Finite slope Triple product $p$-adic $L$-functions over totally real number fields

Let $F$ be a totally real number field with integer ring $\mathcal{O}_{F}$ and fix a real embedding $\tau_{0}$. Let $p>2$ be a prime number and let $\mathfrak{p}_{0}$ the prime over $p$ associated to $\tau_{0}$ under a fixed embedding $\bar{\operatorname{\mathbb{Q}}}\hookrightarrow\operatorname{\mathbb{C}}_{p}$. During all this paper we will suppose that $F_{\mathfrak{p}_{0}}=\operatorname{\mathbb{Q}}_{p}$.

Let $B$ be a quaternion algebra over $F$ that splits at any prime over $p$, and it is ramified at any real place but $\tau_{0}$. In [3], we construct $p$-adic families of automorphic forms on $(B\otimes\operatorname{\mathbb{A}})^{\times}$ as global sections of certain $p$-adic sheaves on certain unitary Shimura curves, extending the previous work of [6]. Such families are parametriced by the $(d+1)$-dimensional Iwasawa algebra $\Lambda=\operatorname{\mathbb{Z}}_{p}[[(\mathcal{O}_{F}\otimes\operatorname{\mathbb{Z}}_{p})^{\times}\times\operatorname{\mathbb{Z}}_{p}^{\times}]]$.

Let $\mu_{1},\mu_{2},\mu_{3}$ be three finite slope $p$-adic families of automorphic eigenforms for $B$ and we denote by $\Lambda_{1}$, $\Lambda_{2}$ and $\Lambda_{3}$ the rings over which they are defined, respectively. Let $x,y,z$ be a triple of classical points corresponding to $((\underline{k}_{1},\nu_{1}),(\underline{k}_{2},\nu_{2}),(\underline{k}_{3},\nu_{3}))$, where $\mu_{i}\in\operatorname{\mathbb{N}}$ and $\underline{k}_{i}\in\operatorname{\mathbb{N}}^{[F:\operatorname{\mathbb{Q}}]}$. We denote by $\pi_{x}$ the automorphic representation of $(B\otimes\operatorname{\mathbb{A}}_{F})^{\times}$ generated by the automorphic form obtained from the specialization of $\mu_{1}$ at $x$, and $\Pi_{x}$ the corresponding cuspidal automorphic representation of $\mathrm{GL}_{2}(\operatorname{\mathbb{A}}_{F})$. We also denote by $\alpha_{x}^{\mathfrak{p}}$ and $\beta_{x}^{\mathfrak{p}}$ the roots of the Hecke polynomial at $\mathfrak{p}$. In the same way we obtain $\pi_{y}$, $\Pi_{y}$, $\alpha_{y}^{\mathfrak{p}}$, $\beta_{y}^{\mathfrak{p}}$, $\pi_{z}$, $\Pi_{z}$, $\alpha_{z}^{\mathfrak{p}}$, $\beta_{z}^{\mathfrak{p}}$.

We have (see Lemma 6.8 and Theorem 6.10 for more details):

As mentioned before this result is analogous to [3, Theorem 1.2] where the triple product $p$-adic L-function $\mathcal{L}_{p}(\mu_{1},\mu_{2},\mu_{3})$ is constructed in the ordinary setting. We have been able to extend this result to the finite slope setting thanks to the strategy to $p$-adically interpolate of the integral powers of the Gauss-Manin connexion. We perform such $p$-adic interpolations as power series of endomorphisms in a well chosen space of nearly overconvergent modular forms.

Let $\mathcal{O}_{B}\subset B$ be a maximal order and let $\mathcal{O}_{D}:=\mathcal{O}_{B}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{E}$. The order $\mathcal{O}_{D}$ is maximal (locally) at every prime not dividing ${\rm disc}(B){\rm disc}(D)^{-1}$. We denote by $G_{D}$ the algebraic group attached to $\mathcal{O}_{D}$, namely, $G_{D}(R):=(\mathcal{O}_{D}\otimes_{\operatorname{\mathbb{Z}}}R)^{\times}$ for any $\operatorname{\mathbb{Z}}$-algebra $R$.

Let $\mathfrak{n}$ be an integral ideal of $F$ prime to $\mathrm{disc}(B)$ and consider the open compact subgroups of $G_{D}(\hat{\operatorname{\mathbb{Z}}})$ and $G^{\prime}(\operatorname{\mathbb{A}}_{f})$:

We define the unitary Shimura curve over $\operatorname{\mathbb{C}}$ of level $K_{1}^{\prime}(\mathfrak{n})$ as:

Let $\mathcal{O}_{B,\mathfrak{m}}\subseteq\mathcal{O}_{B}$ be an Eichler order with a well chosen level $\mathfrak{m}\mid\mathfrak{n}$ so that the conditions of [3, Lemma 4.2] are satisfied. Let $L/E$ be a finite extension such that $B\otimes_{F}L=\mathrm{M}_{2}(L)$. By [7, §2.3] the Riemann surface $X(\operatorname{\mathbb{C}})$ has a model (also denoted $X$) defined over $L$. This curve solves the following moduli problem: if $R$ is a $L$-algebra then $X(R)$ corresponds to the set of the isomorphism classes of tuples $(A,\iota,\theta,\alpha)$ where:

• $(i)$

  $A$ is an abelian scheme over $R$ of relative dimension $4d$.

• $(ii)$

  $\iota:\mathcal{O}_{\mathfrak{m}}\rightarrow\mathrm{End}_{R}(A)$ gives an action of the ring $\mathcal{O}_{\mathfrak{m}}$ on $A$ such that ${\rm Lie}(A)^{-,1}$ is of rank 1 and the action of $\mathcal{O}_{F}$ factors through $\mathcal{O}_{F}\subset E\subseteq L$.

• $(iii)$

  $\theta$ is a $\mathcal{O}_{\mathfrak{m}}$-invariant homogeneous polarization of $A$ such that the Rosati involution sends $\iota(d)$ to $\iota(d^{\ast})$.

• $(iv)$

  $\alpha$ is a class modulo $K_{1}^{{}^{\prime}}(\mathfrak{n})$ of $\mathcal{O}_{\mathfrak{m}}$-linear symplectic similitudes $\alpha:\hat{T}(A)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\hat{\mathcal{O}}_{\mathfrak{m}}$.

We introduce the sheaves which give rise to the modular forms for $G^{\prime}$. Let $L_{0}/F$ be an extension such that $B\otimes_{F}L_{0}=\mathrm{M}_{2}(L_{0})$, write $L=L_{0}(\sqrt{\lambda})\supset E$, and denote by $X_{L}$ the base change to $L$ of the unitary Shimura curve $X$. Using the universal abelian variety $\pi:A\rightarrow X_{L}$ we define the following coherent sheaves on $X_{L}$:

The sheaf $\mathcal{H}$ is endowed with a Gauss-Manin connection $\bigtriangledown:\mathcal{H}\rightarrow\mathcal{H}\otimes\Omega^{1}_{X_{L}}$. The natural $\mathcal{O}_{D}$-equivariant exact sequence:

induces the Hodge exact sequence (see [10, §2.3.1]) $0\rightarrow\omega\rightarrow\mathcal{H}\rightarrow\omega_{-}^{\vee}\rightarrow 0$.

If $L$ contains the Galois closure of $F$ then the natural decomposition $F\otimes_{\operatorname{\mathbb{Q}}}L\cong L^{\Sigma_{F}}$ induces: $\omega=\bigoplus_{\tau\in\Sigma_{F}}\omega_{\tau}$ and $\mathcal{H}=\bigoplus_{\tau\in\Sigma_{F}}\mathcal{H}_{\tau}$. As the sheaves $(\Omega^{1}_{A/X_{L}})^{+,1}$ and $(\Omega^{1}_{A/X_{L}})^{+,2}$ are isomorphic then condition $(ii)(2)$ of the moduli problem of $X$ imply that $\omega_{\tau_{0}}$ is locally free of rank 1, while $\omega_{\tau}$ is of rank $2$ for $\tau\neq\tau_{0}$. Moreover, $\omega_{-}$ is locally free of rank 1. Thus, $\omega_{\tau}=\mathcal{H}_{\tau}$ for each $\tau\neq\tau_{0}$, and we have the exact sequence

Let $\underline{k}=(k_{\tau})\in\operatorname{\mathbb{N}}[\Sigma_{F}]$, we introduce the modular sheaves over $X_{L}$ considered in this paper:

In [3, §4.3] an isomorphism $\varphi_{\tau_{0}}:\omega_{\tau_{0}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\omega_{-}$ is provided by the polarization. Thus, the Kodaira-Spencer isomorphism (see [10, Lemme 2.3.4]) induces the isomorphism:

If $\tau\neq\tau_{0}$, also the polarization provides an isomorphism:

Let $R_{0}$ be a $L$-algebra and let $f\in H^{0}(X/R_{0},\omega^{\underline{k}})$. If $R$ is a $R_{0}$-algebra, $(A,\iota,\theta,\alpha)$ is a tuple corresponding to a point of $X(\mathrm{Spec}(R))$ and $w=(f_{0},(f_{\tau},e_{\tau})_{\tau\neq\tau_{0}})$ is a $R$-basis of $\omega_{A}=\left(\Omega^{1}_{A/R}\right)^{+,2}$, then there exists $f(A,\iota,\theta,\alpha,w)\in\bigotimes_{\tau\neq\tau_{0}}\mathrm{Sym}^{k_{\tau}}\left(R^{2}\right)^{\vee}$ such that

Thus a section $f\in H^{0}(X/R_{0},\omega^{\underline{k}})$ is characterized as a rule that assigns to any $R_{0}$-algebra $R$ and $(A,\iota,\theta,\alpha,w)$ over $R$ a linear form

satisfying:

• (A1)

  The element $f(A,\iota,\theta,\alpha,w)$ depends only on the $R$-isomorphism class of $(A,\iota,\theta,\alpha)$.

• (A2)

  Formation of $f(A,\iota,\theta,\alpha,w)$ commutes with extensions $R\rightarrow R^{\prime}$ of $R_{0}$-algebras.

• (A3)

  If $(t,\underline{g})\in R^{\times}\times\operatorname{\mathrm{GL}}_{2}(R)^{\Sigma_{F}\setminus\{\tau_{0}\}}$:

  $$f(A,\iota,\theta,\alpha,w(t,\underline{g}))=t^{-k_{\tau_{0}}}\cdot\left(\underline{g}^{-1}f(A,\iota,\theta,\alpha,w)\right).$$

  where $(t,\underline{g})$ acts on $w$ in a natural way (see [3, §4.4] for further details).

Let $\mathfrak{n}$ be an integral ideal of $F$ prime to $p$ and $\mathrm{disc}(B)$. Write $K_{1}^{\prime}\left(\mathfrak{n},\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}\mathfrak{p}\right):=K_{1}^{\prime}\left(\mathfrak{n}\right)\cap K_{0}^{\prime}\left(\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}\mathfrak{p}\right)$, and denote by $X$ the unitary Shimura curve of level $K_{1}^{\prime}\left(\mathfrak{n},\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}\mathfrak{p}\right)$ similarly as in §3.1. Since $p$ is coprime to the discriminant of $B$, by [7, §5.3] $X$ admits a canonical model $X_{\mathrm{int}}$ over $\mathcal{O}_{\mathfrak{p}_{0}}$, representing the analogue moduli problem described in §3.1 but exchanging an $E$-algebra by an $\mathcal{O}_{\mathfrak{p}_{0}}$-algebra $R$. Namely, it classifies quadruples $(A,\iota,\theta,\alpha^{\mathfrak{p}_{0}})$ over $R$, where $\alpha^{\mathfrak{p}_{0}}$ is a class of $\mathcal{O}_{D}$-linear symplectic similitudes outside $\mathfrak{p}_{0}$. By [7, §5.4], $X_{\mathrm{int}}$ has good reduction. Let $\pi:{\bf A}\rightarrow X_{\mathrm{int}}$ be the universal abelian variety, whose existence is guaranteed by the moduli interpretation of $X_{\mathrm{int}}$. Notice that the universal abelian variety $A\rightarrow X_{L}$ introduced in §3.2 is the generic fibre of ${\bf A}\rightarrow X_{\mathrm{int}}$. Since we have added $\Gamma_{0}(\mathfrak{p})$-structure for all $\mathfrak{p}\neq\mathfrak{p}_{0}$, ${\bf A}$ is endowed with a subgroup $C_{\mathfrak{p}}\subset{\bf A}[\mathfrak{p}]^{-,1}$ isomorphic to $\mathcal{O}_{\mathfrak{p}}/\mathfrak{p}$ by Remark 3.1.

Let $\mathfrak{X}$ be denote the formal scheme over $\mathrm{Spf}(\mathcal{O}_{\mathfrak{p}_{0}})$ obtained as the completion of $X_{\mathrm{int}}$ along its special fiber which is denoted by $\bar{X}_{\mathrm{int}}$ .

The $p$-divisible group ${\bf A}[p^{\infty}]$ over $X_{\mathrm{int}}$ is decomposed as:

We are interested in the $p$-divisible groups $\mathcal{G}_{0}:={\bf A}[\mathfrak{p}_{0}^{\infty}]^{-,1}$ and $\mathcal{G}_{\mathfrak{p}}:={\bf A}[\mathfrak{p}^{\infty}]^{-,1}$ if $\mathfrak{p}\neq\mathfrak{p}_{0}$, which are defined over $X_{\mathrm{int}}$ and endowed with actions of $\mathcal{O}_{\mathfrak{p}_{0}}$ and $\mathcal{O}_{\mathfrak{p}}$ respectively. The sheaves of invariant differentials of the corresponding Cartier dual $p$-divisible groups are denoted by $\omega_{0}:=\omega_{\mathcal{G}_{0}^{D}}$ and $\omega_{\mathfrak{p}}:=\omega_{\mathcal{G}_{\mathfrak{p}}^{D}}$ if $\mathfrak{p}\neq\mathfrak{p}_{0}$. This fits with the definition of $\omega$ given in §3.2 since, as explained in [3, §5.1], $\left(\pi_{\ast}\Omega^{1}_{{\bf A}/X_{\rm int}}\right)^{+,2}\simeq\left(\pi_{\ast}\Omega^{1}_{{\bf A}^{\vee}/X_{\rm int}}\right)^{-,1}$ by means of the polarization.

As explained in [3, §5.1], the universal polarization provides an isomorphism of sheaves of invariant differentials $\omega_{0}\simeq\omega_{\mathcal{G}_{0}}$ compatible with $\varphi_{\tau_{0}}$. Moreover, for any ring $\hat{\mathcal{O}}$ containing all the $p$-adic embeddings of $\mathcal{O}_{F}\hookrightarrow\mathcal{O}_{\operatorname{\mathbb{C}}_{p}}$, if we extend our base ring $\mathcal{O}_{\mathfrak{p}_{0}}$ to $\hat{\mathcal{O}}$ then we have a decomposition $\omega_{\mathfrak{p}}=\oplus_{\tau\in\Sigma_{\mathfrak{p}}}\omega_{\mathfrak{p},\tau}$, where each $\omega_{\mathfrak{p},\tau}$ has rank $2$ and we have an isomorphism $\varphi_{\tau}:\bigwedge^{2}\omega_{\mathfrak{p},\tau}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathcal{O}_{X_{\rm int}}$. The sheaves $\omega_{\mathfrak{p},\tau}$ and $\omega_{0}$ are the integral models of $\omega_{\tau}$ and $\omega_{\tau_{0}}$, respectively.

There is a dichotomy in $X_{\mathrm{int}}$ which says that any point in the generic fiber $\bar{X}_{\mathrm{int}}$ is ordinary or supersingular (with respect to $\mathcal{G}_{0}$), and there are finitely many supersingular points in $\bar{X}_{\mathrm{int}}$. From [14, Proposition 6.1] there exists $\mathrm{Ha}\in H^{0}\left(\bar{X}_{\mathrm{int}},\omega_{0}^{p-1}\right)$ that vanishes exactly at supersingular geometric points and these zeroes are simple. This is called the Hasse invariant.

We denote by $\overline{\rm Hdg}$ the locally principal ideal of $\mathcal{O}_{\bar{X}_{\mathrm{int}}}$ described as follows: for each $U=\mathrm{Spec}(R)\subset\bar{X}_{\mathrm{int}}$ if $\omega_{0}\mid_{U}=Rw$ and $\mathrm{Ha}\mid_{U}=Hw^{\otimes(p-1)}$ then $\overline{\rm Hdg}\mid_{U}=HR\subseteq R$. Let $\rm Hdg$ the inverse image of $\overline{\rm Hdg}$ in $\mathcal{O}_{\mathfrak{X}}$, which is also a locally principal ideal. Note that $\mathrm{Ha}^{p^{n}}$ extends canonically to a section of $H^{0}(\mathfrak{X},\omega_{0}^{p^{n}(p-1)}\otimes\operatorname{\mathbb{Z}}/p^{n+1}\operatorname{\mathbb{Z}})$, indeed, for any two extensions $\mathrm{Ha}_{1}$ and $\mathrm{Ha}_{2}$ of $\mathrm{Ha}$ we have $\mathrm{Ha}_{1}^{p^{n}}=\mathrm{Ha}_{2}^{p^{n}}$ modulo $p^{n+1}$ by the binomial formula.

We introduce the corresponding neighborhoods of the ordinary locus. For each integer $r\geq 1$ we denote by $\mathfrak{X}_{r}$ the formal scheme over $\mathfrak{X}$ which represents the functor that classifies for each $p$-adically complete $\hat{\mathcal{O}}$-algebra $R$:

here the brackets means the equivalence class given by $(h,\eta)\equiv(h^{\prime},\eta^{\prime})$ if $h=h^{\prime}$ and $\eta=\eta^{\prime}(1+pu)$ for some $u\in R$ (see [2, Definition 3.1]).

If we write $\Omega_{0}\subseteq\omega_{0}$ for the subsheaf generated by the lifts of the image of the Hodge-Tate map, by [2, Proposition A.3] we have that $\Omega_{0}=\underline{\delta}\omega_{0}$. Thus, we obtain a morphism

where $\mathcal{I}_{n}:=p^{n}\mathrm{Hdg}^{-\frac{p^{n}}{p-1}}$.

By the moduli interpretation, the $p$-divisible group $\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}\mathcal{G}_{\mathfrak{p}}\rightarrow\mathfrak{X}_{r}$ is étale isomorphic to $\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}(F_{\mathfrak{p}}/\mathcal{O}_{\mathfrak{p}})^{2}$. Assume that $r\geq n$. We denote by $\mathfrak{X}_{r,n}\rightarrow\mathfrak{X}_{r}$ the formal scheme obtained by adding to the moduli interpretation a point of order $\mathfrak{p}^{n}$ for each $\mathfrak{p}\neq\mathfrak{p}_{0}$ whose multiples generate $\mathcal{G}_{\mathfrak{p}}[\mathfrak{p}]/C_{\mathfrak{p}}$ (see Remark 3.1). It is clear that the extension $\mathfrak{X}_{r,n}\rightarrow\mathfrak{X}_{r}$ is étale and its Galois group contains $\prod_{\mathfrak{p}\neq\mathfrak{p}_{0}}(\mathcal{O}_{\mathfrak{p}}/p^{n}\mathcal{O}_{\mathfrak{p}})^{\times}$ as a subgroup. Moreover, $\mathfrak{X}_{r,n}$ has also good reduction (see [7, §5.4]). Now we trivialize the subgroup $D_{n}$: Let $\mathcal{X}_{r,n}$ be the adic generic fiber of $\mathfrak{X}_{r,n}$. By [2, Corollaire A.2], the group scheme $D_{n}\rightarrow\mathcal{X}_{r,n}$ is also étale isomorphic to $p^{{-n}}\mathcal{O}_{\mathfrak{p}_{0}}/\mathcal{O}_{\mathfrak{p}_{0}}$. We denote by $\mathcal{IG}_{r,n}$ the adic space over $\mathcal{X}_{r,n}$ that trivialize $D_{n}$. Then the map $\mathcal{IG}_{r,n}\rightarrow\mathcal{X}_{r,n}$ is a finite étale with Galois group $(\mathcal{O}_{\mathfrak{p}_{0}}/p^{n}\mathcal{O}_{\mathfrak{p}_{0}})^{\times}$. We denote by $\mathfrak{IG}_{r,n}$ the normalization $\mathcal{IG}_{r,n}$ in $\mathfrak{X}_{r,n}$ which is finite over $\mathfrak{X}_{r,n}$ and it is also endowed with an action of $(\mathcal{O}_{{\mathfrak{p}_{0}}}/p^{n}\mathcal{O}_{{\mathfrak{p}_{0}}})^{\times}$. These constructions are captured by the following tower of formal schemes:

endowed with a natural action of $(\mathcal{O}/p^{n}\mathcal{O})^{\times}\simeq\prod_{\mathfrak{p}}(\mathcal{O}_{\mathfrak{p}}/p^{n}\mathcal{O}_{\mathfrak{p}})^{\times}$.

Let us consider the morphism

The following result is completely analogous to [1, Lemma 3.3]:

We briefly recall in this subsection constructions performed in [1, §2] and [3, §6]. Let $S$ be a formal scheme, $\mathcal{I}$ its (invertible) ideal of definition and $\mathcal{E}$ a locally free $\mathcal{O}_{S}$-module of rank $n$. We write $\bar{S}$ the scheme with structural sheaf $\mathcal{O}_{S}/\mathcal{I}$ and put $\overline{\mathcal{E}}$ the corresponding $\mathcal{O}_{\overline{S}}$-module. We fix marked sections $s_{1},\cdots,s_{m}$ of $\overline{\mathcal{E}}$, namely, the sections $s_{1},\cdots,s_{m}$ define a direct sum decomposition $\overline{\mathcal{E}}=\mathcal{O}_{\bar{S}}^{m}\oplus Q$, where $Q$ is a locally free $\mathcal{O}_{\bar{S}}$-module of rank $n-m$.

Let $S-\mathrm{Sch}$ be the category of the formal $S$-schemes. There exists a formal scheme $\operatorname{\mathbb{V}}(\mathcal{E})$ over $S$ called the formal vector bundle attached to $\mathcal{E}$ which represents the functor, denoted by the same symbol, $S-\mathrm{Sch}\rightarrow\mathrm{Sets}$, given by $\operatorname{\mathbb{V}}(\mathcal{E})(t:T\rightarrow S):=H^{0}(T,t^{\ast}(\mathcal{E})^{\vee})={\rm Hom}_{\mathcal{O}_{T}}(t^{\ast}(\mathcal{E}),\mathcal{O}_{T})$. Crucial in [1] is the construction of the so called formal vector bundles with marked sections which is the formal scheme $\operatorname{\mathbb{V}}_{0}(\mathcal{E},s_{1},\cdots,s_{m})$ over $\operatorname{\mathbb{V}}(\mathcal{E})$ that represents the sub-functor $S-\mathrm{Sch}\rightarrow\mathrm{Sets}$

here $\bar{\rho}$ is the reduction of $\rho$ modulo $\mathcal{I}$.

Given the fixed decomposition $\overline{\mathcal{E}}=Q\oplus\langle s_{i}\rangle_{i}$, let us consider now the sub-functor $\operatorname{\mathbb{V}}_{Q}(\mathcal{E},s_{1},\cdots,s_{m})$ that associates to any formal $S$-scheme $t:T\rightarrow S$ the subset of sections $\rho\in\operatorname{\mathbb{V}}_{0}(\mathcal{E},s_{1},\cdots,s_{m})(T)$ whose reduction $\bar{\rho}$ modulo $\mathcal{I}$ also satisfies $\bar{\rho}(t^{\ast}(m))=0$ for every $m\in Q$.