisfies the functional equations:

(i) For $\lambda \in K^\times$, we have $\vartheta_{\lambda\mathfrak{a}}(w, \tau) = \vartheta_{\mathfrak{a}}(\lambda w, \tau)$.

(ii) For $\gamma \in \Gamma_1(D)$, we have

$$ \vartheta_{\mathfrak{a}}(j(\gamma, \tau)^{-1}w, \gamma\tau) = \omega_{K/\mathbb{Q}}(\gamma)j(\gamma, \tau)\vartheta_{\mathfrak{a}}(w, \tau). $$

(iii) For $\tau_0 \in \mathbb{H} \cap K$, the function $\vartheta_{\mathfrak{a}}(w, \tau_0) : \mathbb{C} \to \mathbb{C}$, is a theta function, in the sense of Shimura, with respect to a certain lattice $L_{\tau, \mathfrak{a}} \subset K$.

*Proof.* See [F99, p. 15]. □

<figure><img src="image_3.png" /></figure>

Finis's result on the Fourier-Jacobi expansion of the lifted form can be expressed in terms of these intrinsic theta functions:

**Theorem 5 (Finis).** Let $f \in S_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}})$. Then the lifted form $\mathcal{L}_{k,\varepsilon,\chi}(f)$ has Fourier-Jacobi coefficients

$$ g_{n/N(\mathfrak{a}),\mathfrak{a}}(w) = \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\mathfrak{b} \in \mathrm{Cl}_K} (\varepsilon^3 \chi^{-1})(\mathfrak{b}) N(\mathfrak{b})^{k/2} T_n(\mathrm{Tr}(f(\tau)\vartheta_{\mathfrak{b}}(n\delta w, \tau)))|_{\tau=\tau_{\mathfrak{a},\mathfrak{b}}(\mathfrak{a}\bar{\mathfrak{b}})}. $$

Here $\mathfrak{b}$ ranges over a system of representatives for the ideal classes of $K$ such that $\mathfrak{a}\bar{\mathfrak{b}} = \mathbb{Z} + \mathbb{Z}\tau_{\mathfrak{a},\mathfrak{b}}$ for some $\tau_{\mathfrak{a},\mathfrak{b}} \in K \cap \mathbb{H}$ and Tr denotes $\mathrm{Tr}_{\Gamma_1(D)\backslash SL_2(\mathbb{Z})}$. □

*Proof.* See [F99, p. 16]. □

<figure><img src="image_4.png" /></figure>
**Theorem 6.** Let $k, \varepsilon$ and $\chi$ be as before. There exists a period $\Omega_0 \in \mathbb{C}$ only depending on $K$, such that, $\mathcal{L}_{k,\varepsilon,\chi}^{\text{ar}} = \Omega_0^{-k} \mathcal{L}_{k,\varepsilon,\chi}$ is arithmetic in the sense of Shimura, i.e. we have a morphism

$$ \mathcal{L}^{\text{ar}} : S_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}}, \bar{\mathbb{Z}}[1/D]) \to \mathcal{S}^{\text{ar}}(k, L, \chi). $$

*Proof.* See [F99, p. 36]. □

The following definition of Hecke operators for $G = \text{GU}(2, 1)$ is recalled in [K81, p. 325]; in the automorphic context, this is also recalled in [MS07, p. 8]. As Kudla in loc. cit. works in the class number one context, we follow the more general notations of Finis's thesis. For each prime ideal $\mathfrak{q}$ of $\mathcal{O}_K$, lying over the prime $q$ of $\mathbb{Q}$, let us define a (non-normalized) Hecke operator $T_q^{\text{naive}}$ acting on $\mathcal{S}(k, L, \chi)$. For $F \in \mathcal{S}(k, L, \chi)$,

$$ T_q^{\text{naive}} F(g_0) = \int_{S_q} F(gg_0)dg, $$

where $S_q$ is a subset of $G(\mathbb{Q}_q)$ defined as follows: if $\mathfrak{q} = \bar{\mathfrak{q}}$,

$$ S_q = \{g \in G(\mathbb{Q}_q) \mid g(L \otimes \mathbb{Z}_q) \subseteq (L \otimes \mathbb{Z}_q), v_q(\mu(g)) = v_q(\text{N}(\mathfrak{q}))\}, $$

where $v_q$ is the valuation associated to $q$, and if $q$ splits in $K$,

$$ S_q = \{g \in G(\mathbb{Q}_q) \mid g(L \otimes \mathbb{Z}_q) \subseteq (L \otimes \mathbb{Z}_q), \det g \in \mathfrak{q}\bar{\mathfrak{q}}^2 \otimes \mathbb{Z}_q\}. $$

If $F$ is a Hecke eigenform with central character $\chi$ and eigenvalues $\tilde{\lambda}(\mathfrak{q}) := \lambda^{\text{naive}}(\mathfrak{q})$, Shintani defined an $L$-function $\zeta(F, \xi, s)$ for $\xi$ a Hecke character of $K$: $\zeta(F, \xi, s) = \prod_{\mathfrak{q}} P_{\mathfrak{q}}(\xi(\mathfrak{q})\text{N}(\mathfrak{q})^{-s-1})^{-1}$ with (here $N = \text{N}(\mathfrak{q})$)

$$ P_{\mathfrak{q}}(x) := \begin{cases} 1 - \tilde{\lambda}(\mathfrak{q})(\bar{\mathfrak{q}})\chi(\mathfrak{q})^{-1}N^{-1}x + \tilde{\lambda}(\mathfrak{q})\chi(\mathfrak{q})^{-1}N^{-1}x^2 - \chi(\bar{\mathfrak{q}}\mathfrak{q}^{-1})x^3, & \mathfrak{q} \neq \bar{\mathfrak{q}}, \\ (1-x)(1+(N-\tilde{\lambda}(\mathfrak{q})\chi(\mathfrak{q})^{-1})N^{-1}x+x^2), & \mathfrak{q} = q\mathcal{O}_K, \\ (1-x)(1+(N^{1/2}-\tilde{\lambda}(\mathfrak{q})\chi(\mathfrak{q})^{-1})N^{-1}x+x^2), & \mathfrak{q} \mid \delta\mathcal{O}_K. \end{cases} $$

**Proposition 2.** Suppose that $f \in S_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}})$ is a Hecke eigenform with eigenvalues $a_q$, whose lifting $\mathcal{L}_{k,\varepsilon,\chi}(f)$ is non-zero. Then the lift is also a Hecke eigenform with eigenvalues

$$ \lambda^{\text{naive}}(\mathfrak{q}, f) = \lambda^{\text{naive}}(\mathfrak{q}) = \begin{cases} q^{2-k/2}\varepsilon(\bar{\mathfrak{q}})a_q + q(\chi\varepsilon^{-2})(\bar{\mathfrak{q}}), & \mathfrak{q} \neq \bar{\mathfrak{q}}, \\ q^{2-k/2}\varepsilon(\bar{\mathfrak{q}})(a_q + \bar{a}_q) + q(\chi\varepsilon^{-2})(\bar{\mathfrak{q}}), & \mathfrak{q} \mid \delta\mathcal{O}_K, \\ q^{4-k}a_q^2 + 2q^2 + q, & \mathfrak{q} = q\mathcal{O}_K, \end{cases} $$

and we have the relation

$$ \zeta(\mathcal{L}(f), \xi, s) = L(f_K \otimes \varepsilon\chi^{-1}, \xi, s + k/2)L(\varepsilon^{-2}\xi, s + 1), $$

where $f_K$ denotes the base change of $f$ to $\text{GL}(2, K)$, the $L$-function appearing in the first factor in the right hand side is the Dirichlet series attached to the form $f_K$ and $L(\xi, s)$ is the $L$-series for the Hecke character $\xi$.

*Proof.* See [K81, p. 327]. □

**Corollary 7.** (i) The normalized operator $T_q := \text{N}(\mathfrak{q})^{k/2-1}T_q^{\text{naive}}$ is integral.
(ii) The eigenvalues for the normalized operator $T_q$ are:

$$ \lambda(\mathbf{q}) = \lambda^{\text{naive}}(\mathbf{q})\mathbf{N}(\mathbf{q})^{k/2-1} = \begin{cases} \varepsilon(\bar{\mathbf{q}})qa_q + q^{k/2}(\chi\varepsilon^{-2})(\bar{\mathbf{q}}), & \mathbf{q} \neq \bar{\mathbf{q}}, \\ \varepsilon(\bar{\mathbf{q}})q(a_q + \bar{a}_q) + q^{k/2}(\chi\varepsilon^{-2})(\bar{\mathbf{q}}), & \mathbf{q} \mid \delta\mathcal{O}_K, \\ qa_q^2 + 2q^{k-1} + q^{k-2}, & \mathbf{q} = q\mathcal{O}_K. \end{cases} $$

*Remark.* Note that the lifted forms are never $T_q$-ordinary. Furthermore, the normalization introduced above is optimal, as there exist eigenforms in $\mathcal{S}(k, L)$ whose $T_q$-eigenvalues are not divisible by $q$, see [F98, p. 178].

4.2. **The p-modification.** In this subsection, for $r \ge 1$ a fixed integer, we will take two lattices $L := p^r D\mathcal{O}_K + \mathfrak{p}^r \delta\mathcal{O}_K + p^r D\mathcal{O}_K$ and $L' := \mathcal{O}_K + \mathfrak{p}^r \delta\mathcal{O}_K + \mathcal{O}_K$, so that $L \subset L'$. We construct a theta lift:

$$ \mathcal{L}^p : S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}}) \to \mathcal{S}(k, \Gamma, \chi), $$

for a certain subgroup $\Gamma \subset \Gamma(L')$. As for the level one lift, we have a Hecke character $\chi$ with infinity type $(-k/2, k/2)$, but with conductor $pD$. For the definition of the theta kernel, we follow Kudla [K79, p. 12], but we adapt instead the notation and formulation of Finis. For any $h \in L'/L$ and $\mathfrak{b} \in \text{Cl}_K$ coprime to $p$, we define the theta kernel $\theta_{h,\mathfrak{b}} = \theta_{h,\mathfrak{b},k,\varepsilon,\chi}$ on $\mathbb{H} \times G(\mathbb{A}_{\mathbb{Q}})$ by

$$ \overline{\theta_{h,\mathfrak{b}}(\tau, g)} = 2^{k-1} D^{-k/2} y^2 \eta(\mathfrak{3}_0)^{-k} \mu(g_\infty)^{-k/2} \sum_{\substack{X \in \mathfrak{b}g_f L' \\ X \equiv h(L)}} \overline{f_{\tau, \mathfrak{b}}^{\mathfrak{b}g_f}(X)}, $$

where

$$ \overline{f_{\tau, \mathfrak{b}}^{\mathfrak{b}g_f}(X)} = (X, g_{\infty, \mathfrak{3}_0})^k e^{2\pi i |N(\mathfrak{b})\mu(g_f)|_{\mathcal{A}_{\mathbb{Q}}} (2iy|(X, g_{\infty, \mathfrak{3}_0})|^2 (\mu(g_\infty)\eta(\mathfrak{3}_0))^{-1} + \bar{\tau}(X, X))}, $$

so that

$$ \overline{\theta_h(\tau, g)} = \sum_{\substack{\mathfrak{b} \in \text{Cl}_K \\ (\mathfrak{b}, p)=1}} \varepsilon(\det g) |N(\mathfrak{b})\mu(g_f)|_{\mathcal{A}_{\mathbb{Q}}}^{k/2} (\varepsilon^3 \chi^{-1})(\mathfrak{b}) \overline{\theta_{h,\mathfrak{b}}(\tau, g)}. $$

To obtain our formula, we relate our lifting to the lifting from $\text{GU}(2)$ to $\text{GU}(2)$ introduced by Kudla [K79, p. 14], a technique already used by Finis. To this end, consider the two-dimensional hermitian space $V_0 = K^2$ with hermitian form of matrix

$$ \begin{pmatrix} 0 & \delta^{-1} \\ -\delta^{-1} & 0 \end{pmatrix} $$

Then the associated special unitary group is $SU(V_0) \equiv SL_2(\mathbb{R})$. Fix $L'_0 = (p^r D\mathcal{O}_K)^2$ and $L_0 = (\mathcal{O}_K)^2$, and set

$$ \tilde{\Gamma} = \Gamma(p^r D) = \{\gamma \in SL_2(\mathbb{Z}) \mid \gamma L_0 = L_0 \text{ and } \gamma \text{ is trivial on } L'_0/L_0\}. $$

For a fractional ideal $\mathfrak{c}$ of $K$ and $h \in L'_0/L_0$ a theta kernel $\mathbb{H} \times \mathbb{H} \to \mathbb{C}$ is given by

$$ \Theta_{k, \mathfrak{c}, h}^{0}(\tau, z) := 2^{k-1} D^{-k/2} y \sum_{\substack{X \in \bar{\tau} L'_{0} \\ \bar{\tau}^{-1} X \equiv h (L_0)}} \Phi_{k, \mathfrak{c}, h}(X, \tau, z). $$

where

$$ \Phi_{k, \mathfrak{c}, h}(X, \tau, z) := (X, (z, 1))^k \eta^0(z)^{-k} e^{2\pi i N(\mathfrak{c})^{-1} (2iy|(X, (z, 1))|^2 \eta^0(z)^{-1} + \bar{\tau}(X, X))}, $$
with $\eta^0(z) = (2/\sqrt{D})\mathrm{Im}(z)$. For $h = (h_1, h_2) \in L'_0$, call $\gamma_h \in M_2(\mathbb{Z})$ the unique matrix such that

$$ (h_1, h_2) = (1, \tau_K)\gamma_h. $$

For $n \in \mathbb{Z}$, let

$$ \mathcal{H}_h^{=n} = \{ \gamma \in M_2(\mathbb{Z}) \mid \det \gamma = n, \gamma \equiv \gamma_h \quad \mathrm{mod} \ p^r D \}. $$

For each $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{Z})$ such that $\det \gamma < 0$, write $\gamma'' = \begin{pmatrix} d & b \\ -c & -a \end{pmatrix} \in M_2(\mathbb{Z})$. The following result is a restatement of a result of Kudla:

**Proposition 3.** [K79, p. 20] Let $\mathfrak{c} = \mathbb{Z} + \mathbb{Z}\tau_c$, $\tau_c \in \mathbb{H}$, be a fractional ideal of $\mathcal{O}_K$, $k \ge 4$ be an even integer and $h \in L'_0/L_0$. For $f \in S_k(\Gamma(p^r D))$, let $\tilde{\mathcal{L}}_{k,\mathfrak{c},h}^0(f) = \langle f, \Theta_{k,\mathfrak{c},h}^0 \rangle \in S_k(\Gamma(p^r D))$. Writing $\tilde{\mathcal{L}}_h^0(f) = \sum_{n=1}^\infty a_n q^n$, we have

$$ a_n = \mathrm{CN}(\mathfrak{c})^k \sum_{\gamma \in \mathcal{H}_h^{=n} / \Gamma(p^r D)} (f|_k \gamma'') |_{\tau=\tau_c}, $$

where

$$ C = \sum_{t \in \mathcal{O}_K(p^r D \mathcal{O}_K)} e^{-2\pi i (\mathrm{tr}(h_1 \delta^{-1} t))} = |\mathcal{O}_K / p^r D \mathcal{O}_K|. $$

We prefer to use the normalization of the GU(2) lifting by $\mathcal{L}_h^0 = C^{-1}\tilde{\mathcal{L}}_h^0$. In other words, $\mathcal{L}_h^0(f)$ is the inner product of $f$ with $C^{-1}\Theta_{k,\mathfrak{c},h}^0$. Now define

$$ \mathcal{H} = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{Z}) \mid c \equiv 0 \quad \mathrm{mod} \ p^r D, d \equiv 1 \quad \mathrm{mod} \ p^r D \right\} $$

and for each $n \in \mathbb{Z}$,

$$ \mathcal{H}^{=n} := \mathcal{H} \cap \{ \gamma \in M_2(\mathbb{Z}) \mid \det \gamma = n \}. $$

Note that we have a disjoint union:

$$ \mathcal{H}^{=n} = \dot{\bigcup}_{h \in T} \mathcal{H}_h^{=n}, $$

where $T = \{h \in L'_0/L_0 \mid \gamma_h \in \mathcal{H}\}$. For $f \in S_k(\Gamma(p^r D))$, define

$$ \begin{aligned} \mathcal{L}^0(f) &:= [\Gamma_1(p^r D) : \Gamma(p^r D)]^{-1} \sum_{h \in T} \mathcal{L}_h^0(f) \\ &:= [\Gamma_1(p^r D) : \Gamma(p^r D)]^{-1} \sum_{h \in T} C^{-1} \langle f, \Theta_{k, \mathfrak{c}, h}^0 \rangle. \end{aligned} $$

**Theorem 8.** Let $\mathfrak{c} = \mathbb{Z} + \mathbb{Z}\tau_c$, $\tau_c \in \mathbb{H}$, be a fractional ideal of $\mathcal{O}_K$ and $k \ge 4$ be an even integer. For $f \in S_k(\Gamma(p^r D))$, write $\mathcal{L}^0(f) = \sum_{n=1}^\infty a_n q^n \in S_k(\Gamma(p^r D))$. We have

$$ a_n = [\Gamma_1(p^r D) : \Gamma(p^r D)]^{-1} \mathrm{N}(\mathfrak{c})^k T_n(\mathrm{Tr}_{\Gamma(p^r D) \setminus \Gamma_1(p^r D)} f) |_{\tau=\tau_c}, $$

where $T_n$ is the Hecke operator at $n$ of level $\Gamma_1(p^r D)$.

*Proof.* First of all, notice that, for $n \in \mathbb{N}$,

$$ \mathcal{H}^{-n} / \Gamma(p^r D) = \dot{\bigcup}_{h \in T} \mathcal{H}_h^{-n} / \Gamma(p^r D). $$
Consider first the case where $n = q^s$ is the power of a prime $q$. If $\gamma_1, \dots, \gamma_l$ is a set of representatives for $\mathcal{H}^{-n}/\Gamma(p^rD)$, then $\gamma_1'', \dots, \gamma_l''$ is a set of representatives for the left quotient by $\Gamma(p^rD)$ of the set

$$ \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{Z}) \mid ad - bc = n, c \equiv 0 \mod p^rD, a \equiv 1 \mod p^rD \right\}. $$

We have two possibilities:

(i) Suppose $q \nmid p^rD$. Then the above set is clearly equal to the double coset $\Gamma_1(p^rD) \binom{1}{n} \Gamma_1(p^rD)$.

(ii) If $q \mid p^rD$, then for each matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ in the above set we also see that $d \equiv 0 \mod \gcd(n, p^rD)$, so that again the above is just the double coset $\Gamma_1(p^rD) \binom{1}{n} \Gamma_1(p^rD)$.

In both cases, from a set of representatives of $\mathcal{H}^{-n}/\Gamma(p^rD)$, we get a set of representatives for

$$ \Gamma(p^rD) \backslash \Gamma_1(p^rD) \binom{1}{n} \Gamma_1(p^rD). $$

Hence, if $g$ is a modular form of level $\Gamma(p^rD)$ and weight $k$,

$$ \sum_{\gamma \in \mathcal{H}^{-n} / \Gamma(p^rD)} g|_k \gamma'' = T_n(\mathrm{Tr}_{\Gamma(p^rD) \backslash \Gamma_1(p^rD)} g). $$

Then, we have for the $n$-th coefficient of $\mathcal{L}^0(f)$,

$$ a_n = [\Gamma_1(p^rD) : \Gamma(p^rD)]^{-1} \mathrm{N}(\mathfrak{c})^k \sum_{h \in T} \sum_{\gamma \in \mathcal{H}_h^{-n} / \Gamma(p^rD)} (f|_k \gamma'')|_{\tau=\tau_c} $$

But now

$$ \sum_{h \in T} \sum_{\gamma \in \mathcal{H}_h^{-n} / \Gamma(p^rD)} (f|_k \gamma'')|_{\tau=\tau_c} = \sum_{\gamma \in \mathcal{H}^{-n} / \Gamma(p^rD)} (f|_k \gamma'')|_{\tau=\tau_c}, $$

and applying the above argument we get the formula for $n$ power of a prime. The general case follows from the above by using the multiplicativity of the double cosets. $\square$

The following proposition tells us that by putting together the "partial" lifts $\mathcal{L}_h^0(f)$, we gain some additional structure.

**Proposition 4.** The lifted form $\mathcal{L}^0(f)$ belongs to $S_k(\Gamma^1(p^rD))$, where $\Gamma^1(p^rD) \subset \mathrm{SL}_2(\mathbb{Z})$ is the congruence subgroup of matrices that are in the lower unipotent modulo $p^rD$, i.e., for $\gamma \in \Gamma^1(p^rD)$,

$$ \gamma \equiv \begin{pmatrix} 1 & 0 \\ * & 1 \end{pmatrix} \mod p^rD. $$

*Proof.* For $\gamma \in \Gamma^1(p^rD)$, the computation in [K81, p. 329] tells us that

$$ \Theta_{k,c,h}^0|_k \gamma = \Theta_{k,c,\gamma^{-1}h}^0 $$

and notice that $\gamma^{-1} \in \Gamma^1(p^rD)$ induces a permutation on the fixed set of indices $T$. $\square$
Remark 2. As

$$ \begin{pmatrix} 1/p^r D & 0 \\ 0 & 1 \end{pmatrix} \Gamma^1(p^r D) \begin{pmatrix} p^r D & 0 \\ 0 & 1 \end{pmatrix} = \Gamma_1(p^r D), $$

we get a Hecke equivariant isomorphism $S_k(\Gamma^1(p^r D)) \cong S_k(\Gamma_1(p^r D))$ by $f(z) \mapsto f(p^r Dz)$. Thus we will think of our lifts as lying in the more classical space $S_k(\Gamma_1(p^r D))$.

**Corollary 9.** The lift $\mathcal{L}^0$ restricts to an algebraic Hecke equivariant endomorphism of $S_k(\Gamma_1(p^r D))$, and the formula for the coefficients of $\mathcal{L}^0(f) \in S_k(\Gamma_1(p^r D))$ is

$$ a_n = \mathrm{N}(\mathfrak{c})^k T_n f \mid_{\tau=\tau_c}. $$

*Proof.* This follows easily from the above remark, and by considering a basis of Hecke eigenforms for $S_k(\Gamma_1(p^r D))$. $\square$

For $f \in S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}})$ define

$$ \mathcal{L}^p(f) = \sum_{h \in T} C^{-1} \langle f, \theta_h \rangle, $$

where we recall that $C = |\mathcal{O}_K/p^r D \mathcal{O}_K|$ is the constant defined in Proposition 3. Consider the group of matrices in $G(L)_f \cap G(\mathbb{Q})$ of the form

$$ \begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix} \mod p^r D. $$

Given the clear analogy, we will denote simply by $\Gamma_1(p^r D)$ this congruence subgroup, with slight abuse of notation.

**Proposition 5.** For $f \in S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}})$, we have $\mathcal{L}^p(f) \in \mathcal{S}(k, \Gamma_1(p^r D), \chi)$.

*Proof.* Define, by analogy with the elliptic case, $\Gamma^1(p^r D) \subset \mathrm{GU}(2, 1)(\mathbb{Q})$ the stabilizer of $T$. Then we know that $\mathcal{L}^p(f) \in \mathcal{S}(k, \Gamma^1(p^r D), \chi)$. Setting $\mathcal{U}_f^0 \subset G(\mathbb{A}_{\mathbb{Q},f})$ for the adelic level subgroup corresponding to $\Gamma^1(p^r D)$, we see that for

$$ \gamma = \begin{pmatrix} p^r D & & \\ & p^r \omega & \\ & & 1 \end{pmatrix} \times \begin{pmatrix} p^r D & & \\ & -\omega & \\ & & 1 \end{pmatrix} \in \mathrm{GL}_3(\hat{\mathbb{Z}}) \times \mathrm{GL}_3(\hat{\mathbb{Z}}), $$

where we have taken the image of $\delta \in \mathcal{O}_K$ in $\hat{\mathbb{Z}} \times \hat{\mathbb{Z}}$ to be $(\omega, -\omega)$. Then

$$ \gamma^{-1} \mathcal{U}_f^0 \gamma = \mathcal{U}_{f,0}, $$

where $\mathcal{U}_{f,0}$ is the adelic level subgroup associated to $\Gamma_1(p^r D)$. Hence, we see, after switching to $\mathcal{U}_{f,0}$ by conjugation by $\gamma$, $\mathcal{L}^p(f) \in \mathcal{S}(k, \Gamma_1(p^r D), \chi)$. $\square$

**Theorem 10.** Let $k \ge 6$ and $f \in S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}})$. Then the lifted form $\mathcal{L}_{k,\varepsilon,\chi}^p(f)$ has Fourier-Jacobi coefficients

$$ g_{n/N(\mathfrak{a}),\mathfrak{a}}(w) = \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \mathrm{Cl}_K \\ (\mathfrak{b},p)=1}} (\varepsilon^3 \chi^{-1})(\mathfrak{b}) N(\mathfrak{b})^{k/2} T_n(f(\tau)\vartheta_{\delta\bar{p}^r\bar{\mathfrak{b}}}(n\delta w, \tau))|_{\tau=\tau_{\mathfrak{a},\mathfrak{b}}}. $$

Here $\mathfrak{b}$ ranges over a system of representatives for the ideal classes of $K$ such that $\mathfrak{a}\mathfrak{b} = \mathbb{Z} + \mathbb{Z}\tau_{\mathfrak{a},\mathfrak{b}}$ for some $\tau_{\mathfrak{a},\mathfrak{b}} \in K \cap \mathbb{H}$.
Proof. We adapt Finis's argument to our case. To compute the Fourier-Jacobi coefficients $g_{n/N(a),a}$ of $F = \mathcal{L}^p(f)$ set

$$g_f = \begin{pmatrix} \bar{a}_f & 1 \\ & a_f^{-1} \end{pmatrix} \in G(\mathbb{A}_{\mathbb{Q},f})$$

for $a_f \in \mathbb{A}_{K,f}^\times$ with corresponding fractional ideal $\mathfrak{a}$, and consider the Fourier-Jacobi expansion of $F_{g_f}$:

$$F_{g_f}(\mathfrak{z}) = \sum_{n \in \mathbb{Q}} g_{n,\mathfrak{a}}(w)q^n.$$

We have

$$F_{g_f}(\mathfrak{z}) = \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \text{Cl}_K \\ (\mathfrak{b},p)=1}} (\varepsilon^3 \chi^{-1})(\mathfrak{b}) N(\mathfrak{b})^{-k/2} F_{\mathfrak{a},\mathfrak{b}}(\mathfrak{z}),$$

where $F_{\mathfrak{a},\mathfrak{b}}(f)(\mathfrak{z})$ is the lifting of $f$ against the theta kernel $\Theta_{\mathfrak{a},\mathfrak{b}}$ defined as

$$\Theta_{\mathfrak{a},\mathfrak{b}}(\tau, \mathfrak{z}) = 2^{k-1} D^{-k/2} y^2 C^{-1} \sum_{h \in T} \sum_{\substack{X \in \mathfrak{b}g_f L' \\ (\mathfrak{b}g_f)^{-1} X \in T}} (X, \mathfrak{z})^k \eta(\mathfrak{z})^{-k} e^{2\pi i N(\mathfrak{b})^{-1} |\mu(g_f)|_{\mathbb{Q}} (2iy\eta(\mathfrak{z})^{-1} |(X, \mathfrak{z})|^2 + \bar{\tau}(X, X))}$$

For $h \in T$, let $\Theta_{\mathfrak{a},\mathfrak{b},h}$ be the theta kernel

$$\Theta_{\mathfrak{a},\mathfrak{b},h}(\tau, \mathfrak{z}) = 2^{k-1} D^{-k/2} y^2 \sum_{\substack{X \in \mathfrak{b}g_f L' \\ (\mathfrak{b}g_f)^{-1} X \equiv h (L)}} (X, \mathfrak{z})^k \eta(\mathfrak{z})^{-k} e^{2\pi i N(\mathfrak{b})^{-1} |\mu(g_f)|_{\mathbb{Q}} (2iy\eta(\mathfrak{z})^{-1} |(X, \mathfrak{z})|^2 + \bar{\tau}(X, X))},$$

so that

$$\Theta_{\mathfrak{a},\mathfrak{b}} = C^{-1} \sum_{h \in T} \Theta_{\mathfrak{a},\mathfrak{b},h}$$

and $F_{\mathfrak{a},\mathfrak{b},h} := C^{-1}\langle f, \Theta_{\mathfrak{a},\mathfrak{b},h} \rangle$. By considering Taylor coefficients at zero, we are reduced to computing the Fourier expansions of the scalar valued functions

$$\phi_{\mathfrak{a},\mathfrak{b},h}^{(m)}(z) := \frac{1}{m!} \left. \frac{\partial^m}{\partial w^m} F_{\mathfrak{a},\mathfrak{b},h}(z, w) \right|_{w=0}.$$

In a first step, we express these functions in terms of the so-called two-dimensional liftings $\mathcal{L}_h^0$. Evidently $\phi_{\mathfrak{a},\mathfrak{b},h}^{(m)} = \langle f, \Theta_{\mathfrak{a},\mathfrak{b},h}^{(m)} \rangle$ with

$$\Theta_{\mathfrak{a},\mathfrak{b},h}^{(m)}(\tau, z) = \frac{1}{m!} \left. \frac{\partial^m}{\partial w^m} \Theta_{\mathfrak{a},\mathfrak{b},h}(\tau, (z, w)) \right|_{w=0}.$$

Explicitly, we have

$$\overline{\Theta_{\mathfrak{a},\mathfrak{b},h}(\tau, \mathfrak{z})} = 2^{k-1} D^{-k/2} y^2 \sum_{\substack{X \in \bar{\mathfrak{a}} \oplus \delta \mathfrak{p}^r \mathfrak{b} \oplus \mathfrak{a}^{-1} \mathfrak{b} \\ (g_f \mathfrak{b})^{-1} X \equiv h (L)}} (\overline{(X^0, (z, 1))} - X_2 \overline{w})^k (\eta^0(z) - |w|^2)^{-k} e^{-2\pi i N(\mathfrak{b})^{-1}(-2iy|(X^0, (z, 1)) - \overline{X_2 w}|^2(\eta^0(z) - |w|^2)^{-1} + \tau((X^0, X^0) - N(X_2)))},$$
with $X^0 := (X_1, X_3) \in \mathcal{O}_K^2$. Taking holomorphic derivatives with respect to $w$ yields

$$ \overline{\Theta_{a,b,h}^{(m)}(\tau, z)} = 2^{k-1} D^{-k/2} y^{2+m} N(\mathfrak{a})^{-k-m} \frac{1}{m!} \left(\frac{4\pi}{N(\mathfrak{b})}\right)^m \left(\sum_{Y \in \tilde{c}L'_0} \Phi_{k+m,\mathfrak{c},h}(Y, \tau, z') \right) \left(\sum_{X_2 \in \delta_{\mathfrak{p}^r\mathfrak{b}}} \bar{X}_2^m e^{2\pi i N(b)^{-1} N(X_2)\tau} \right), $$

where we set $Y = (X_1, N(\mathfrak{a})X_3)$, $z' = N(\mathfrak{a})^{-1}z$ and $\mathfrak{c} = \mathfrak{a}\bar{\mathfrak{b}}$. Now notice how the first summation just corresponds to the conjugation of the theta kernel $\Theta_{k+m,\mathfrak{c},h}^0(\tau, z')$ and the second summation can be rewritten as $\frac{\partial^m}{\partial w^m} \vartheta_{\delta_{\mathfrak{p}^r\mathfrak{b}}^{(m)}}$, for $\vartheta_{\delta_{\mathfrak{p}^r\mathfrak{b}}^{(m)}}$ the theta function defined in the previous section, 4.1. By results of Hecke, see [P82, p.237], $\vartheta_{\delta_{\mathfrak{p}^r\mathfrak{b}}^{(m)}}$ is a modular form of weight $n+1$ and character $\omega_{K/\mathbb{Q}}$ for the group $\Gamma_0(D)$. After some computations which are carried exactly in the same way in [F99, p. 19-20], we can now rewrite

$$ \phi_{\mathfrak{a},\mathfrak{b},h}^{(m)}(z) = N(\mathfrak{a})^{-k-m} N(\mathfrak{b})^{-m} \frac{(2\pi\sqrt{D})^m}{m!} \tilde{\mathcal{L}}_{k+m,\mathfrak{c},h}^0 \left( f \vartheta_{\delta_{\mathfrak{p}^r\mathfrak{b}}^{(m)}}^{(m)} \right) (z'). $$

Summing over all representatives $\mathfrak{b}$ and $h$, we have computed the $m$-th Taylor coefficient of $g_{n/N(\mathfrak{a}),\mathfrak{a}}$ at $w=0$. The result easily follows by putting together the Taylor expansion $F_{\mathfrak{a},\mathfrak{b},h}$ of $\vartheta_{\delta_{\mathfrak{p}^r\mathfrak{b}}^{(m)}}$:

$$ \sum_{m \ge 0} \phi_{\mathfrak{a}, \mathfrak{b}, h}^{(m)} w^m = \sum_{m \ge 0} N(\mathfrak{a})^{-k-m} N(\mathfrak{b})^{-m} \frac{(2\pi\sqrt{D})^m}{m!} \tilde{\mathcal{L}}_{k+m, \mathfrak{c}, h}^0 \left( f \vartheta_{\delta_{\mathfrak{p}^r \mathfrak{b}}^{(m)}}^{(m)} w^m \right) (z'). $$

<figure><img src="image_5.png" /></figure>

**Corollary 11.** Let $k, \varepsilon$ and $\chi$ be as before. There exists a complex period $\Omega_0$ such that $\mathcal{L}_{k,\varepsilon,\chi}^{p,\text{ar}} = \Omega_0^{-k} \mathcal{L}_{k,\varepsilon,\chi}^p$ is arithmetic, i.e. for $f \in S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}}, \mathbb{Z}[1/D])$, $\mathcal{L}^{p,\text{ar}}(f) \in \mathcal{S}(k, \Gamma_1(p^r D), \chi, \mathbb{Z}[1/D])$.

*Proof.* Finis's arguments, see [F99, p. 36], still apply to our simpler formulas, as the theta functions $\vartheta$ are the same in both cases. $\square$

<figure><img src="image_6.png" /></figure>

We now show that, much as in the level 1 case, the $p$-modified lift is Hecke equivariant, and in fact that a completely analogous statement to Proposition 2 can be obtained. For this we will adapt Kudla's proof given in [K81, p. 329], which draws from Eichler's classical technique [Eic74, 2, Satz 21.3] to relate the actions of Hecke operators and generalized Brandt matrices on spaces of theta functions.

**Theorem 12.** If $f \in S_{k-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}})$ is a Hecke eigenform with eigenvalues $a_p$ and the lifting $\mathcal{L}_{k,\varepsilon,\chi}^p(f)$ is non-zero, then the latter is again a Hecke eigenform with eigenvalues:

$$ \lambda^p(\mathfrak{q}) = \begin{cases} p^2 \epsilon(\bar{\mathfrak{p}}) a_p, & \mathfrak{q} = \mathfrak{p}, \\ \lambda(\mathfrak{q}), & \text{if } q \neq p \text{ is unramified in } K, \end{cases} $$

where $\lambda(\mathfrak{q})$ are the values defined in section 2.
Proof. For $g \in G(\mathbb{Q})$, let $g^t = \mu(g)g^{-1}$. Then $g^t \in G(\mathbb{Q})$, $(gX, Y) = (X, g^tY)$ and $\mu(g^t) = \mu(g)$. Also, $\Gamma^1(p^rD)^t = \Gamma^1(p^rD)$ and, since $L'$ is maximal in the sense of [S64, p. 375], the set $\{g \in G(\mathbb{Q}) \mid gL' \subset L'\}$ is also stable under $\iota$.

Now let $S(\mathfrak{q}) = \bigcup_i \Gamma^1(p^rD)g_i$, and for $X \in L'$ set

$$m(\mathfrak{q}, X) = |\{i \mid X \in g_i^t L'\}|.$$

Note that the the sublattices $g_i^t L' \subset L'$ are independent of the choice of coset representatives, and that the multiplicities $m(\mathfrak{q}, X)$ only depend on the $\Gamma^1(p^rD)$ orbit of $X$, since $\Gamma^1(p^rD)$ only permutes the sublattices in question.

Given a fractional ideal $\mathfrak{b}$ of $K$, for convenience write

$$\theta_{\mathfrak{b}}(\tau, \mathfrak{z}) = y^2 \sum_{\substack{X \in L' \\ X(L) \in T}} f_{\tau, \mathfrak{z}}^{\mathfrak{b}}(X),$$

where $\theta_{\mathfrak{b}} = \sum_{h \in T} \theta_{\mathfrak{b},h}$, and the $\theta_{\mathfrak{b},h}$'s are the theta kernels defined in the beginning of the section. As the computations in the rest of the proof are independent of the fixed fractional ideal $\mathfrak{b}$, we drop the notation concerning this choice, which amounts to restricting to the case $\mathrm{Cl}_K = 1$. Indeed, all the arguments are of a local nature.

Applying $T(\mathfrak{q})$ to $\overline{\theta(\tau, \mathfrak{z})}$ yields:

$$ \begin{align*} T(\mathfrak{q})(\bar{\theta}) &= \mathrm{N}(\mathfrak{q})^{k-1} \sum_i j(g_i, \mathfrak{z})^{-k} y^2 \sum_{\substack{X \in L' \\ X(L) \in T}} \overline{f_{\tau, g_i(\mathfrak{z})}(X)} \\ & \quad + \mathrm{N}(\mathfrak{q})^{k-1} \sum_i \mu(g_i)^{-k} y^2 \sum_{\substack{X \in L' \\ X(L) \in T}} \overline{f_{\mu(g_i)^{-1}\tau, \mathfrak{z}}(g_i^t X)} \\ & \quad - p^{-1} y^2 \sum_{\substack{X \in L' \\ X(L) \in T}} m(\mathfrak{q}, X) \overline{f_{\mathrm{N}(\mathfrak{q})^{-1}\tau, \mathfrak{z}}(X)}, \end{align*} $$

where we use the identity

$$\overline{g(\mathfrak{z})\eta(g(\mathfrak{g})))^{-1} = \mu(g)^{-1}g\mathfrak{z}\eta(\mathfrak{z})^{-1}j(g, \mathfrak{z})$$

and the fact that $\mu(g_i) = \mathrm{N}(\mathfrak{q})$ for all $i$.

Almost all the rest of the proof is devoted to compute the multiplicities $m(\mathfrak{q}, X)$. For each rational prime $q$, let $K_q = K \otimes_{\mathbb{Q}} \mathbb{Q}_q$, $\mathcal{O}_q = \mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{Z}_q$, $V_q = K_q^3$ and $L'_q = \mathcal{O}_q + \pi^r \delta \mathcal{O}_q + \mathcal{O}_q$. Also let $G_q = G(\mathbb{Q}_q)$ and let

$$\Gamma_q = \{\gamma \in G_q \mid \gamma L'_q = L'_q, \gamma T_q = T_q\}.$$

Then for a prime $\mathfrak{q}$ of $K$ above $q$, we have

$$S_{\mathfrak{q}} = \bigcup_{i} \Gamma_{\mathfrak{q}} g_i$$

so that $S_{\mathfrak{q}} \subset G_q$, $\Gamma_{\mathfrak{q}} S_{\mathfrak{q}} \Gamma_{\mathfrak{q}} = S_{\mathfrak{q}}$ and in fact the above union is still disjoint. For $\ell \neq q$,

$$(g_i^t L)'_\ell = L'_\ell,$$

so that

$$X \in g_i^t L' \Leftrightarrow X \in g_i^t L'_q$$

and

$$m(\mathfrak{q}, X) = m(q, X) = |\{i \mid X \in g_i^t L'_q\}|,$$
where the $g_i^t$ can be any set of coset representatives for $\Gamma_q$ in $S_q$. From this, it immediately follows that if $q$ is coprime to $pD$, then Kudla's computations, [K81, p. §2] apply in the exact same way. Hence we have proved the second part of the theorem, and we can restrict to $q = p$. First of all note that $m(p, X)$ only depends on $X \bmod pL'_p$, cf. [K81, p. 330]. We need the following technical result to conclude:

$$m(p, X) = \begin{cases} p & \text{if } p \mid (X, X) \\ 0 & \text{otherwise.} \end{cases}$$

We postpone its proof to the end of the section. By the above we have,

$$T(\mathfrak{p})(\bar{\theta}) = y^2 \sum_{\substack{X(L_p) \in T \\ p|(X,X)}} \overline{f_{p^{-1}\tau, \mathfrak{z}}(X)}.$$

On the other hand,

$$\overline{T(p)(\bar{\theta})} = p^{-1} \sum_{j=0}^{p-1} \overline{\theta(p^{-1}(\tau + j), \mathfrak{z})} = p^{-2} y^2 \sum_{\substack{X(L_p) \in T \\ p|(X,X)}} \overline{f_{p^{-1}\tau, \mathfrak{z}}(X)},$$

and the result follows easily by the usual adjointness properties of the Hecke operators with respect to the Petersson inner product, cf. [K81, p. 327]; note that the dependence on $\epsilon$ of the final formula is analogous to the result in Finis [F99, p. 11] and is due to the presence of the factor $\epsilon(\det g)$ in the definition of $\theta(\tau, g)$. $\square$

Now we only need to prove the technical lemma:

**Lemma 13.** If $X \in L'_p, X(L_p) \in T$, is such that $(X, X) \equiv 0 \bmod p$ then $m(p, X) = p$, otherwise $m(p, X) = 0$.

*Proof.* It is clear that the function $m(p, X)$ is actually defined on $L'_p/pL'_p$, and is constant on $\Gamma_p$-orbits in this set. Therefore to compute $m(p, X)$ it will be sufficient to find a set of $\Gamma_p$-orbit representatives and a set of coset representatives $g_j$; finally, for each such orbit representative we will count the number of coset representatives which contain the former in their image.

Recall that $p$ splits in $K$, and so take idempotents $e$ and $e^\sigma$ in $K_p$ corresponding to the two primes dividing $p$. Then

$$V_p = eV_p + e^\sigma V_p \cong \mathbb{Q}_p^3 \times \mathbb{Q}_p^3$$

and

$$L'_p = eL'_p + e^\sigma L'_p \cong (\mathbb{Z}_p + p\mathbb{Z}_p + \mathbb{Z}_p) \times \mathbb{Z}_p^3.$$

Set $J_1 = eJ, J_2 = e^\sigma J$. Then

$$G_p = G(\mathbb{Q}_p) \cong \{(g_1, g_2) \in \mathrm{GL}_3(\mathbb{Q}_p) \times \mathrm{GL}_3(\mathbb{Q}_p) \mid {}^t g_1 J_2 g_2 = c J_2, c \in \mathbb{Q}_p^\times\} \\ \cong \mathrm{GL}_3(\mathbb{Q}_p) \times \mathbb{Q}_p^\times,$$

since $g_2 = cg_1^*$ with $g_1^* = J_2^{-1} {}^t g_1^{-1} J_2$. The action of $[g, c] \in \mathrm{GL}_3(\mathbb{Q}_p) \times \mathbb{Q}_p^\times$ on $[X, Y] \in \mathbb{Q}_p^3 \times \mathbb{Q}_p^3$ is given by

$$[g, c][X, Y] = [gX, cg^*Y].$$

The above action is equivalent to the standard action

$$[g, c]'[X, Y] = [gX, c^t g^{-1}Y]$$
under the isomorphism $1 \times J_2 : \mathbb{Q}_p^3 \times \mathbb{Q}_p^3 \to \mathbb{Q}_p^3 \times \mathbb{Q}_p^3$. Given a set of representatives $\tilde{g}_i$ for $S_{\bar{p}}/\Gamma_1(p^rD)_p$, since $S_{\bar{p}} = S_p^t$, we know that to get our $g_i^t$ we just need to conjugate the $\tilde{g}_i$ by the diagonal matrix

$$ \gamma_{p^r} = \begin{pmatrix} p^r & & \\ & p^r & \\ & & 1 \end{pmatrix} \times \begin{pmatrix} p^r & & \\ & 1 & \\ & & 1 \end{pmatrix} \in \mathrm{GL}_3(\mathbb{Q}_p) \times \mathrm{GL}_3(\mathbb{Q}_p). $$

By [K81, p. 333], the images of these $g_i^t$'s in $\mathrm{GL}_3(\mathbb{Q}_p) \times \mathrm{GL}_3(\mathbb{Q}_p)$ under the isomorphism corresponding to the above standard action will be, cf. [K81, p. 334],

$$ \left[ \begin{pmatrix} p & \mu b & p^r a \\ & 1 & \\ & & 1 \end{pmatrix}, \begin{pmatrix} 1 & & \\ -\mu b & p & \\ -p^r a & & p \end{pmatrix} \right], $$

where $a$ and $b$ run over a set of representatives for $\mathbb{Z}_p/p\mathbb{Z}_p$ and the image of $\delta$ in $\mathbb{Q}_p \times \mathbb{Q}_p$ is $[\mu, -\mu]$.

The reduction modulo $p$ of these elements gives representatives in $M_3(\mathbb{F}_p) \times M_3(\mathbb{F}_p)$:

$$ \left[ \begin{pmatrix} 0 & \mu b & \\ & 1 & \\ & & 1 \end{pmatrix}, \begin{pmatrix} 1 & & \\ -\mu b & 0 & \\ & & 0 \end{pmatrix} \right]. $$

As we can see, the spaces spanned by the column of the first entry in the representatives are always orthogonal to the span of the columns of the second entry, and thus we see that the images of the $g_i^t$ clearly only contain vectors whose norm is divisible by $p$, so that $m(p, X) = 0$ if $p \nmid (X, X)$.

Now suppose that $p \mid (X, X)$. Note that the above reductions are independent of $a$, and so $m(p, X)$ will be divisible by $p$. We can also easily see that every vector $X \in L'_p, X(L_p) \in T$ such that $(X, X) \equiv 0 \mod p$ has its first coordinate divisible by $p$, and this is sufficient to establish that it will be in the image of exactly $p$ coset representatives, since for each such vector we have to fix a value of $b$. $\square$

## 5. THE $p$-ADIC INTERPOLATION OF THE KUDLA LIFT

In this section, we write $\mathcal{L}$ for $\mathcal{L}^{p,\mathrm{ar}}$, and use the same convention for the Fourier-Jacobi coefficients, to simplify the notation. In the following, we show that the lift defined in section 4 induces a lift of $\Lambda$-adic elliptic modular forms to $\Lambda$-adic Picard modular forms. First of all, we recall the notions of $p$-adic and $\Lambda$-adic modular forms both in the elliptic and in the Picard case, following [dSG16, p. 60] for the latter.

A formal $q$-expansion $f \in \bar{\mathbb{Z}}_p[[q]]$ is a $p$-adic modular form (resp. cusp form) of tame level $\Gamma_1(D)$, if it is the $p$-adic limit of classical modular forms $f_i$ of tame level $\Gamma_1(D)$. As is well known, the weights $k_i$ of the modular forms $f_i$ also converge to a character $\kappa \in \mathrm{Hom}_{\mathrm{cont}}(\mathbb{Z}_p^\times, \mathbb{Z}_p^\times)$. We will write $f \in M_{\kappa-1}^{\mathrm{Serre}}(\Gamma_1(D), \bar{\mathbb{Z}}_p)$ (resp. $S_{\kappa-1}^{\mathrm{Serre}}(\Gamma_1(D), \bar{\mathbb{Z}}_p)$), to underline that it is a $p$-adic modular form (resp. cusp form) in the sense of Serre.
5.1. **On Λ-adic elliptic cusp forms.** We start by reviewing Hida's construction of elliptic cuspidal Λ-adic modular forms of tame level $D$. Let $\Gamma = 1+p\mathbb{Z}_p$ be the maximal torsion-free subgroup of $\mathbb{Z}_p^\times$. We fix a topological generator $\gamma \in \Gamma$ such that $\Gamma = \langle\gamma\rangle$. Let $\Lambda = \mathbb{Z}_p[[\Gamma]]$ and $\Lambda_1 = \mathbb{Z}_p[[\mathbb{Z}_p^\times]]$ be the completed group rings on $\Gamma$ and on $\mathbb{Z}_p^\times$ over $\mathbb{Z}_p$, respectively. Clearly $\Lambda_1$ has a natural Λ-algebra structure induced from $\Lambda_1 \cong \Lambda[\mu_{p-1}]$, where $\mu_{p-1}$ denotes the maximal torsion subgroup of $\mathbb{Z}_p^\times$ consisting of $(p-1)$-th roots of unity.

**Definition 4 (Λ-adic weight spaces).** For each $\Lambda_1$-algebra $R$ finite flat over $\Lambda$, we define the Λ-adic weight space $\mathcal{X}(R)$ associated with $R$ as

$$ \mathcal{X}(R) := \operatorname{Hom}_{\operatorname{cont}}(R, \overline{\mathbb{Q}}_p), $$

on which the following arithmetic notions are introduced:

(i) A point $P \in \mathcal{X}(R)$ is said to be arithmetic if there exists an integer $k \ge 2$ such that the restriction of $P$ to $\mathcal{X}(\Lambda) \cong \operatorname{Hom}_{\operatorname{cont}}(\Gamma, \overline{\mathbb{Q}}_p^\times)$ corresponds to a continuous character $P_k : \Gamma \to \overline{\mathbb{Q}}_p^\times$ satisfying $P_k(\gamma) = \gamma^k$. We denote by $\mathcal{X}_{\operatorname{alg}}(R)$ the set of all arithmetic points in $\mathcal{X}(R)$.

(ii) An arithmetic point $P \in \mathcal{X}_{\operatorname{alg}}(R)$ is said to have signature $(k, \epsilon)$ if there exists an integer $k \ge 2$ and a finite character $\epsilon : \mathbb{Z}_p^\times \to \overline{\mathbb{Q}}_p^\times$ such that $P$ lies over the point $P_{k,\epsilon} \in \mathcal{X}_{\operatorname{alg}}(\Lambda_1) \cong \operatorname{Hom}_{\operatorname{cont}}(\mathbb{Z}_p^\times, \overline{\mathbb{Q}}_p^\times)$ corresponding to the character $P_{k,\epsilon}(y) = y^k \epsilon(y)$ on $\mathbb{Z}_p^\times$. For simplicity, we just write $P = (k, \epsilon)$ and often refer to it as the arithmetic point of weight $k$ and Nebentypus $\epsilon\omega^{-k}$, where $\omega$ denotes the Teichmüller character.

We note that $\mathcal{X}(\Lambda)$ has a natural analytic structure induced from the identification $\operatorname{Hom}_{\operatorname{cont}}(\Lambda, \overline{\mathbb{Q}}_p) \cong \operatorname{Hom}_{\operatorname{cont}}(\Gamma, \overline{\mathbb{Q}}_p^\times)$. Moreover, for a given $R$ the natural restriction to $\Lambda$ gives a finite-to-one mapping

$$ \pi : \mathcal{X}(R) \to \mathcal{X}(\Lambda_1) \to \mathcal{X}(\Lambda), $$

which allows us to define some analytic charts around all points of $\mathcal{X}_{\operatorname{alg}}(R)$. Indeed, it is established by Hida that each $P \in \mathcal{X}_{\operatorname{alg}}(R)$ is unramified over $\mathcal{X}(\Lambda)$, and consequently there exists a natural section of $\pi$

$$ s_P : U_P \subseteq \mathcal{X}(\Lambda) \to \mathcal{X}(R) $$

defined on a neighborhood $U_P$ of $\pi(P)$. These local sections endow $\mathcal{X}(R)$ with analytic charts around points in $\mathcal{X}_{\operatorname{alg}}(R)$. For each $P \in \mathcal{X}_{\operatorname{alg}}(R)$, a function $f : \mathfrak{U} \to \overline{\mathbb{Q}}_p$ defined on $\mathfrak{U} = s_P(U_P) \subset \mathcal{X}(R)$ is said to be analytic if $f \circ s_P : U_P \to \overline{\mathbb{Q}}_p$ is analytic. In parallel, an open subset $\mathfrak{U} \subset \mathcal{X}(R)$ containing some $P \in \mathcal{X}_{\operatorname{alg}}(R)$ is called an analytic neighborhood of $P$ if $\mathfrak{U} = s_P(U_P)$. For instance, we easily see that each element $\mathbf{a} \in R$ gives rise to an analytic function $\mathbf{a} : \mathcal{X}_{\operatorname{alg}}(R) \to \overline{\mathbb{Q}}_p$ defined by $\mathbf{a}(P) = P(\mathbf{a})$. In general, if $P \in \mathcal{X}(R)$ unramified over $\mathcal{X}(\Lambda)$, then each element $\mathbf{a} \in R_{(P)}$ gives rise to an analytic function defined on some analytic neighborhood of $P$, where $R_{(P)}$ denotes the localization of $R$ at $P$, and it gives rise to a discrete valuation ring finite and unramified over $\Lambda$. According to the standard usage, we refer to the evaluation $\mathbf{a}(P)$ at $P \in \mathcal{X}_{\operatorname{alg}}(R)$ as the specialization of $\mathbf{a}$ at $P$.

For each integer $r \ge 1$, let

$$ X_1(p^r D) = \Gamma_1(p^r D) \setminus (\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q})) $$
be the compactified modular curve, and $V_r = H^1(X_1(p^rD), \mathbb{Z}_p)$ the singular cohomology group of $X_1(p^rD)$ with values in $\mathbb{Z}_p$. It is well-known that $V_r$ is canonically isomorphic to the parabolic cohomology group $H_{\text{par}}^1(\Gamma_1(p^rD), \mathbb{Z}_p) \subset H^1(\Gamma_1(p^rD), \mathbb{Z}_p)$, which is defined to be the image of the compactly supported cohomology group under the natural map, cf. [H93, p. 358]. We denote the abstract $\Lambda$-adic Hecke algebra of tame level $D$ by the free polynomial algebra

$$ \mathbb{T} := \Lambda_1[T_n \mid 1 \le n \in \mathbb{Z}] $$

generated by the Hecke operators $T_n$ over $\Lambda_1$. Since $\mathbb{T} \cong \mathbb{Z}_p[T_n, \mathbb{Z}_p^\times]$, a natural action of $\mathbb{T}$ on $V_r$ is defined by regarding the generator $T_n$ act via the $n$-th Hecke correspondence and elements of $\mathbb{Z}_p^\times$ act via the usual action of the diamond operators. For each pair of positive integer $r_1 \ge r_2$ we get a corestriction map $V_{r_1} \to V_{r_2}$, induced by $\Gamma_1(p^{r_1}) \hookrightarrow \Gamma_1(p^{r_2})$, which commutes with the action of $\mathbb{T}$. Hence the projective limit

$$ V_{\infty} := \lim_{r \ge 1} V_r $$

is naturally endowed with a $\mathbb{T}$-algebra structure. We denote by $V_\infty^{\text{ord}}$ the direct factor of $V_\infty$ cut out by Hida's idempotent $e_{\text{ord}} = \lim_{m \to \infty} U_p^{m!}$, that is,

$$ V_{\infty}^{\text{ord}} = e_{\text{ord}} \cdot V_{\infty}, $$

on which $U_p$ acts invertibly. We note that $V_\infty^{\text{ord}}$ is a $\Lambda$-algebra free of finite rank [H93, p. 210]. Moreover, for $\mathcal{L} = \text{Frac}(\Lambda)$, Hida constructed an idempotent $e_{\text{prim}}$ in the image of $\mathbb{T} \otimes_\Lambda \mathcal{L}$ in $\text{End}_\mathcal{L}(V_\infty^{\text{ord}} \otimes_\Lambda \mathcal{L})$, which can be regarded as an analogue of the projection to the space of primitive Hecke eigenforms in Atkin-Lehner theory, [H96, p. 250-252]. Then we define the universal ordinary parabolic cohomology group of tame level $D$ by the $\mathbb{T}$-algebra

$$ \mathbb{V}^{\text{ord}} := V_{\infty}^{\text{ord}} \cap e_{\text{prim}}(V_{\infty}^{\text{ord}} \otimes_{\Lambda} \mathcal{L}), $$

which is a reflexive $\Lambda$-algebra of finite rank and is consequently locally free of finite rank over $\Lambda$. Then the universal $p$-ordinary Hecke algebra of tame level $D$ is defined to be the image $\mathbb{R}^{\text{ord}}$ of the homomorphism

$$ h : \mathbb{T} \to \text{End}_{\Lambda_1}(\mathbb{V}^{\text{ord}}). $$

Each morphism $\phi : \mathbb{T}^{\text{ord}} \to \Lambda_1$ is naturally equipped with a formal $q$-expansion

$$ \mathbf{f}_{\phi}^{\text{ord}} = \sum_{n \ge 1} \mathbf{a}_n q^n \in \mathbb{R}^{\text{ord}}[[q]], \quad \mathbf{a}_n = \phi(T_n), $$

which is called a universal $p$-stabilized ordinary form of tame level $D$.

**Definition 5** (Ordinary $p$-stabilized newforms). For given integers $k \ge 2$ and $r \ge 1$, a Hecke eigenform $f_k^* \in S_k(\Gamma_1(p^rD), \omega_{K/\mathbb{Q}})$ is called an ordinary $p$-stabilized newform if one of the following conditions holds true:

(i) $f_k^*$ is a $p$-ordinary Hecke eigenform in $S_k^{\text{new}}(\Gamma_1(p^rD), \omega_{K/\mathbb{Q}})$, where the latter is the subspace of $S_k(\Gamma_1(p^rD), \omega_{K/\mathbb{Q}})$ consisting of newforms.

(ii) If $r = 1$, then there exists a normalized ordinary Hecke eigenform $f_k \in S_k(\Gamma_1(D), \omega_{K/\mathbb{Q}})$ such that

$$ f_k^*(z) = f_k(z) - \beta_p(f_k)f_k(pz), $$

where $\beta_p(f_k)$ is the non-unit root of the polynomial $X^2 - a_p(f_k)X + p^{k-1}$.
Remark 3. It follows from the definition that the ordinary *p*-stabilized new-forms are literally *p*-ordinary Hecke eigenforms. Indeed, this assertion is trivial in the case of (i). If $f_k^* \in S_k(\Gamma_1(pD), \omega_{K/\mathbb{Q}})$ is taken as in (ii), then for each prime $\ell$, we have

$$a_{\ell}(f_k^*) = \begin{cases} a_{\ell}(f_k) & \text{if } \ell \neq p, \\ \alpha_p(f_k) & \text{if } \ell = p, \end{cases}$$

where $\alpha_p(f_k)$ is the unit root of the polynomial $X^2 - a_p(f_k)X + p^{k-1} = (X - \alpha_p(f_k))(X - \beta_p(f_k))$. For a given *p*-ordinary Hecke eigenform in $S_k(\Gamma_1(D), \omega_{K/\mathbb{Q}})$, this type of *p*-adic normalization process is called the ordinary *p*-stabilization. However, note that each ordinary *p*-stabilized newform $f_k^* \in S_k(\Gamma_1(D), \omega_{K/\mathbb{Q}})$ is actually an oldform except for $k=2$. Indeed, if $f \in f_k^* \in S_k^{\text{new}}(\Gamma_1(pD))$, then we have

$$|a_p(f_k^*)| = p^{k/2-1}.$$

The following theorem has been established by Hida:

**Theorem 14** (cf. Theorem 2.6 in [GS93]). Let $r$ be a fixed positive integer and $f^{\text{ord}} = \sum_{n \ge 1} a_n q^n \in \Lambda_1[[q]]$ be a universal ordinary $p$-stabilized form of tame level $D$ and character $\omega_{K/\mathbb{Q}}$, in the sense of [H04, p. 91]. Then for each $P \in \mathcal{X}_{\text{alg}}(\Lambda_1)$, the specialization

$$f^{\text{ord}}(P) = \sum_{n \ge 1} a_n(P)q^n \in \overline{\mathbb{Q}}_p[[q]]$$

induces a one-to-one correspondence

$$\{P = (k, \epsilon) \in \mathcal{X}_{\text{alg}}(\Lambda_1) \mid 2 \le k \in \mathbb{Z}, \epsilon : \mathbb{Z}_p^\times \to \overline{\mathbb{Q}}_p^\times \text{ character of order } p^r\} \\ \longleftrightarrow \{f_k^* \in S_k(\Gamma_1(p^r D), \epsilon \omega^{-k} \omega_{K/\mathbb{Q}})\} \mid \text{ordinary } p\text{-stabilized newform.}$$

By applying the above theorem for $r=1$, each $P = (2k, \omega^{2k}) \in \mathcal{X}_{\text{alg}}(\Lambda_1)$ corresponds to an ordinary $p$-stabilized newform $f_{\text{ord}}(P) = f_{2k}^* \in S_{2k}(\Gamma_0(pD), \omega_{K/\mathbb{Q}})$ associated with a $p$-ordinary normalized Hecke eigenform $f_{2k} \in S_{2k}(\Gamma_1(D), \omega_{K/\mathbb{Q}})$ via the ordinary $p$-stabilization. Therefore, for a fixed $P_0(2k_0, \omega^{2k_0}) \in \mathcal{X}_{\text{alg}}(\Lambda_1)$ with $k_0 \gg 0$, we may regard $f_{\text{ord}} \in \Lambda_1[[q]]$ as a $\Lambda$-adic elliptic cusp form, and we consequently obtain a $p$-adic analytic family of ordinary $p$-stabilized newforms $\{f_{\text{ord}}(P) = f_{2k}^*\}$ parametrized by $P = (2k, \omega^{2k}) \in \mathcal{X}_{\text{alg}}(\Lambda_1)$ with $k_0 \le k \in \mathbb{Z}$.

**5.2. On $\Lambda$-adic Picard modular forms.** In the following, write $K_p = K \otimes \mathbb{Z}_p$ and $\mathcal{O}_p = \mathcal{O}_K \otimes \mathbb{Z}_p$.

**Definition 6.** [dSG16, p. 60] Let

$$\mathcal{X}_p = \lim_{\leftarrow \mathbb{Z}/(p^2 - 1)p^m\mathbb{Z}}$$

be the space of $p$-adic weights.

Recall the embedding $\bar{\sigma}: K \hookrightarrow \mathbb{C}$. If $\kappa \in \mathcal{X}_p$, then $\bar{\sigma}^{\kappa}$ is a well-defined locally $\mathbb{Q}_p$-analytic homomorphism of $\mathcal{O}_p^\times$ to itself, but note that not every such homomorphism is a $\bar{\sigma}^{\kappa}$ for some $\kappa \in \mathcal{X}_p$. For $\kappa \in \mathcal{X}_p$, a $p$-adic Picard modular form à la Serre is defined as a formal Fourier-Jacobi expansion which is the limit of $q$-expansions of classical Picard modular forms of integral weight, cf.
[dSG16, p. 60]. For a tame level $N$, we denote the space of $p$-adic Picard modular forms of level $N$ and weight $\kappa \in \mathfrak{X}_p$ as $\mathcal{M}^{\text{Serre}}(\kappa, N, K_p)$. There is another possible construction of $p$-adic modular forms, which is more geometric, à la Katz. Here is an overview of the construction: first of all, recall that as $p$ is split in $K$, the ordinary locus of the Picard modular surface is open and dense [H04, p. 352]. Let

$$V_r^{(m)} = H^0(\overline{\text{Ig}}(p^r)^{(m)}, \mathcal{O}_{\overline{\text{Ig}}(p^r)^{(m)}})$$

be the ring of regular functions on the truncated Igusa scheme of level $p^r$ over the completion of the ordinary locus, $\overline{\text{Ig}}(p^r)^{(m)}$. Here $\mathcal{O}_{\overline{\text{Ig}}(p^r)^{(m)}}$ is the sheaf of regular functions on $\overline{\text{Ig}}(p^r)$. The latter object's definition is detailed in [dSG16, p. 57], but let us just recall that this is the étale cover of the truncation modulo $p^m$ of the ordinary locus which classifies ordinary points on the Picard scheme, together with a trivialization of their group of $p^r$-torsion points. Now we also let

$$V^{(m)} = \lim_{\underset{r}{\rightarrow}} V_r^{(m)}, \quad V = \lim_{\underset{m}{\leftarrow}} V^{(m)}.$$

We call $V$ the space of Katz $p$-adic modular forms of all weights. Recall that on $\overline{\text{Ig}}(p^r)^{(m)}$, we have an action of $\Delta(p^r) = (\mathcal{O}_K/p^r\mathcal{O}_K)^\times$ via

$$\gamma(\underline{A}, \epsilon_r^{(m)}) = (\underline{A}, \epsilon_r^{(m)} \circ \gamma^{-1}).$$

Consequently we get actions of $\Delta(p^r)$ and $\Delta := \lim_{\leftarrow} \Delta(p^r)$ on $V^{(m)}$ and on $V$, respectively, defined by $\gamma(f) = f \circ \gamma^{-1}$. Let $\kappa \in \mathfrak{X}_p$ and define, for a fixed tame level $N$,

$$\mathcal{M}^{\text{Katz}}(\kappa, N, \mathcal{O}_p) = V(\bar{\sigma}^{\kappa}) = \{f \in V \mid \gamma(f) = \bar{\sigma}^{\kappa}(\gamma) \cdot f \ \forall \gamma \in \Delta\}.$$

**Theorem 15.** For $\kappa \in \mathfrak{X}_p$, there is a natural isomorphism

$$\mathcal{M}^{\text{Serre}}(\kappa, N, \mathcal{O}_p) \cong \mathcal{M}^{\text{Katz}}(\kappa, N, \mathcal{O}_p).$$

*Proof.* See [dSG16, p. 62]. □

Analogously, a $p$-adic cusp form is defined as a limit of classical Picard cusp forms, and Theorem 15 restricts to cuspidal forms. Later, we will usually drop $\mathcal{O}_p$ from the notation, and we will consider, for $\chi_\kappa$ a $p$-adic Hecke character of weight $\kappa$, the spaces

$$\mathcal{S}(\kappa, N, \chi_\kappa) \subset \mathcal{M}(\kappa, N, \chi_\kappa)$$

of $p$-adic Picard modular forms of central character $\chi_\kappa$, meaning that the limit is taken on Picard modular forms of central characters $\chi_{k_i}$ converging to $\chi_\kappa$ $p$-adically.

We introduce the notion of $\Lambda$-adic Picard modular forms from the point of view of Fourier expansions. Recall that in section 3.2 we introduced the notions of arithmetic theta functions, following Shimura.

**Definition 7** ($\Lambda$-adic theta function). For $R$ a $\Lambda_1$-algebra, finite flat over $\Lambda$, define the space of $\Lambda$-adic theta functions $\mathcal{T}_d(R)$ of degree $d$ simply as

$$\mathcal{T}_d^{\text{ar}} \otimes_{\overline{\mathbb{Q}}_p} R.$$

Furthermore, write

$$\mathcal{T}(R) = \mathcal{T}^{\text{ar}} \otimes_{\overline{\mathbb{Q}}_p} R,$$
for the space of theta functions of any degree. If $\mathbf{g} \in \mathcal{T}_d(R)$ and $P \in \mathcal{X}(R)$ then define $\mathbf{g}(P) \in \mathcal{T}_d^{ar}$ by applying $P$ on the second component, that is, if $\mathbf{g} = \theta \otimes_{\overline{\mathbb{Q}}_p} \mathbf{a}$, then

$$ \mathbf{g}(P) = \theta \otimes_{\overline{\mathbb{Q}}_p} \mathbf{a}(P). $$

*Remark 4.* In greater generality, since we have defined our *p*-adic weight space as $\mathcal{X}_p = \mathbb{Z}/(p^2 - 1) \times \mathbb{Z}_p$, we could consider the bigger space $\mathcal{O}_{\mathcal{X}_p}$ instead of $\Lambda_1$, but for our purposes this will suffice. Indeed we have the natural inclusions

$$ \Lambda \hookrightarrow \Lambda_1 \hookrightarrow \mathcal{O}_{\mathcal{X}_p}. $$

**Definition 8** ($\Lambda$-adic Picard modular form). Let $R$ be a $\Lambda_1$-algebra, finite flat over $\Lambda$. Pick $P_0 = (k_0, \omega^{k_0}) \in \mathcal{X}_{alg}(R)$. A formal Fourier-Jacobi expansion

$$ \mathbf{F} = (\mathbf{F}_{\mathfrak{a}})_{\mathfrak{a} \in \mathcal{A}} = \left( \sum_{n \ge 0} \mathbf{g}_{n/N(\mathfrak{a}), \mathfrak{a}}(w) q^n \right)_{\mathfrak{a} \in \mathcal{A}} \in \mathcal{T}(R_{(P_0)})([q]) $$

is called a $\Lambda$-adic Picard modular form of tame level $D$ if there exists an analytic neighborhood $\mathfrak{U}_0$ of $P_0$ such that for each arithmetic point $P = (k, \epsilon\omega^k) \in \mathfrak{U}_0$ with $k \ge k_0$ and $\epsilon : \mathbb{Z}_p^\times \to \overline{\mathbb{Q}}_p^\times$ a finite character, the specialization

$$ \mathbf{F}(P) := \left( \sum_{n \ge 0} \mathbf{g}_{n/N(\mathfrak{a}), \mathfrak{a}}(w)(P)q^n \right)_{\mathfrak{a}} \in \overline{\mathbb{Q}}_p([q]) $$

gives rise to the Fourier expansion of a holomorphic Picard modular form in $\mathcal{M}(k, \Gamma_1(p^r D))$, for some $r$. In particular, a $\Lambda$-adic Picard modular form $\mathbf{F}$ is said to be cuspidal if $\mathbf{F} \in \mathcal{S}(k, \Gamma_1(p^r D))$ for almost all $P \in \mathfrak{U}_0$.

If there exists a $\Lambda$-adic Picard modular form $\mathbf{F} \in \mathcal{T}(R_{(P_0)})([q])$, then every coefficient $\mathbf{g}_{n/N(\mathfrak{a}), \mathfrak{a}}(w) \in \mathcal{T}_{n/N(\mathfrak{a})}(R_{(P_0)})$ of $\mathbf{F}$ gives rise to an analytic function $\mathfrak{U}_0 \to \mathcal{T}_{n/N(\mathfrak{a})}^{ar}$. Hence every specialization $\mathbf{F}(P)$ gives a holomorphic Picard modular form whose Fourier-Jacobi coefficients are $p$-adic analytic functions on $\mathfrak{U}_0$, cf. §5.1 for the analogous elliptic case.

**5.3. The p-adic lift.** For a $p$-adic elliptic modular form $f$, let $\mathcal{L}^*(f)$ be the formal series with Fourier-Jacobi coefficients

$$ g_{n/N(\mathfrak{a}), \mathfrak{a}}^*(w) = \kappa(\Omega_0)^{-1} \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \text{Cl}_K \\ (\mathfrak{b}, p)=1}} (\varepsilon^3 \chi_{\kappa}^{-1})(\mathfrak{b}) g_{n/N(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^*(w), $$

where

$$ g_{n/N(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^*(w) = \kappa(\mathcal{N}(\mathfrak{b}))^{1/2} T_n(f(\tau) \vartheta_{\delta p^r \mathfrak{b}}(n\delta w, \tau))|_{\tau=\tau_{\mathfrak{a}, \mathfrak{b}}}, $$

and $\kappa(\Omega_0)$ is the $p$-adic period associated to the $p$-adic weight $\kappa$ by [GS93, p. 436.]

**Proposition 6.** *The rule $f \mapsto \mathcal{L}^*(f)$ defines a morphism of Hecke modules*

$$ \mathcal{L}^* : S_{\kappa-1}^{\text{Serre}}(\Gamma_1(D), \omega_{K/\mathbb{Q}}, \overline{\mathbb{Z}}_p) \to \mathcal{S}^{\text{Serre}}(\kappa, \Gamma_1(D), \chi_{\kappa}). $$

We postpone the proof of this proposition to the subsection 5.4.
5.4. **The Λ-adic lift.** Now suppose that $f \in S_{k-1}(\Gamma_1(D), \omega_{K/\mathbb{Q}}, \bar{\mathbb{Z}}_p)$ is a normalized $T_p$-ordinary Hecke eigenform. By the discussion in the previous section, its ordinary $p$-stabilization $f^*$ is a Hecke eigenform with the same eigenvalues of $f$ at all primes except $p$, where $f^*$ has eigenvalue $\alpha_p$, the latter being the unique root of the polynomial $X^2 - a_pX + p^{k-2}$ which is a $p$-adic unit. Additionally, we know that $f^*$ is part of a Hida family $\mathfrak{F}$, which means that $P_{k-1} \circ \mathfrak{F} = f^*$. We want to define a Λ-adic lift of $\mathfrak{F}$, i.e. a formal Fourier-Jacobi expansion with coefficients in $\Lambda$, whose specializations at an arithmetic prime $P_k$ is the lift of the specialization of $\mathfrak{F}$ at $P_{k-1}$. Concretely, for each positive even integer $j$, define the Λ-adic Fourier-Jacobi expansion $\mathcal{L}_{\omega_j}(\mathfrak{F})$ by

$$ \mathfrak{g}_{n/\mathrm{N}(\mathfrak{a}), \mathfrak{a}}^{j}(w) = \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \mathrm{Cl}_K \\ (\mathfrak{b}, p)=1}} (\varepsilon^3 \Xi^{-1})(\mathfrak{b}) \mathfrak{g}_{n/\mathrm{N}(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^{j}(w), \quad (1) $$

where

$$ \mathfrak{g}_{n/\mathrm{N}(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^{j}(w) = \kappa(\langle \mathrm{N}(\mathfrak{b}) \rangle^{1/2}) \omega^{j/2}(\mathrm{N}(\mathfrak{b})) T_n(\mathfrak{F}(\tau) \star \vartheta_{\delta_{\overline{\mathfrak{p}}^\tau \mathfrak{b}}}(n\delta w, \tau))|_{\tau=\tau_{\mathfrak{a},\mathfrak{b}}}, $$

$T_n$ now are the universal Hecke operators [H93, p. 209], acting on the space of ordinary Λ-adic modular forms and $\star$ is the convolution introduced by Hida, cf. [H93, p. 200]. Recall also that by [H04, p. 93], it is possible to evaluate a given ordinary Λ-adic form at a CM-point: indeed, the underlying elliptic curve is ordinary, as $p$ splits in $K$.

At last, to interpolate the Hecke characters $\chi_k$, we adapt the strategy of Collins in [Col20, p. 23]:

**Lemma 16.** Let $k_0$ be a positive even integer and $\chi_{k_0} : \mathbb{I}_K/K^\times \to \mathbb{Z}_{p^e}^\times$ an algebraic Hecke character for $K$ of weight $k_0$, where $\mathbb{Z}_{p^e}$ is the ring of integers of a finite extension $F/\mathbb{Q}_p$. Then there exists a family of characters $\Xi$ with values in $\mathcal{O}_F[[X]]$ such that if we set $\xi_k = P_k \circ \Xi$ for a positive even integer, we have:

(i) For every $k$, $\xi_k$ is a character of weight $k$.

(ii) $\xi_{k_0} = \chi_{k_0}$.

(iii) For $k \equiv k_0 \bmod 2(p-1)$, $\xi_k$ has the same conductor and same finite type as $\xi_{k_0} = \chi_{k_0}$; for $k \not\equiv k_0 \bmod 2(p-1)$ the finite types differ by powers of the Teichmüller characters $\omega_p$ and $\omega_{\bar{p}}$.

The proof of this lemma will follow once we have defined a certain $p$-adic Hecke character, and the family passing through it. Let $\alpha : \mathbb{I}_K/K^\times \to \bar{\mathbb{Q}}_p^\times$ be the algebraic Hecke character such that, for $x = (x_v) \in \prod \mathcal{O}_{K_v}^\times$, $\alpha(x) = x_p^{-1}\omega_p(x_p)$, where $\omega_p : (\mathcal{O}_K/\mathfrak{p})^\times \to \mathbb{Z}_p^\times$ is the canonical Teichmüller character for $\mathcal{O}_K$. The character $\alpha$ has infinity type (1,0), conductor $\mathfrak{p}$ and finite type $\omega_p^{-1}$ (this is a consequence of the convention adopted), see [Col20, p. 24] for details.

Remark that if we use $\bar{\mathfrak{p}}$ instead of $\mathfrak{p}$, we get a character of infinity type $(0,1)$, conductor $\bar{\mathfrak{p}}$ and finite type $\omega_{\bar{p}}^{-1}$. Indeed, this is simply $\bar{\alpha} := \alpha \circ \tilde{c}$, where $\tilde{c} : \mathbb{I}_K/K^\times \to \mathbb{I}_K/K^\times$ is the automorphism on ideles induced by the non-trivial automorphism $c$ of $K$. Clearly $\bar{\alpha}(\mathfrak{a}) = \alpha(\bar{\mathfrak{a}})$.

We may now use the character $\alpha$ to define a family of characters. In general, if we are given a character $\mathbb{I}_K/K^\times \to \Gamma$ for $\Gamma$ the $p$-profinite group of units
that are congruent to 1 modulo the uniformizer, i.e. principal units, in a finite extension $L$ of $\mathbb{Q}_p$, we can just define a character $\mathbb{I}_K/K^\times \to \mathcal{O}_L[[\Gamma]]^\times$ by composing with the tautological embedding $\Gamma \hookrightarrow \mathcal{O}_L[[\Gamma]]^\times$. Now recall that we have $\alpha : \mathbb{I}_K/K^\times \to (1+\pi)^{\mathbb{Z}_p} \subset \mathcal{O}_F^\times$ for some finite extension $F/\mathbb{Q}_p$, with $(1+\pi)^{p^e} = (1+p)$ for some positive integer $e$, and hence take $\mathcal{I} = \mathcal{O}_F[[\Gamma']]$ for $\Gamma' = (1+\pi)^{\mathbb{Z}_p}$. As opposed to what Collins does, we will actually work with a square root of the character defined by $\alpha$. To this purpose, choose the unique square root of $\alpha$ that still takes values in $\Gamma'$, and compose this with the tautological embedding. We thus have defined the character $\mathcal{A} := \iota \circ \alpha^{1/2}$. As $\alpha$, its square root and the tautological embedding are all continuous by construction, this $\mathcal{I}$-adic character $\mathcal{A}$ is also continuous. In the same way, we construct a character $\bar{\mathcal{A}} := \iota \circ \bar{\alpha}^{1/2}$ by starting with $\bar{\alpha}$ instead of $\alpha$, or, equivalently, by precomposing $\mathcal{A}$ with the automorphism $\tilde{c}$ of $\mathbb{I}_K/K^\times$ defined above. By definition, we have $P_k \circ \mathcal{A} = \alpha^{k/2}$ and $P_k \circ \bar{\mathcal{A}} = \bar{\alpha}^{k/2}$.

*Proof of the lemma.* We set

$$ \Xi := \chi_{k_0} \alpha^{k_0/2} (\bar{\alpha})^{-k_0/2} \mathcal{A}^{-1} \bar{\mathcal{A}}. $$

By construction, $P_k \circ \Xi = \chi_{k_0} \alpha^{(k_0-k)/2} (\bar{\alpha})^{(k-k_0)/2}$. From this formula, if $k = k_0$ we recover the character $\chi_{k_0}$ we started with, and the weight of $\xi_k$ is $k_0 + (k - k_0) = k$. Finally, the conductor of $\xi_k$ is clearly the conductor of $\chi_{k_0}$ and the finite type of $\xi_k$ is just the product of the finite type of $\chi_{k_0}$ with the finite type of $\alpha^{(k_0-k)/2} (\bar{\alpha})^{(k-k_0)/2}$, which by construction is a product of powers of $\omega_p$ and $\omega_{\bar{p}}$. $\square$

As done in [Col20, p. 25], we can also specialize our $\Lambda$-adic Hecke character at weights $P_{k,\epsilon} = (k, \epsilon)$, for a finite order character $\epsilon : \mathbb{Z}_p^\times \to \mathbb{Q}_p^\times$. By construction and Definition 4 we get

$$ P_{k,\epsilon} \circ \Xi = \chi_k. $$

Indeed, the image of $\alpha$ is contained in the kernel of $\epsilon$. We can finally prove Proposition 6:

*Proof of Proposition 6.* We recall that the Fourier-Jacobi coefficients of $\mathcal{L}^*(f)$ have been defined as

$$ g_{n/N(\mathfrak{a}),\mathfrak{a}}^*(w) = \kappa(\Omega_0)^{-1} \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \text{Cl}_K \\ (\mathfrak{b},p)=1}} (\varepsilon^3 \chi_{\kappa}^{-1})(\mathfrak{b}) g_{n/N(\mathfrak{a}),\mathfrak{a},\mathfrak{b}}^*(w), $$

where

$$ g_{n/N(\mathfrak{a}),\mathfrak{a},\mathfrak{b}}^*(w) = \kappa(\mathcal{N}(\mathfrak{b}))^{1/2} T_n(f(\tau)\vartheta_{\overline{\delta p^r \mathfrak{b}}}(n\delta w, \tau))|_{\tau=\tau_{\mathfrak{a},\mathfrak{b}}}, $$

First of all, the Hecke equivariance follows from the continuity of the Hecke operators on the space of $p$-adic modular forms and by Theorem 12. Take a sequence of modular forms $f_i \in S_{k_i-1}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}}, \bar{\mathbb{Z}}_p)$ such that $f_i \to f$, $p$-adically. Up to multiplying by powers of $E_{p-1}$, the normalized Eisenstein series of weight $p-1$, we may suppose that the weights $k_i$ go to $\infty$ in the archimedean norm. Consider the sequence of lifts $\mathcal{L}(f_i) \in \mathcal{S}(k_i, \Gamma_1(p^r D), \chi_{k_i}, \bar{\mathbb{Z}}_p)$. Clearly $\mathcal{N}(\mathfrak{b})^{k_i/2}$ converges to $\kappa(\mathcal{N}(\mathfrak{b}))^{1/2}$, as $\mathcal{N}(\mathfrak{b})$ is prime to $p$. Also, as $p$ divides the level of the $f_i$, for $p \mid n$ the Hecke operator $T_n$ appearing in the $n$-th Fourier coefficient is continuous with respect to the $p$-adic topology on the space of $p$-adic modular forms. Finally, we see by the above lemma that
there exists a p-adic analytic family of Hecke characters passing through the $\chi_{k_i}$, so that, in particular, the limit exists and still defines a character in the same family. $\square$

We are in position to state and prove our main result:

**Theorem 17.** The $\Lambda$-adic Fourier-Jacobi expansion $\mathcal{L}_{\omega j}(\mathfrak{F})$ introduced in (1) defines a $\Lambda$-adic Picard modular form. More precisely, for each positive integer $k \gg 0$ and a finite character $\epsilon : \mathbb{Z}_p^\times \to \overline{\mathbb{Q}}_p^\times$ of order $p^r$ such that

$$P_{k-1,\epsilon} \circ \mathfrak{F} = f_{k-1} \in S_{k-1}^{\text{ord}}(\Gamma_1(p^r D), \omega_{K/\mathbb{Q}}),$$

we have

$$\Omega_0^{-k} P_{k,\epsilon} \circ \mathcal{L}_{\omega j}(\mathfrak{F}) = \mathcal{L}_{k,\epsilon, \chi_k \tilde{\omega}^{k-j}}^*(f_{k-1}),$$

where $\tilde{\omega}$ is the composition of the Teichmüller character and the square root of the norm character.

*Proof.* The $\Lambda$-adic Fourier-Jacobi coefficients of $\mathcal{L}_{\omega j}(\mathfrak{F})$ have been defined by

$$\mathfrak{g}_{n/\mathbb{N}(\mathfrak{a}), \mathfrak{a}}^j(w) = \varepsilon(\bar{\mathfrak{a}}/\mathfrak{a}) \sum_{\substack{\mathfrak{b} \in \mathbb{C}I_K \\ (\mathfrak{b}, p)=1}} (\varepsilon^3 \Xi^{-1})(\mathfrak{b}) \mathfrak{g}_{n/\mathbb{N}(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^j(w),$$

where

$$\mathfrak{g}_{n/\mathbb{N}(\mathfrak{a}), \mathfrak{a}, \mathfrak{b}}^j(w) = \kappa(\langle N(\mathfrak{b}) \rangle^{1/2}) \omega^{j/2}(N(\mathfrak{b})) T_n(\mathfrak{F}(\tau) \star \vartheta_{\delta \mathfrak{p}^r \mathfrak{b}}(n \delta w, \tau))|_{\tau=\tau_{\mathfrak{a}, \mathfrak{b}}},$$

By the definition of $\star$, we have

$$P_{k,\epsilon} \circ (\mathfrak{F}(\tau) \star \vartheta_{\delta \mathfrak{p}^r \mathfrak{b}}(n \delta w, \tau)) = (P_{k-1,\epsilon} \circ \mathfrak{F}(\tau)) \vartheta_{\delta \mathfrak{p}^r \mathfrak{b}}(n \delta w, \tau) = f_{k-1} \vartheta_{\delta \mathfrak{p}^r \mathfrak{b}}(n \delta w, \tau).$$

We also see that

$$\omega^{j/2}(N(\mathfrak{b})) P_{k,\epsilon} \circ \kappa(\langle N(\mathfrak{b}) \rangle^{1/2}) = N(\mathfrak{b})^{k/2} \omega^{(j-k)/2}(N(\mathfrak{b})).$$

The $T_n$ are well-defined on families, see [H93, (1), p. 209]. Finally, we apply Lemma 16 to get

$$P_{k,\epsilon} \circ \Xi^{-1}(\mathfrak{b}) = \chi_k^{-1}(\mathfrak{b}).$$

$\square$

## ACKNOWLEDGEMENTS

I am deeply grateful to Professor Marc-Hubert Nicole for his uninterrupted guidance throughout the drafting of this work. I also wish to thank Maria Rosaria Pati for numerous discussions.

## REFERENCES

[Bel02] Joël Bellaïche, Congruences endoscopiques et représentations galoisiennes, Ph.D. thesis, 2002.

[Cog85] James Wesley Cogdell, Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus, *Journal für die reine und angewandte Mathematik* 357 (1985), 115-137.

[Col20] Daniel J. Collins, Anticyclotomic p-adic L-functions and Ichino's formula, *Annales mathématiques du Québec* 44 (2020), no.1, 27-89.

[dSG16] Ehud de Shalit, Eyal Zvi Goren, A theta operator on Picard modular forms modulo an inert prime, *Research in the Mathematical Sciences* 3 (2016), Paper No. 28, 65 pp.
[Eic74] Martin Eichler, Quadratische Formen and orthogonale Gruppen, *Grundlehren der mathematischen Wissenschaften*, Bond 63, Springer-Verlag, Berlin Heidelberg, New York, 1974.

[F98] Tobias Finis, Some computational results on Hecke eigenvalues of modular forms on a unitary group, *Manuscripta Mathematica* 96 (1998), 149-180.

[F99] Tobias Finis, Arithmetic properties of a theta lift from GU(2) to GU(3), PhD thesis, unpublished, 1999.

[GeR91] Stephen Gelbart, Jonathan David Rogawski, *L*-functions and Fourier-Jacobi coefficients for the unitary group U(3), *Inventiones Mathematicae* 105 (1991), no.3, 445-472.

[GS93] Ralph Greenberg, Glenn Stevens, *p*-adic *L*-functions and *p*-adic periods of modular forms, *Inventiones Mathematicae* 111 (1993), 407-447.

[HLS05] Michael Harris, Jian-Shu Li, Christopher McLean Skinner, The Rallis inner product formula and *p*-adic *L*-functions, in *Automorphic Representations, *L*-Functions and Applications: Progress and Prospects: Proceedings of a conference honoring Steve Rallis on the occasion of his 60th birthday*, The Ohio State University, March 27-30, 2003, edited by James W. Cogdell, Dihua Jiang, Stephen S. Kudla, David Soudry and Robert J. Stanton, Berlin, New York: De Gruyter, 2005, pp. 225-256. 4

[H86] Haruzo Hida, Galois representations into GL₂(Zₚ[[X]]) attached to ordinary cusp forms, *Inventiones Mathematicae* 85 (1986), no.3, 545-613.

[H93] Haruzo Hida, *Elementary Theory of *L*-functions and Eisenstein Series*, London Mathematical Society Student Texts, 26 Cambridge University Press, Cambridge, 1993. xii+386 pp.

[H96] Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, *Annales Scientifiques de l'École Normale Supérieure*, IV. Sér. 19, 231-273 (1986).

[H04] Haruzo Hida, *p*-adic automorphic forms on *Shimura varieties*, Springer Monographs in Mathematics Springer-Verlag, New York, 2004. xii+390 pp.

[HZ76] Friedrich Hirzebruch, Don Zagier, Intersection Numbers of Curves on Hilbert Modular Surfaces and Modular Forms of Nebentypus. *Inventiones Mathematicae* 36 (1976): 57-114.

[Ka73] Nicolas Katz, *p*-adic properties of modular schemes and modular forms, in *Modular functions of one variable, III* (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69190 Lecture Notes in Math., Vol. 350 Springer-Verlag, Berlin-New York, 1973.

[K78] Stephen S. Kudla, Intersection Numbers for Quotients of the Complex 2-Ball and Hilbert Modular Forms. *Inventiones Mathematicae* 47 (1978): 189-208.

[K79] Stephen S. Kudla, On certain arithmetic automorphic forms for SU(1, q), *Inventiones Mathematicae* 52(1979), no.1, 1-25.

[K81] Stephen S. Kudla, On certain Euler products for SU(2, 1), *Compositio Mathematica* 42, no.3, 321-344.

[KM90] Stephen S. Kudla, John James Milsson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, *Institut des Hautes Études Scientifiques, Publications Mathématiques* (1990), no. 71, 121-172.

[Lan12] Kai-Wen Lan, *Arithmetic compactifications of PEL-type Shimura varieties*, London Mathematical Society Monographs, vol. 36, Princeton University Press, Princeton, 2013.

[Lar88] Michael Jeffrey Larsen, Unitary groups and *l*-adic representations, Ph.D. thesis, Princeton University, ProQuest LLC, Ann Arbor, 1988, 72 pp.

[Lar92] Michael Jeffrey Larsen, Arithmetic compactification of some Shimura surfaces, in *The zeta functions of Picard modular surfaces*, 31-45, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1992.

[L19] David Loeffler, Spherical varieties and norm relations in Iwasawa theory, *Journal de théorie des nombres de Bordeaux*, Volume 33 (2021) no. 3.2, pp. 1021-1043.

[MS07] Atsushi Murase, Takashi Sugano, On the Fourier-Jacobi expansion of the unitary Kudla lift, *Compositio Mathematica* 143 (2007), no.1, 1-46.
[P82] Hans Petersson, Modulfunktionen und quadratische Formen, *Ergebnisse der Mathe- matik und ihrer Grenzgebiete*, 100, Springer-Verlag, Berlin, 1982, x+307 pp.

[S64] Goro Shimura, The arithmetic of unitary groups. *Annals of Mathematics* 79 (1964), 369-409.

[S76] Goro Shimura, Theta functions with complex multiplication, *Duke Mathematical Journal* J. 43 (1976), 673-696.

[S78] Goro Shimura, The arithmetic of automorphic forms with respect to a unitary group, *Annals of Mathematics* 107 (1978), no.3, 569-605.

[TW83] Yue Lin Lawrence Tong, Shu Ping Wang, Theta Functions Defined by Geodesic Cycles in Quotients of SU($p$, 1). *Inventiones mathematicae* 71 (1983): 467-500.

NORMANDIE UNIV, UNICAEN, CNRS, LMNO, 14000 CAEN, FRANCE

Email address: francesco-maria.iudica@unicaen.fr