2.1 Global datum

Let $E/\mathbb{Q}$ a quadratic imaginary field and denote by $\overline{\bullet }$ the complex conjugation of $E$ . Let $(V=E^{3},\unicode[STIX]{x1D713})$ be the hermitian space of dimension $3$ over E, of signature $(2,1)$ at infinity given by the matrix

$$\begin{eqnarray}J=\left(\begin{array}{@{}ccc@{}} & & 1\\ & 1 & \\ 1 & & \end{array}\right).\end{eqnarray}$$

Let us then denote by

$$\begin{eqnarray}\displaystyle G & = & \displaystyle GU(V,\unicode[STIX]{x1D713})=GU(2,1)\nonumber\\ \displaystyle & = & \displaystyle \{(g,c(g))\in \operatorname{GL}(V)\times \mathbb{G}_{m,\mathbb{Q}}:\forall x,y\in V,\unicode[STIX]{x1D713}(gx,gy)=c(g)\unicode[STIX]{x1D713}(x,y)\}\subset \operatorname{GL}_{V}\times \mathbb{G}_{m}\nonumber\end{eqnarray}$$

the reductive group over $\mathbb{Q}$ of unitary similitudes of $(V,\unicode[STIX]{x1D713})$ .

Let $p$ be a prime number, unramified in $E$ . If $p=v\overline{v}$ is split in $E$ , then

$$\begin{eqnarray}V\otimes _{\mathbb{Q}}\mathbb{Q}_{p}=V\otimes _{E}E_{v}\oplus \overline{V\otimes _{E}E_{v}},\end{eqnarray}$$

where the action of $E_{v}$ is by $\overline{v}$ on $\overline{V\otimes _{E}E_{v}}$ . Moreover, the complex conjugation exchanges $V\otimes _{E}E_{v}$ and $\overline{V\otimes _{E}E_{v}}$ . In particular, $G\otimes _{\mathbb{Q}}\mathbb{Q}_{p}\simeq \operatorname{GL}(V\otimes _{E}E_{v})\times \mathbb{G}_{m}$ (this isomorphism depends on the choice of $v$ over $p$ ).

We will be particularly interested in the case where $p$ is inert in $E$ ; the case when $p$ split has been studied before (see for example [Reference BrascaBra16]).

Remark 2.1. We could more generally work in the setting of $(B,\star )$ a simple $E$ -algebra of rank $9$ with an involution of the second kind, such that $(B\otimes \mathbb{Q}_{p},\star )$ is isomorphic to $(\operatorname{End}(V\otimes \mathbb{Q}_{p}),\star _{V})$ ,Footnote ³ and replace $G$ with the group,

$$\begin{eqnarray}G_{B}=\{g\in B^{\times }:g^{\star }g=c(g)\in \mathbb{G}_{m,\mathbb{Q}}\}.\end{eqnarray}$$

The construction of the eigenvarieties in the case where $B$ is not split is easier as the associated Shimura varieties are compact, but some non-tempered automorphic form, for example the one constructed by Rogawski and studied in [Reference Bellaïche and ChenevierBC04] and the second part of this article, will never be automorphic for such non-split $B$ .

The Shimura datum we consider is given by

$$\begin{eqnarray}h:\begin{array}{@{}ccc@{}}\mathbb{S} & \longrightarrow & G_{\mathbb{R}}\\ z=x+iy & \longmapsto & \left(\begin{array}{@{}ccc@{}}x & & iy\\ & z & \\ iy & & x\end{array}\right).\end{array}\end{eqnarray}$$

2.2 Complex Picard modular forms and automorphic forms

Classically, Picard modular forms are introduced using the unitary group $U(2,1)$ , but we can treat the case of $GU(2,1)$ similarly. Let $G(\mathbb{R})=GU(2,1)(\mathbb{R})$ the group stabilizing (up to scalar) the signature matrix $J$ and let

$$\begin{eqnarray}Y=\{z=(z_{1},z_{2})\in \mathbb{C}^{2}:2\Im (z_{1})+|z_{2}|^{2}<0\}\end{eqnarray}$$

be the symmetric space associated to $G(\mathbb{R})$ , it is isomorphic to the two-dimensional complex unit ball

$$\begin{eqnarray}B=\{(z_{1},z_{2})\in \mathbb{C}^{2}||z_{1}|^{2}+|z_{2}|^{2}<1\}.\end{eqnarray}$$

On $Y$ , there is an action of $G(\mathbb{R})$ through

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}A & b\\ c & d\end{array}\right)z=\frac{1}{c\cdot z+d}(Az+b)\in B,\quad A\in M_{2\times 2}(\mathbb{C}).\end{eqnarray}$$

This action is transitive and identifies $Y$ with $G(\mathbb{R})/K_{\infty }$ where $K_{\infty }=\operatorname{Stab}((i,0))\subset$ $\{(A,e)\in GU(2)(\mathbb{R})\times GU(1)(\mathbb{R})\}$ can be identified with $\{(A,e)\in GU(2)(\mathbb{R})\times \mathbb{C}^{\times }:c(A)=N(e)\}$ . We write $(i,0)=x_{0}$ .

The subgroup $K_{\infty }$ is not compact but can be written $Z(\mathbb{R})^{0}(U(2)(\mathbb{R})\times U(1)(\mathbb{R}))$ , with $Z$ the center of $GU(2,1)$ . Let $L$ be the $\mathbb{C}$ -points of $K_{\infty }$ . Then $L\simeq (\operatorname{GL}_{2}\times \operatorname{GL}_{1})\times \operatorname{GL}_{1}(\mathbb{C})$ is the Levi of a parabolic $P$ in $\operatorname{GL}_{3}\times \operatorname{GL}_{1}(\mathbb{C})$ . For any $\unicode[STIX]{x1D705}=(k_{1},k_{2},k_{3},r)\in \mathbb{Z}^{4}$ such that $k_{1}\geqslant k_{2}$ , there is an associated (irreducible) representation $S_{\unicode[STIX]{x1D705}}(\mathbb{C})$ of $P$ , of highest weight

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}t_{1} & & \\ & t_{2} & \\ & & t_{3}\end{array}\right),c\in (\operatorname{GL}_{2}\times \operatorname{GL}_{1})\times \operatorname{GL}_{1}(\mathbb{C})\longmapsto t_{1}^{k_{1}}t_{2}^{k_{2}}t_{3}^{k_{3}}c^{r}.\end{eqnarray}$$

The subgroup $K_{\infty }$ embeds in $L\subset P$ by $(A,e)\mapsto ((\overline{A},e),N(e))$ .

Following [Reference HarrisHar90b, Reference HarrisHar84, Reference MilneMil88], such a representation gives $\unicode[STIX]{x1D6FA}^{\unicode[STIX]{x1D705}}$ a locally free sheaf with $G(\mathbb{C})$ -action on $G(\mathbb{C})/P$ , whose structure as sheaf does not depend on $r$ . Restricting it to $G(\mathbb{R})/K_{\infty }=Y$ we get a sheaf $\unicode[STIX]{x1D6FA}^{\unicode[STIX]{x1D705}}$ whose sections over $X$ can be seen as holomorphic functions,

$$\begin{eqnarray}f:G(\mathbb{R})/K_{\infty }\longmapsto S_{\unicode[STIX]{x1D705}}(\mathbb{C}),\end{eqnarray}$$

such that $f(gk)=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D705}}(k)^{-1}f(g)$ , for $g\in G(\mathbb{R}),k\in K_{\infty }$ , which we call (meromorphic at infinity) modular forms of weight $\unicode[STIX]{x1D705}$ . In an informal way, the choice of the previous integer $r$ normalizes the action of the Hecke operators and corresponds to normalizing the (norm of the) central character of the modular forms. We will not use this description of the sheaves, and instead introduced a modular description of these automorphic sheaves.

Fix $\unicode[STIX]{x1D70F}_{\infty }:E\longrightarrow \mathbb{C}$ an embedding, and $\unicode[STIX]{x1D70E}\neq 1\in \operatorname{Gal}(E/\mathbb{Q})$ ; thus $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}_{\infty }=\overline{\unicode[STIX]{x1D70F}_{\infty }}$ is the other embedding of $E$ . Over $\mathbb{C}$ , for any sufficiently small compact open $K$ in $G(\mathbb{A}_{f})$ , the Picard variety $Y_{K}(\mathbb{C})$ of level $K$ can be identified with a (disjoint union of some) quotient of $B=GU(2,1)/K_{\infty }$ , but also with the moduli space parametrizing quadruples $(A,\unicode[STIX]{x1D704},\unicode[STIX]{x1D706},\unicode[STIX]{x1D702})$ where $A$ is an abelian scheme of genus $3$ , $\unicode[STIX]{x1D704}:{\mathcal{O}}_{E}\longrightarrow \operatorname{End}(A)$ is an injection, $\unicode[STIX]{x1D706}$ is a polarization for which Rosati involution corresponds to the conjugation $\overline{\cdot }$ on ${\mathcal{O}}$ , and $\unicode[STIX]{x1D702}$ is a type- $K$ -level structure such that the action of ${\mathcal{O}}_{E}$ on the conormal sheaf $\unicode[STIX]{x1D714}_{A}=e_{A/S}^{\ast }\unicode[STIX]{x1D6FA}_{A/S}^{1}$ Footnote ⁴ decomposes under the embeddings $\unicode[STIX]{x1D70F}_{\infty },\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}_{\infty }$ into two direct factors of respective dimensions $1$ and $2$ . This is done, for example, explicitly in [Reference de Shalit and GorendSG16, § 1.2.2–1.2.4], and we will be especially interested in the description by ‘moving lattice’ given in [Reference de Shalit and GorendSG16, § 1.2.4]. Thus, to every $x=(z_{1},z_{2})\in B$ , we can associate a complex abelian variety,

$$\begin{eqnarray}A_{x}=\mathbb{C}^{3}/L_{x},\end{eqnarray}$$

where $L_{x}$ is the ${\mathcal{O}}_{E}$ -module given in [Reference de Shalit and GorendSG16, (1.25)], and the action of ${\mathcal{O}}_{E}$ on $A_{x}$ is given by

$$\begin{eqnarray}a\in {\mathcal{O}}_{E}\longmapsto \left(\begin{array}{@{}ccc@{}}\overline{\unicode[STIX]{x1D70F}}_{\infty }(a) & & \\ & \overline{\unicode[STIX]{x1D70F}}_{\infty }(a) & \\ & & \unicode[STIX]{x1D70F}_{\infty }(a)\end{array}\right)\in M_{3}(\mathbb{C}).\end{eqnarray}$$

There is, moreover, $\unicode[STIX]{x1D702}_{x}$ a canonical ( $K$ -orbit of) level- $N$ -structure (for $K(N)\subset K\subset G(\mathbb{A}_{f})$ ). Over $Y_{K}(\mathbb{C})$ we thus have a sheaf $\unicode[STIX]{x1D714}_{A}$ that can be decomposed $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F},A}\oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},A}$ according to the action of ${\mathcal{O}}_{E}$ , and we can consider the sheaf

$$\begin{eqnarray}\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}:=\operatorname{Sym}^{k_{1}-k_{2}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},A}\otimes \det ^{\otimes k_{2}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},A}\otimes \det ^{\otimes k_{3}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F},A},\end{eqnarray}$$

for $(k_{1},k_{2},k_{3})$ a dominant (i.e. $k_{1}\geqslant k_{2}$ ) weight. Using the previous description, if we denote by $\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2},\unicode[STIX]{x1D701}_{3}$ the coordinates on $\mathbb{C}^{3}$ , $\unicode[STIX]{x1D714}_{A_{x},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ is generated by $d\unicode[STIX]{x1D701}_{1},\,d\unicode[STIX]{x1D701}_{2}$ and $\unicode[STIX]{x1D714}_{A_{x},\unicode[STIX]{x1D70F}}$ by $d\unicode[STIX]{x1D701}_{3}$ .

There is also $X_{K}(\mathbb{C})$ a toroidal compactification of $Y_{K}(\mathbb{C})$ , [Reference LarsenLar92] and [Reference BellaïcheBel06b], on which $\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}$ extends as $\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}$ (the canonical sheaf of Picard modular forms) and $\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}(-D)$ (the sheaf of cuspidal forms) where $D$ here is the closed subscheme $X_{K}\backslash Y_{K}$ together with its reduced structure.

We sometimes say ‘classical’ if we want to emphasis the difference with overconvergent modular forms defined later. Denote also by $V^{\unicode[STIX]{x1D705}}$ the representation of $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ given by

$$\begin{eqnarray}(A,e)\longmapsto \operatorname{Sym}^{k_{1}-k_{2}}(\overline{A})\otimes \det ^{k_{2}}\overline{A}\otimes \det ^{k_{3}}e.\end{eqnarray}$$

Definition 2.4. For all $g\in G(\mathbb{R})=GU(2,1)(\mathbb{R})$ , write

$$\begin{eqnarray}g=\left(\begin{array}{@{}cc@{}}A & b\\ c & d\end{array}\right),\quad A=\left(\begin{array}{@{}cc@{}}a_{1} & a_{2}\\ a_{3} & a_{4}\end{array}\right)\end{eqnarray}$$

and for $x=(z_{1},z_{2})\in B$ , following [Reference ShimuraShi78], define

$$\begin{eqnarray}\unicode[STIX]{x1D705}(g,x)=\left(\begin{array}{@{}cc@{}}\overline{a_{1}}-\overline{a_{3}}z_{1} & \overline{c_{2}}z_{1}-\overline{c_{1}}\\ \overline{a_{3}}z_{2}-\overline{b_{1}} & \overline{d}-\overline{c_{2}}z_{2}\end{array}\right),\quad \text{and}\quad j(g,x)=(cx+d).\end{eqnarray}$$

Finally, define

$$\begin{eqnarray}J(g,x)=(\unicode[STIX]{x1D705}(g,x),j(g,x))\in \operatorname{GL}_{2}\times \operatorname{GL}_{1}(\mathbb{C}).\end{eqnarray}$$

The following proposition is well known (see [Reference HsiehHsi14, Lemma 3.7]) and probably already in [Reference ShimuraShi78], but we rewrite it to fix the notations,

Proof. For all $\unicode[STIX]{x1D6FE}\in G(\mathbb{Z})$ , there is an isomorphism between $(A_{x},\unicode[STIX]{x1D702}_{x})$ and $(A_{\unicode[STIX]{x1D6FE}x},\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FE}x}\circ \unicode[STIX]{x1D6FE}^{\unicode[STIX]{x1D70E}})$ , for example described in [Reference de Shalit and GorendSG16, § 1.2.2] or in [Reference GordonGor92], which sends $(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2},d\unicode[STIX]{x1D701}_{3})$ to $\unicode[STIX]{x1D6FE}^{\ast }(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2},d\unicode[STIX]{x1D701}_{3})=(\unicode[STIX]{x1D6FE}^{\ast }(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2}),\unicode[STIX]{x1D6FE}^{\ast }d\unicode[STIX]{x1D701}_{3})$ as $\unicode[STIX]{x1D6FE}$ preserve the action of ${\mathcal{O}}_{E}$ ; $\unicode[STIX]{x1D6FE}^{\ast }d\unicode[STIX]{x1D701}_{3}$ is calculated in [Reference de Shalit and GorendSG16, Proposition 1.15] and given by

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{\ast }d\unicode[STIX]{x1D701}_{3}=j(\unicode[STIX]{x1D6FE},x)^{-1}d\unicode[STIX]{x1D701}_{3}.\end{eqnarray}$$

Moreover, by the Kodaira–Spencer isomorphism $\unicode[STIX]{x1D714}_{A_{x},\unicode[STIX]{x1D70F}}\otimes \unicode[STIX]{x1D714}_{A_{x},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FA}^{1}$ (see [Reference de Shalit and GorendSG16, Proposition 1.22] for example), we only need to determine the action of $\unicode[STIX]{x1D6FE}$ on $dz_{1},dz_{2}$ . But this is done in [Reference ShimuraShi78, 1.15] (or an explicit calculation), given by $c(\unicode[STIX]{x1D6FE})^{t}\unicode[STIX]{x1D705}^{-1}(\unicode[STIX]{x1D6FE},x)j(\unicode[STIX]{x1D6FE},x)$ , and we get

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{\ast }(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2}){=^{t}\unicode[STIX]{x1D705}(\unicode[STIX]{x1D6FE},x)}^{-1}(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2}).\end{eqnarray}$$

Thus, setting $F(x,k)=f(A_{x},\unicode[STIX]{x1D702}_{x}\circ k^{\unicode[STIX]{x1D70E}},(d\unicode[STIX]{x1D701}_{1},d\unicode[STIX]{x1D701}_{2},d\unicode[STIX]{x1D701}_{3}))$ , we get,

Thus, to $f\in H^{0}(Y_{K}(\mathbb{C}),\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}})$ we can associate a function, $\unicode[STIX]{x1D6F7}_{f}:G(\mathbb{Q})\backslash G(\mathbb{A})\longrightarrow V^{\unicode[STIX]{x1D705}}$ , by

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{f}(g)=c(g_{\infty })^{-k_{1}-k_{2}-k_{3}}J(g_{\infty },x_{0})^{-1}\cdot F(g_{\infty }x_{0},g_{f}),\end{eqnarray}$$

where the action $\cdot$ is the one on $V^{\unicode[STIX]{x1D705}}$ , and we use the decomposition $g=g_{\mathbb{Q}}g_{\infty }g_{f}\in G(\mathbb{Q})G(\mathbb{R})G(\mathbb{A}_{f})$ . We can check that this expression does not depend on the choice in the decomposition. This association commutes with Hecke operators. The function $\unicode[STIX]{x1D6F7}_{f}$ does not necessarily have a unitary central character, but it is nevertheless normalized by $\unicode[STIX]{x1D705}$ . Indeed, for $z_{\infty }\in \mathbb{C}^{\times }=Z(\mathbb{R})$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{f}(z_{\infty }g)=N(z_{\infty })^{-k_{1}-k_{2}-k_{3}}\overline{z_{\infty }}^{k_{1}+k_{2}}{z_{\infty }}^{k_{3}}\unicode[STIX]{x1D6F7}_{f}(g)=\overline{z}_{\infty }^{-k_{3}}{z_{\infty }}^{-k_{1}-k_{2}}\unicode[STIX]{x1D6F7}_{f}(g).\end{eqnarray}$$

Let $L:V^{\unicode[STIX]{x1D705}}\longrightarrow \mathbb{C}$ a non-zero linear form. Define the injective map of right- $K_{\infty }$ -modules,

$$\begin{eqnarray}{\mathcal{L}}:\begin{array}{@{}ccc@{}}V^{\unicode[STIX]{x1D705}} & \longrightarrow & \mathit{Fonct}(K_{\infty },\mathbb{C})\\ v & \longmapsto & L(J(k,x_{0})^{-1}v).\end{array}\end{eqnarray}$$

We denote by $L_{0}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}),\mathbb{C})$ the set of cuspidal $L^{2}$ -automorphic forms for $G$ . We have the following well-known proposition

Using the previous proposition, to every $f\in H_{\text{cusp}}^{0}(X_{K},\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}})$ , an eigenvector for the Hecke algebra, we will be able to attach a automorphic form $\unicode[STIX]{x1D711}_{f}$ , and an automorphic representation $\unicode[STIX]{x1D6F1}_{f}$ (with same central character).

2.3 Local groups

In this subsection, we describe the local group at an inert prime. Let $p$ be a prime, inert in $E$ . Let $E_{p}\supset \mathbb{Q}_{p}$ its $p$ -adic completion. Recall that $V\otimes \mathbb{Q}_{p}=E_{p}^{3}$ and that the hermitian form is given by the matrix,

$$\begin{eqnarray}J=\left(\begin{array}{@{}ccc@{}} & & 1\\ & 1 & \\ 1 & & \end{array}\right).\end{eqnarray}$$

The diagonal maximal torus $T$ of $G_{\mathbb{Q}_{p}}$ is isomorphic to $E_{p}^{\times }\times E_{p}^{\times }$ ,

$$\begin{eqnarray}T(\mathbb{Q}_{p})=\left\{\!\left(\begin{array}{@{}ccc@{}}a & & \\ & e & \\ & & N(e)\overline{a}^{-1}\end{array}\right),\quad a,e\in E_{p}^{\times }\right\}\end{eqnarray}$$

and contains $T^{1}$ , isomorphic to $E_{p}^{\times }\times E_{p}^{1}$ , where $E_{p}^{1}=\{x\in E_{p}:x\overline{x}=1\}=({\mathcal{O}}_{E_{p}})^{1}$ , the torus of $U(E^{3},J)$ ,

$$\begin{eqnarray}T^{1}(\mathbb{Q}_{p})=\left\{\!\left(\begin{array}{@{}ccc@{}}a & & \\ & e & \\ & & \overline{a}^{-1}\end{array}\right),\quad a\in E_{p}^{\times },e\in E_{p}^{1}\right\}.\end{eqnarray}$$

We also have the Borel subgroups $B=B_{\operatorname{GL}_{3}(E)}\cap G_{\mathbb{Q}_{p}}$ of upper-triangular matrices,

$$\begin{eqnarray}B(\mathbb{Q}_{p})=\left\{\!\left(\begin{array}{@{}ccc@{}}a & x & y\\ & e & \overline{x}\overline{a}^{-1}e\\ & & N(e)\overline{a}^{-1}\end{array}\right),\quad a,e,x,y\in E_{p}^{\times }\text{and }\operatorname{Tr}(\overline{a}^{-1}y)=N(a^{-1}x)\right\},\end{eqnarray}$$

and $B^{1}$ the corresponding Borel subgroup, for $U(E^{3},J)$ ,

$$\begin{eqnarray}B^{1}(\mathbb{Q}_{p})=\left\{\!\left(\begin{array}{@{}ccc@{}}a & x & y\\ & e & \overline{x}\overline{a}^{-1}e\\ & & \overline{a}^{-1}\end{array}\right),\quad a,x,y\in E_{p}^{\times },e\in E_{p}^{1}\text{ and }\operatorname{Tr}(\overline{a}^{-1}y)=N(a^{-1}x)\right\}.\end{eqnarray}$$

5.1 Classical modular sheaves and geometric modular forms

On $X$ , there is a sheaf $\unicode[STIX]{x1D714}$ , the conormal sheaf of ${\mathcal{A}}$ , the universal (semi-)abelian scheme, along its unit section, and $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F}}\oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ .

For any $\unicode[STIX]{x1D705}=(k_{1},k_{2},k_{3})\in \mathbb{Z}^{3}$ such that $k_{1}\geqslant k_{2}$ , there is associated a ‘classical’ modular sheaf,

$$\begin{eqnarray}\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}=\operatorname{Sym}^{k_{1}-k_{2}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes (\det \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}})^{k_{2}}\otimes \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F}}^{k_{3}}.\end{eqnarray}$$

Write $\unicode[STIX]{x1D705}^{\prime }=(-k_{2},-k_{1},-k_{3})$ ; this is still a dominant weight, and $\unicode[STIX]{x1D705}\mapsto \unicode[STIX]{x1D705}^{\prime }$ is an involution. There is another way to see the classical modular sheaves.

Write ${\mathcal{T}}=\operatorname{Hom}_{X,{\mathcal{O}}}({\mathcal{O}}_{X}^{2}\otimes {\mathcal{O}}_{X},\unicode[STIX]{x1D714})$ where ${\mathcal{O}}$ acts by $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}$ on the first two-dimensional factor and $\unicode[STIX]{x1D70F}$ on the other one. Write ${\mathcal{T}}^{\times }=\operatorname{Isom}_{X,{\mathcal{O}}}({\mathcal{O}}_{X}^{2}\otimes {\mathcal{O}}_{X},\unicode[STIX]{x1D714})$ , the $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ -torsor of trivializations of $\unicode[STIX]{x1D714}$ as a ${\mathcal{O}}$ -module. There is an action on ${\mathcal{T}}$ of $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ by $g\cdot w=w\circ g^{-1}$ .

Denote by $\unicode[STIX]{x1D70B}:{\mathcal{T}}^{\times }\longrightarrow X$ the projection. For any dominant $\unicode[STIX]{x1D705}$ as before, define

$$\begin{eqnarray}\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{T}}^{\times }}[\unicode[STIX]{x1D705}^{\prime }],\end{eqnarray}$$

the subsheaf of $\unicode[STIX]{x1D705}^{\prime }$ -equivariant functions for the action of the upper triangular Borel subgroup $B\subset \operatorname{GL}_{2}\times \operatorname{GL}_{1}$ . As the notation suggests, there is an isomorphism, if $\unicode[STIX]{x1D705}=(k_{1},k_{2},k_{3})$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{T}}^{\times }}[\unicode[STIX]{x1D705}^{\prime }]\simeq \operatorname{Sym}^{k_{1}-k_{2}}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes (\det \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}})^{k_{2}}\otimes \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F}}^{k_{3}}.\end{eqnarray}$$

In the sequel we will be interested in the case $K_{p}=I$ , the Iwahori subgroup when we speak about modular forms on the rigid space associated to $X$ . But for moduli-theoretic reason we will not directly consider an integral model of $X_{I}^{\text{rig}}$ , but only an integral model of a particular open subset (the one given by the canonical filtration). Indeed, in this particular setting ( $U(2,1)$ with $p$ inert), the canonical filtration identifies a strict neighborhood of the $\unicode[STIX]{x1D707}$ -ordinary locus in $X$ with an open in the level $I$ -Picard variety.

5.2 Local constructions

Let $G$ be the $p$ -divisible group of the universal abelian scheme over $Y\subset X$ . Later we will explain how to extend our construction to all $X$ . The group $G$ is endowed with an action of ${\mathcal{O}}$ , and we have that its signature is given by

$$\begin{eqnarray}\left\{\begin{array}{@{}lc@{}}p_{\unicode[STIX]{x1D70F}}=1, & q_{\unicode[STIX]{x1D70F}}=2,\\ p_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=2, & q_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=1,\end{array}\right.\end{eqnarray}$$

which means that if we write $\unicode[STIX]{x1D714}_{G}=\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\oplus \unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}$ , the two pieces have respective dimensions $p_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=2$ and $p_{\unicode[STIX]{x1D70F}}=1$ . Moreover, $G$ carries a polarization $\unicode[STIX]{x1D706}$ , such that $\unicode[STIX]{x1D706}:G\overset{{\sim}}{\longrightarrow }G^{D,(\unicode[STIX]{x1D70E})}$ is ${\mathcal{O}}$ -equivariant.

The main result of [Reference HernandezHer18], see also [Reference Goldring and NicoleGN17], is the following,

Definition 5.5. There exists sections,

$$\begin{eqnarray}\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}\in H^{0}(Y\otimes {\mathcal{O}}/p,\det (\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}})^{\otimes (p^{2}-1)})\quad \text{and}\quad \widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}\in H^{0}(Y\otimes {\mathcal{O}}/p,(\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}})^{\otimes (p^{2}-1)}),\end{eqnarray}$$

such that $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}$ is given by (the determinant of) $V^{2}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}\overset{V}{\longrightarrow }\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{(p)}\overset{V}{\longrightarrow }\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}^{(p^{2})},\end{eqnarray}$$

and $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D70F}}$ is given by a division by $p$ on the Dieudonne crystal of $G$ of $V^{2}$ , restricted to a lift of the Hodge Filtration $\unicode[STIX]{x1D714}_{G^{D},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ .

On $Y\otimes {\mathcal{O}}/p$ , the sheaves $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F}}$ (respectively $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ ) are isomorphic. In fact, the previous sections extends to all $X$ , for example by Koecher’s principle, see [Reference LanLan17, Theorem 8.7].

Remark 5.6. (i) These sections are Cartier divisors on $X$ , i.e. they are invertible on an open and dense subset (cf. [Reference HernandezHer18, Proposition 3.22] and [Reference WedhornWed99]).

(ii) Because of the ${\mathcal{O}}$ -equivariant isomorphism $\unicode[STIX]{x1D706}:G\simeq G^{D,(\unicode[STIX]{x1D70E})}$ , and the compatibility of $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}$ with duality (see [Reference HernandezHer18, § 1.10]), we deduce that

$$\begin{eqnarray}\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}(G)=\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}(G^{D})=\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}(G^{(\unicode[STIX]{x1D70E})})=\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}(G).\end{eqnarray}$$

Thus, we could use only $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}$ or $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}$ and define it in this case without using any crystalline construction. We usually write $\widetilde{\text{}^{\unicode[STIX]{x1D707}}\operatorname{ha}}=\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}\otimes \widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}$ , but because of the this remark, we will only use $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}$ in this article (which is then reduced, see the appendix).

(iii) We use the notation $\widetilde{\bullet }$ to denote the previous global sections, but if we have $G/\operatorname{Spec}({\mathcal{O}}_{\mathbb{C}_{p}}/p)$ a $p$ -divisible ${\mathcal{O}}$ -module of signature $(2,1)$ , we will also use the notation $\operatorname{ha}_{\unicode[STIX]{x1D70F}}(G)=v(\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}(G)})$ , where the valuation $v$ on ${\mathcal{O}}_{\mathbb{C}_{p}}$ is normalized such that $v(p)=1$ and truncated by  $1$ .

Let us recall the main theorem of [Reference HernandezHer19] in the simple case of Picard varieties. Recall that $p$ still denotes a prime, inert in $E$ , and suppose $p>2$ .

Theorem 5.8. Let $n\in \mathbb{N}^{\times }$ . Let $H/\operatorname{Spec}({\mathcal{O}}_{L})$ , where $L$ is a valued extension of $\mathbb{Q}_{p}$ , a truncated $p$ -divisible ${\mathcal{O}}$ -module of level $n+1$ and signature $(p_{\unicode[STIX]{x1D70F}}=1,p_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=2)$ . Suppose

$$\begin{eqnarray}\operatorname{ha}_{\unicode[STIX]{x1D70F}}(H)<\frac{1}{4p^{n-1}}.\end{eqnarray}$$

Then there exists a unique filtration (so-called ‘canonical’ of height $n$ ) of $H[p^{n}]$ ,

$$\begin{eqnarray}0\subset H_{\unicode[STIX]{x1D70F}}^{n}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}\subset H[p^{n}],\end{eqnarray}$$

by finite flat sub- ${\mathcal{O}}$ -modules of $H[p^{n}]$ , of ${\mathcal{O}}$ -heights $n$ and $2n$ respectively. Moreover,

$$\begin{eqnarray}\deg _{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n})+p\deg _{\unicode[STIX]{x1D70F}}(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n})\geqslant n(p+2)-\frac{p^{2n}-1}{p^{2}-1}\operatorname{ha}_{\unicode[STIX]{x1D70F}}(H),\end{eqnarray}$$

and

$$\begin{eqnarray}\deg _{\unicode[STIX]{x1D70F}}(H_{\unicode[STIX]{x1D70F}}^{n})+p\deg _{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}(H_{\unicode[STIX]{x1D70F}}^{n})\geqslant n(2p+1)-\frac{p^{2n}-1}{p^{2}-1}\operatorname{ha}_{\unicode[STIX]{x1D70F}}(H).\end{eqnarray}$$

In particular, the groups $H_{\unicode[STIX]{x1D70F}}^{n}$ and $H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ are of high degree. In addition, points of $H_{\unicode[STIX]{x1D70F}}^{n}$ coincide with the kernel of the Hodge–Tate map $\unicode[STIX]{x1D6FC}_{H[p^{n}],\unicode[STIX]{x1D70F},n-((p^{2n}-1)/(p^{2}-1))\operatorname{ha}_{\unicode[STIX]{x1D70F}}(H)}$ and $H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ with the one of $\unicode[STIX]{x1D6FC}_{H[p^{n}],\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},n-((p^{2n}-1)/(p^{2}-1))\operatorname{ha}_{\unicode[STIX]{x1D70F}}(H)}$ . They also coincide with steps of the Harder–Narasimhan filtration and are compatible with $p^{s}$ -torsion ( $s\leqslant n$ ) and quotients.

Proof. This is exactly [Reference HernandezHer19] if $p\geqslant 5$ . Here the bound on $p$ is slightly better than the one of [Reference HernandezHer19]. The reason is that we construct $H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ using [Reference HernandezHer19, Théorème 8.3 and Remarque 8.4] (with our bound for $\operatorname{ha}_{\unicode[STIX]{x1D70F}}$ ). We can then construct $D_{\unicode[STIX]{x1D70F}}^{n}\subset G[p^{n}]^{D}$ of ${\mathcal{O}}$ -height $2n$ by the same reason. Then setting

$$\begin{eqnarray}H_{\unicode[STIX]{x1D70F}}^{n}:=(D_{\unicode[STIX]{x1D70F}}^{n})^{\bot }=(G[p^{n}]^{D}/D_{\unicode[STIX]{x1D70F}}^{n})^{D}\subset G[p^{n}],\end{eqnarray}$$

we can have the asserted bound on $\deg _{\unicode[STIX]{x1D70F}}H_{\unicode[STIX]{x1D70F}}^{n}$ . For the assertion on the Kernel of the Hodge–Tate map, we get an inclusion with the bound on the degree. We can then use [Reference HernandezHer19, Proposition 7.6]. Then by [Reference HernandezHer19, Remarque 8.4], these groups are steps of the Harder–Narasimhan filtration (the classical one), and thus we get the assertion of the inclusion $H_{\unicode[STIX]{x1D70F}}^{n}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ . By unicity of the Harder–Narasimhan filtration, we get that $(H_{\unicode[STIX]{x1D70F}}^{n})^{\bot }=H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ .◻

Definition 5.9. Let $H/\operatorname{Spec}({\mathcal{O}}_{L})$ as before, with $n=2m$ . Then we can consider inside $H[2m]$ the finite flat subgroup,

$$\begin{eqnarray}K_{m}=H_{\unicode[STIX]{x1D70F}}^{2m}+H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}.\end{eqnarray}$$

It coincides, after reduction to $\operatorname{Spec}({\mathcal{O}}_{L}/\unicode[STIX]{x1D70B}_{L})$ (the residue field of $L$ ) with the kernel of $F^{2m}$ of $H[p^{2m}]$ (see [Reference HernandezHer19, § 2.9.1]).

Recall that we denoted by $X/\operatorname{Spec}({\mathcal{O}})$ the (schematic) Picard surface. Denote by $X^{\text{rig}}$ the associated rigid space over $E_{p}$ , there is a specialization map,

$$\begin{eqnarray}\operatorname{sp}:X^{\text{rig}}\longrightarrow \overline{X},\end{eqnarray}$$

and we denote by $X^{\text{ord}}\subset X^{\text{rig}}$ the open subspace defined by $\operatorname{sp}^{-1}(\overline{X}^{\text{ord}})$ .

Let us denote, for $v\in (0,1]$ ,

$$\begin{eqnarray}X(v)=\{x\in X^{\text{rig}}:\operatorname{ha}_{\unicode[STIX]{x1D70F}}(x)=v(\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}(G_{x}))<v\}\quad \text{and}\quad X(0)=X^{\text{ord}},\end{eqnarray}$$

the strict neighborhoods of $X^{\text{ord}}$ . The previous theorem and technics introduced in [Reference FarguesFar10] (see [Reference HernandezHer19, § 2.9]) implies, if $v\leqslant 1/4p^{n-1}$ , that we have a filtration in families over the rigid space $X(v)$ ,

$$\begin{eqnarray}0\subset H_{\unicode[STIX]{x1D70F}}^{n}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}\subset G[p^{n}].\end{eqnarray}$$

Let us explain how to get this result. On $Y^{\text{rig}}(v)$ this is simply [Reference HernandezHer19, Théorème 9.1] which is essentially [Reference FarguesFar10, Théorème 4] (again, see proof of Theorem 5.8 about the bound). The problem is that the $p$ -divisible group $G$ does not extend to the boundary. But by results of Stroh [Reference StrohStr10, § 3.1], there exists

$$\begin{eqnarray}\overline{U}\longrightarrow X^{\text{rig}},\end{eqnarray}$$

which is an étale covering of $X^{\text{rig}}$ (actually $\overline{U}$ is algebraic and exists also integrally) together with $\overline{R}\overset{p_{1}}{\underset{p_{2}}{\rightrightarrows }}\overline{U}$ étale maps, such that

$$\begin{eqnarray}X^{\text{rig}}\simeq [\overline{U}/\overline{R}].\end{eqnarray}$$

Over $\overline{U}$ there is a Mumford $1$ -motive $M=[L\longrightarrow \widetilde{G}]$ such that $M[p^{n}]=A[p^{n}]$ ( $A$ is the semi-abelian scheme), and thus there is a canonical $\widetilde{G}$ semi-abelian scheme with an action of ${\mathcal{O}}_{E}$ of (locally) constant toric rank and thus $\widetilde{G}[p^{n}]$ is finite flat. Thus applying to $\widetilde{G}[p^{n}]$ the results of [Reference HernandezHer19], there exists on $\overline{U}(v)=\overline{U}\times _{X}X(v)$ the two groups $H_{\unicode[STIX]{x1D70F}}^{n}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ . Moreover, over $R(v)\overset{p_{1}}{\underset{p_{2}}{\rightrightarrows }}U(v)$ , we have $pr_{1}^{\ast }H_{\star }^{n}=pr_{2}^{\ast }H_{\star }^{n}$ as $H_{\star }^{n}$ descend to $Y(v)$ . Over $\overline{R}$ we have an isomorphism $pr_{1}^{\ast }G=pr_{2}^{\ast }G$ [Reference StrohStr10, 3.1.5], and thus we have over $R(v)$ an isomorphism

$$\begin{eqnarray}G/pr_{1}^{\ast }H_{\star }^{n}\overset{{\sim}}{\longrightarrow }G/pr_{2}^{\ast }H_{\star }^{n}.\end{eqnarray}$$

As $\overline{R}$ is normal (because smooth) and $R$ is dense in $\overline{R}$ , by [Reference StrohStr10, Théorème 1.1.2] (due to Faltings and Chai) there is an isomorphism over $\overline{R}$

$$\begin{eqnarray}G/pr_{1}^{\ast }H_{\star }^{n}\overset{{\sim}}{\longrightarrow }G/pr_{2}^{\ast }H_{\star }^{n}.\end{eqnarray}$$

Thus, by faithfully flat descent, $H_{\unicode[STIX]{x1D70F}}^{n}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{n}$ exist on $X(v)$ . In particular, $K_{m}=H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}+H_{\unicode[STIX]{x1D70F}}^{2m}$ also exists on $X(v)$ .

A priori, this filtration does not extend to a formal model of $X(v)$ , but as $X/\operatorname{Spec}({\mathcal{O}})$ is a normal scheme, we will be able to use the following proposition.

6.1 Hodge–Tate map and image sheaves

Let $p$ a prime, inert in $E$ , and ${\mathcal{O}}={\mathcal{O}}_{E,p}$ , a degree- $2$ unramified extension of $\mathbb{Z}_{p}$ . Let $K$ be a valued extension of $E_{p}$ . Let $m\in \mathbb{N}^{\times }$ and $v<1/4p^{2m-1}$ . Let $S=\operatorname{Spec}(R)$ where $R$ is an object of $\mathfrak{NAdm}/{\mathcal{O}}_{K}$ , and $G\longrightarrow S$ a truncated $p$ -divisible ${\mathcal{O}}$ -module of level $2m$ and signature,

$$\begin{eqnarray}\left\{\begin{array}{@{}lc@{}}p_{\unicode[STIX]{x1D70F}}=1, & q_{\unicode[STIX]{x1D70F}}=2,\\ p_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=2, & q_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}=1,\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}:{\mathcal{O}}\longrightarrow {\mathcal{O}}_{S}$ is the fixed embedding. Suppose moreover that for all $x\in S^{\text{rig}}$ , $\operatorname{ha}_{\unicode[STIX]{x1D70F}}(x)\leqslant v$ . According to the previous section, there exists on $S^{\text{rig}}$ a filtration of $G[p^{2m}]$ by finite flat ${\mathcal{O}}$ -modules,

$$\begin{eqnarray}0\subset H_{\unicode[STIX]{x1D70F}}^{2m}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{2m}\subset G[p^{2m}],\end{eqnarray}$$

of ${\mathcal{O}}$ -heights $2m$ and $4m$ respectively. Moreover, we have on $S$ a subgroup $K_{m}\subset G[p^{2m}]$ , finite flat of ${\mathcal{O}}$ -height $3m$ , étale-locally isomorphic (on $S^{\text{rig}}$ ) to ${\mathcal{O}}/p^{2m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{m}{\mathcal{O}}$ , and on $S^{\text{rig}}$ , $K_{m}=H_{\unicode[STIX]{x1D70F}}^{2m}+H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{2m}[p^{m}]$ .

Proposition 6.1. Let $w_{\unicode[STIX]{x1D70F}},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\in v({\mathcal{O}}_{K})$ such that $w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}<m-((p^{2m}-1)/(p^{2}-1))v$ and $w_{\unicode[STIX]{x1D70F}}<2m-((p^{4m}-1)/(p^{2}-1))v$ . Then, the morphism of sheaves on $S$ $\unicode[STIX]{x1D70B}:\unicode[STIX]{x1D714}_{G}\longrightarrow \unicode[STIX]{x1D714}_{K_{m}}$ , induce by the inclusion $K_{m}\subset G$ , induces isomorphisms,

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70F}}:\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}}\overset{{\sim}}{\longrightarrow }\unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}}\quad \text{and}\quad \unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}\overset{{\sim}}{\longrightarrow }\unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}.\end{eqnarray}$$

Proof. If $G/\operatorname{Spec}({\mathcal{O}}_{C})$ ( $C$ a complete algebraically closed extension of $\mathbb{Q}_{p}$ ), the degrees of the canonical filtration of $G$ assure that

$$\begin{eqnarray}\deg _{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}(G[p^{m}]/H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m})\geqslant \frac{p^{2m}-1}{p^{2}-1}v\quad \text{and}\quad \deg _{\unicode[STIX]{x1D70F}}(G[p^{2m}]/H_{\unicode[STIX]{x1D70F}}^{2m})\geqslant \frac{p^{4m}-1}{p^{2}-1}v,\end{eqnarray}$$

and there is thus an isomorphism,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{G[p^{m}],\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}}\overset{{\sim}}{\longrightarrow }\unicode[STIX]{x1D714}_{H_{\unicode[STIX]{x1D70F}}^{n},\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

and also for $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}$ and $G[p^{2m}]$ . But there are inclusions $H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}=H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{2m}[p^{m}]\subset K_{m}\subset G$ and $H_{\unicode[STIX]{x1D70F}}^{2m}\subset K_{m}\subset G$ such that the composite,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}\longrightarrow \unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}\longrightarrow \unicode[STIX]{x1D714}_{H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

is an isomorphism, which implies that the first one is. The same reasoning applies for $\unicode[STIX]{x1D70F}$ . We can thus conclude the following for general $S$ as in [Reference Andreatta, Iovita and PilloniAIP15, Proposition 4.2.1]. Up to reducing $R$ we can suppose $\unicode[STIX]{x1D714}_{G}$ is a free $R/p^{2m+1}$ -module, and look at the surjection $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:R^{2}{\twoheadrightarrow}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}{\twoheadrightarrow}\unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}$ ; it is enough to prove that for any $(x_{1},x_{2})$ in $\ker \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ we have $x_{i}\in p^{w_{\unicode[STIX]{x1D70F}}}R$ , but, as $R$ is normal, it suffices to do it for $R_{\mathfrak{p}}$ , and even for $\widehat{R_{\mathfrak{p}}}$ , for all codimension- $1$ prime ideals $\mathfrak{p}$ that contain $(p)$ . But now $\widehat{R_{\mathfrak{p}}}$ is a complete, discrete valuation ring of mixed characteristic, and this reduces to the preceding assertion.◻

Proposition 6.2. Suppose there is an isomorphism $K_{m}^{D}(R)\simeq {\mathcal{O}}/p^{m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{2m}{\mathcal{O}}$ . Then the cokernel of the $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}$ -Hodge–Tate map,

$$\begin{eqnarray}\operatorname{HT}_{K_{m}^{D},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes \,1:K_{m}^{D}(R)[p^{m}]\otimes _{{\mathcal{O}}}R\longrightarrow \unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}},\end{eqnarray}$$

is killed by $p^{(p+v)/(p^{2}-1)}$ , and the cokernel of the $\unicode[STIX]{x1D70F}$ -Hodge–Tate map,

$$\begin{eqnarray}\operatorname{HT}_{K_{m}^{D},\unicode[STIX]{x1D70F}}\otimes \,1:K_{m}^{D}(R)[p^{m}]\otimes _{{\mathcal{O}}}R\longrightarrow \unicode[STIX]{x1D714}_{K_{m},\unicode[STIX]{x1D70F}},\end{eqnarray}$$

is killed by $p^{v/(p^{2}-1)}$ .

Proof. This is true for $G/\operatorname{Spec}({\mathcal{O}}_{C})$ by [Reference HernandezHer19, Théorème 6.10(2)] with the previous proposition (because $(p+v)/(p^{2}-1)<1-v$ , already for $m=1$ ). For a general normal $R$ , we can reduce to previous case (see also [Reference Andreatta, Iovita and PilloniAIP15, Proposition 4.2.2]): up to reducing $\operatorname{Spec}(R)$ , we have a diagram,

and $\operatorname{Fitt}^{1}(\unicode[STIX]{x1D6FE})$ (which is just a determinant here) annihilates the cokernel of $\unicode[STIX]{x1D6FE}$ , and it suffices to prove that $p^{(p+v)/(p^{2}-1)}\in \operatorname{Fitt}^{1}(\unicode[STIX]{x1D6FE})$ . But as $R$ is normal, it suffices to prove that $p^{(p+v)/(p^{2}-1)}\in \operatorname{Fitt}^{1}(\unicode[STIX]{x1D6FE})R_{\mathfrak{p}}$ for every codimension- $1$ prime ideal $\mathfrak{p}$ that contains $(p)$ . But by the previous case, we can conclude. The same works for $\unicode[STIX]{x1D70F}$ .◻

Proposition 6.3. Suppose we have an isomorphism $K_{m}^{D}(R)\simeq {\mathcal{O}}/p^{m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{2m}{\mathcal{O}}$ . Then there exist on $S=\operatorname{Spec}R$ locally free subsheaves ${\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}},{\mathcal{F}}_{\unicode[STIX]{x1D70F}}$ of $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}$ respectively, of ranks $2$ and $1$ , which contain $p^{(p+v)/(p^{2}-1)}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ and $p^{v/(p^{2}-1)}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}$ , and which are equipped, for all $w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}<m-((p^{2m}-1)/(p^{2}-1))v$ and $w_{\unicode[STIX]{x1D70F}}<2m-((p^{4m}-1)/(p^{2}-1))v$ , with maps,

$$\begin{eqnarray}\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}:K_{m}^{D}(R)\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}R_{w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}},\quad \text{and}\quad \operatorname{HT}_{\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}}:K_{m}^{D}(R)\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70F}}\otimes _{R}R_{w_{\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

which are surjective after tensoring $K_{m}^{D}(R)$ with $R$ over ${\mathcal{O}}$ .

More precisely, via the projection,

$$\begin{eqnarray}K_{m}^{D}(R){\twoheadrightarrow}(H_{\unicode[STIX]{x1D70F}}^{2m})^{D}(R_{K}),\end{eqnarray}$$

we have induced isomorphisms,

$$\begin{eqnarray}\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}:K_{m}^{D}(R_{K})\otimes _{{\mathcal{O}}}R_{w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}}\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}R_{w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

and

$$\begin{eqnarray}\operatorname{HT}_{\unicode[STIX]{x1D70F},w_{\unicode[STIX]{x1D70F}}}:(H_{\unicode[STIX]{x1D70F}}^{2m})^{D}(R_{K})\otimes _{{\mathcal{O}}}R_{w_{\unicode[STIX]{x1D70F}}}\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70F}}\otimes _{R}R_{w_{\unicode[STIX]{x1D70F}}}.\end{eqnarray}$$

Moreover the construction of the sheaves ${\mathcal{F}}$ is functorial in the following sense.

Proposition 6.4. Suppose given $G,G^{\prime }$ two truncated $p$ -divisible ${\mathcal{O}}$ -module such that for all $x\in S^{\text{rig}}$

$$\begin{eqnarray}\operatorname{ha}_{\unicode[STIX]{x1D70F}}(G_{x}),\operatorname{ha}_{\unicode[STIX]{x1D70F}}(G_{x}^{\prime })<v,\end{eqnarray}$$

and an ${\mathcal{O}}_{E}$ -isogeny,

$$\begin{eqnarray}\unicode[STIX]{x1D719}:G\longrightarrow G^{\prime }.\end{eqnarray}$$

Assume moreover that we are given trivializations of the points of $K_{m}^{D}(G)$ and $K_{m}^{D}(G^{\prime })$ . Then $\unicode[STIX]{x1D719}^{\ast }$ induces maps

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }:{\mathcal{F}}_{\unicode[STIX]{x1D70F}}^{\prime }\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70F}}^{\prime }\quad \text{and}\quad \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{\ast }:{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{\prime }\longrightarrow {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}},\end{eqnarray}$$

that are compatible with inclusion in $\unicode[STIX]{x1D714}$ , reduction modulo $p^{w}$ and the Hodge–Tate maps of $K_{m}^{D}$ .

6.2 The torsors

To simplify the notations, fix $w=w_{\unicode[STIX]{x1D70F}}=w_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}<m-((p^{2m}-1)/(p^{2}-1))v$ to use the previous propositions. Let $R\in {\mathcal{O}}_{K}-\mathfrak{NAdm}$ and $S=\operatorname{Spf}(R)$ . In rigid fiber, we have a subgroup of $K_{m}[p^{m}]/S^{\text{rig}}$ , $H_{\unicode[STIX]{x1D70F}}^{m}\subset K_{m}[p^{m}]$ which induces a filtration,

$$\begin{eqnarray}0\subset (H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}(R_{K})\subset K_{m}^{D}(R),\end{eqnarray}$$

of cokernel isomorphic to $(H_{\unicode[STIX]{x1D70F}}^{m})^{D}(R_{K})$ .

Suppose we are given a trivialization,

$$\begin{eqnarray}\unicode[STIX]{x1D713}:{\mathcal{O}}/p^{m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{2m}{\mathcal{O}}\simeq K_{m}^{D}(R),\end{eqnarray}$$

which induces trivializations (first coordinate and quotient),

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}(R_{K})\simeq {\mathcal{O}}/p^{m}{\mathcal{O}}\quad \text{and}\quad \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}:(H_{\unicode[STIX]{x1D70F}}^{2m})^{D}(R_{K})\simeq {\mathcal{O}}/p^{2m}{\mathcal{O}}.\end{eqnarray}$$

Let ${\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\longrightarrow S$ be the Grassmanian of locally direct factor sheaves of rank $1$ , $\operatorname{Fil}^{1}{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\subset {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ . Let ${\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{+}\longrightarrow {\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ be the $\mathbb{G}_{m}^{2}$ -torsor of trivializations of $\operatorname{Fil}^{1}{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ and ${\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}/\text{Fil}^{1}{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ . Also let ${\mathcal{G}}r_{\unicode[STIX]{x1D70F}}^{+}\longrightarrow S$ be the $\mathbb{G}_{m}$ -torsor of trivializations of ${\mathcal{F}}_{\unicode[STIX]{x1D70F}}$ .

Definition 6.6. We say that a $A$ -point of ${\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ (respectively ${\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{+}$ or ${\mathcal{G}}r_{\unicode[STIX]{x1D70F}}^{+}$ )

$$\begin{eqnarray}\operatorname{Fil}^{1}({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes A)\quad (\text{respectively }(\operatorname{Fil}^{1}({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes A),P_{1}^{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}},P_{2}^{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}})\text{ or }P^{\unicode[STIX]{x1D70F}})\end{eqnarray}$$

is $w$ -compatible with $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ if, respectively:

1. (i) $\operatorname{Fil}^{1}({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes A)\otimes _{R}R_{w}=\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}((H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}(R_{K})\otimes _{{\mathcal{O}}}R_{w})\otimes _{R}A$ ;

2. (ii) $P_{1}^{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}R_{w}=\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}\circ (\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}}R_{w})\otimes _{R}A$ ;

3. (iii) $P_{2}^{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}R_{w}=\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}\circ (\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}}R_{w})\otimes _{R}A$ ;

4. (iv) $P^{\unicode[STIX]{x1D70F}}\otimes _{R}R_{w}=\operatorname{HT}_{\unicode[STIX]{x1D70F},w}\circ (\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}}R_{w})\otimes _{R}A$ .

We can define the functors,

$$\begin{eqnarray}\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}:\begin{array}{@{}ccc@{}}R-\mathfrak{Adm} & \longrightarrow & SET\\ A & \longmapsto & \{w-\text{compatible}\operatorname{Fil}^{1}({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}A)\in {\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}(A)\},\end{array}\end{eqnarray}$$

$$\begin{eqnarray}\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{+}:\begin{array}{@{}ccc@{}}R-\mathfrak{Adm} & \longrightarrow & SET\\ A & \longmapsto & \{w-\text{compatible}(\operatorname{Fil}^{1}({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{R}A),P_{1}^{\unicode[STIX]{x1D70F}},P_{2}^{\unicode[STIX]{x1D70F}})\in {\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{+}(A)\},\end{array}\end{eqnarray}$$

$$\begin{eqnarray}\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70F},w}^{+}:\begin{array}{@{}ccc@{}}R-\mathfrak{Adm} & \longrightarrow & SET\\ A & \longmapsto & \{w-\text{compatible}P^{\unicode[STIX]{x1D70F}}\in {\mathcal{G}}r_{\unicode[STIX]{x1D70F}}^{+}(A)\}.\end{array}\end{eqnarray}$$

The previous functors are representable by formal schemes, affine over $S=\operatorname{Spf}(R)$ , and locally isomorphic to

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}1 & \\ p^{w}\mathfrak{B}(0,1) & 1\end{array}\right)\times _{\operatorname{Spf}({\mathcal{O}}_{K})}\operatorname{Spf}(R)\quad \text{for }\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w},\quad 1+p^{w}\mathfrak{B}(0,1)\quad \text{for }\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70F},w}^{+}\end{eqnarray}$$

where $\mathfrak{B}(0,1)=\operatorname{Spf}({\mathcal{O}}_{K}\langle T\rangle )$ is the one-dimensional formal unit ball, and

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}1+p^{w}\mathfrak{B}(0,1) & \\ p^{w}\mathfrak{B}(0,1) & 1+p^{w}\mathfrak{B}(0,1)\end{array}\right)\times _{\operatorname{Spf}(O_{K})}\operatorname{Spf}(R)\quad \text{for }\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{+}.\end{eqnarray}$$

We also define $\mathfrak{I}\mathfrak{W}_{w}^{+}=\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70F},w}^{+}\times _{S}\mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{+}$ . The previous constructions are independent of $n=2m$ (because ${\mathcal{F}}_{\unicode[STIX]{x1D70F}},{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ are).

Let $T^{1}=\operatorname{Res}_{{\mathcal{O}}/\mathbb{Z}_{p}}\mathbb{G}_{m}\times U(1)$ the torus of $U(2,1)$ over $\mathbb{Z}_{p}$ whose $\mathbb{Z}_{p}$ -points are ${\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}$ . Its scalar extension $T=T^{1}\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}$ is isomorphic to $\mathbb{G}_{m}^{3}$ , and ${\mathcal{G}}r^{+}={\mathcal{G}}r_{\unicode[STIX]{x1D70F}}^{+}\times {\mathcal{G}}r_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{+}\longrightarrow {\mathcal{G}}r={\mathcal{G}}r_{\unicode[STIX]{x1D70F}}$ is a $T$ -torsor. Denote by $\mathfrak{T}\longrightarrow \operatorname{Spf}({\mathcal{O}})$ the formal completion of $T$ along its special fiber, and $\mathfrak{T}_{w}$ the torus defined by

$$\begin{eqnarray}\mathfrak{T}_{w}(A)=\operatorname{Ker}(\mathfrak{T}(A))\longrightarrow \mathfrak{T}(A/p^{w}A).\end{eqnarray}$$

Then $\mathfrak{I}\mathfrak{W}_{w}^{+}\longrightarrow \mathfrak{I}\mathfrak{W}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}$ is a $\mathfrak{T}_{w}$ -torsor.

Denote by ${\mathcal{I}}{\mathcal{W}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w},{\mathcal{I}}{\mathcal{W}}_{\unicode[STIX]{x1D70F},w}^{+},{\mathcal{I}}{\mathcal{W}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{+},{\mathcal{I}}{\mathcal{W}}_{w}^{+},{\mathcal{T}}$ the generic fibers of the previous formal schemes.

7.1 Constructing automorphic sheaves

Let us consider the datum $(E,V,\unicode[STIX]{x1D713},{\mathcal{O}}_{E},\unicode[STIX]{x1D6EC}={\mathcal{O}}_{E}^{3},h)$ , the PEL datum introduced in § 2. Let $p$ be a prime, inert in $E$ and $G$ the reductive group associated over $\mathbb{Z}_{p}$ . We fix $K^{p}$ a compact open subgroup of $G(\mathbb{A}_{f}^{p})$ sufficiently small and $\mathfrak{C}=G(\mathbb{Z}_{p})$ an hyperspecial subgroup at $p$ . Let $X=X_{K^{p}\mathfrak{C}}/\text{Spec}({\mathcal{O}})$ the (integral) Picard variety associated to the previous datum (cf. [Reference KottwitzKot92, Reference LanLan13, Reference Langlands and RamakrishnanLR92]).

Let $K/{\mathcal{O}}[1/p]$ be a finite extension (that we will choose sufficiently large) and still write $X=X_{{\mathcal{O}}_{K}}=X\times _{\operatorname{Spec}{\mathcal{O}}}\operatorname{Spec}({\mathcal{O}}_{K})$ .

Denote by $A$ the universal semi-abelian scheme, $X^{\text{rig}}$ the rigid fiber of $X$ , $X^{\text{ord}}$ the ordinary locus and for $v\in v(K)$ , $X(v)$ the rigid-analytic open $\{x\in X^{\text{rig}}:\operatorname{ha}_{\unicode[STIX]{x1D70F}}(x)<v\}$ . Denote also $\mathfrak{X}\longrightarrow \operatorname{Spf}({\mathcal{O}}_{K})$ the formal completion of $X$ along its special fiber, $\widetilde{\mathfrak{X}(v)}$ the admissible blow up of $\mathfrak{X}$ along the ideal $(\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}},p^{v})$ and $\mathfrak{X}(v)$ its open subscheme where $(\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}},p^{v})$ is generated by $\widetilde{\operatorname{ha}_{\unicode[STIX]{x1D70F}}}$ .

Proof. For $R$ a (noetherian) ring denote by $R^{\text{norm}}$ its integral closure in its total ring of fractions. Write, for all $x\in A_{K}$ ,

$$\begin{eqnarray}v_{A}(x)=\sup \{n\in \mathbb{Z}:\unicode[STIX]{x1D70B}_{K}^{-n}x\in A\}\quad \text{and}\quad |x|_{A}=\unicode[STIX]{x1D70B}_{K}^{-v_{A}(x)}.\end{eqnarray}$$

Then we can check that $|f+g|_{A}\leqslant \sup (|f|_{A},|g|_{A})$ and that $|f^{n}|_{A}=|f|_{A}^{n}$ . Indeed, $\unicode[STIX]{x1D70B}_{K}^{-v_{A}(f)}f\in A\backslash \unicode[STIX]{x1D70B}_{K}A$ . Thus, as $A/\unicode[STIX]{x1D70B}_{K}A$ is reduced, $(\unicode[STIX]{x1D70B}_{K}^{-v_{A}(f)}f)^{n}\in A\backslash \unicode[STIX]{x1D70B}_{K}A$ and thus $v_{A}(f^{n})=nv_{A}(f)$ . For this norm, we have that

$$\begin{eqnarray}A=\{x\in A_{K}:|x|_{A}\leqslant 1\}.\end{eqnarray}$$

Now let us verify that $A$ is normal. Let $x\in A^{\text{norm}}$ , in particular, $x\in A_{K}^{\text{norm}}$ . However, as $A_{K}$ is normal, $x\in A_{K}$ . Now write, for $a_{i}\in A,0\leqslant i\leqslant n$ and $a_{n}=1$ ,

$$\begin{eqnarray}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x=-a_{0}.\end{eqnarray}$$

Then $\sup _{1\leqslant i\leqslant n}|x^{i}a_{n-i}|_{A}=|a_{0}|_{A}\leqslant 1$ , thus $|x^{n}|_{A}=|x|_{A}^{n}\leqslant 1$ , and $|x|_{A}\leqslant 1$ . Thus $x\in A$ .◻

By the previous sections, we have on $X(v)$ a filtration of $G$ by finite flat ${\mathcal{O}}$ -modules,

$$\begin{eqnarray}0\subset H_{\unicode[STIX]{x1D70F}}^{2m}\subset H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{2m}\subset G[p^{2m}],\end{eqnarray}$$

locally isomorphic to ${\mathcal{O}}/p^{2m}{\mathcal{O}}$ and $({\mathcal{O}}/p^{2m}{\mathcal{O}})^{2}$ . Moreover, the subgroup $K_{m}=H_{\unicode[STIX]{x1D70F}}^{2m}+H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{2m}[p^{m}]$ extends to $\mathfrak{X}(v)$ by Proposition 5.11, and over $X(v)$ is locally isomorphic to

$$\begin{eqnarray}{\mathcal{O}}/p^{2m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{m}{\mathcal{O}}.\end{eqnarray}$$

Definition 7.3. We write

$$\begin{eqnarray}X_{1}(p^{2m})(v)=\operatorname{Isom}_{X(v),\text{pol}}(K_{m}^{D},{\mathcal{O}}/p^{m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{2m}{\mathcal{O}}),\end{eqnarray}$$

where the condition pol means that we are looking at isomorphisms $\unicode[STIX]{x1D713}=(\unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2})$ which induce an isomorphism ‘in first coordinate’,

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{1,1}=(\unicode[STIX]{x1D713}_{1})_{|(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}}:(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}\simeq {\mathcal{O}}/p^{m}{\mathcal{O}},\end{eqnarray}$$

such that $(\unicode[STIX]{x1D713}_{1,1})^{D}=((\unicode[STIX]{x1D713}_{1,1})^{(\unicode[STIX]{x1D70E})})^{-1}$ , and such that the quotient morphism,

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{2,1}=\unicode[STIX]{x1D713}_{1}/(\unicode[STIX]{x1D713}_{1})_{|(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D}}:(H_{\unicode[STIX]{x1D70F}}^{2m})^{D}\longrightarrow {\mathcal{O}}/p^{m}{\mathcal{O}},\end{eqnarray}$$

is zero.

Remark 7.4. The map $\unicode[STIX]{x1D713}_{1,1}$ is automatically an isomorphism. Moreover,

$$\begin{eqnarray}(\unicode[STIX]{x1D713}_{1,1})^{D}:{\mathcal{O}}/p^{m}{\mathcal{O}}\longrightarrow H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m}\overset{\unicode[STIX]{x1D706}}{\simeq }(H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D,(\unicode[STIX]{x1D70E})},\end{eqnarray}$$

where the last morphism is induced by $\unicode[STIX]{x1D706}$ , the polarization of $A$ .

Denote by $B_{n}$ the subgroup of $\operatorname{GL}({\mathcal{O}}/p^{m}{\mathcal{O}}\oplus {\mathcal{O}}/p^{2m}{\mathcal{O}})$ of matrices,

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}a & p^{m}b\\ 0 & d\end{array}\right)\end{eqnarray}$$

such that $a^{-1}=a^{(\unicode[STIX]{x1D70E})}$ i.e. $a\in ({\mathcal{O}}/p^{m}{\mathcal{O}})^{1}$ . We can map ${\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}$ (diagonally) to $B_{n}$ .

$$\begin{eqnarray}B_{n}\simeq \left(\begin{array}{@{}cc@{}}({\mathcal{O}}/p^{m}{\mathcal{O}})^{1} & {\mathcal{O}}/p^{m}{\mathcal{O}}\\ & ({\mathcal{O}}/p^{2m}{\mathcal{O}})^{\times }\end{array}\right).\end{eqnarray}$$

Write also

$$\begin{eqnarray}B_{\infty }(\mathbb{Z}_{p})=\left(\begin{array}{@{}cc@{}}{\mathcal{O}}^{1} & {\mathcal{O}}\\ 0 & {\mathcal{O}}^{\times }\end{array}\right)\end{eqnarray}$$

which surjects to $B_{n}$ and that we can embed into $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ (even in its upper triangular Borel subgroup) via

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}a & b\\ 0 & d\end{array}\right)\longmapsto \left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(a) & \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(b) & \\ & \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(d) & \\ & & \unicode[STIX]{x1D70F}(d)\end{array}\right).\end{eqnarray}$$

We denote by $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ the inverses of the induced isomorphisms,

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:{\mathcal{O}}/p^{m}{\mathcal{O}}\simeq (H_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}^{m}/H_{\unicode[STIX]{x1D70F}}^{m})^{D},\end{eqnarray}$$

and the quotient,

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D713}^{-1}/\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:{\mathcal{O}}/p^{2n}{\mathcal{O}}\simeq (H_{\unicode[STIX]{x1D70F}}^{2m})^{D}.\end{eqnarray}$$

We also denote by $\mathfrak{X}_{1}(p^{2m})(v)$ the normalization of $\mathfrak{X}(v)$ in $X_{1}(p^{2m})(v)$ . Over $\mathfrak{X}_{1}(p^{2m})(v)$ , we have, by the previous section, locally free subsheaves of ${\mathcal{O}}_{\mathfrak{X}_{1}(p^{2m})(v)}$ -modules ${\mathcal{F}}_{\unicode[STIX]{x1D70F}},{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ of $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ together with morphisms,

$$\begin{eqnarray}\operatorname{HT}_{\unicode[STIX]{x1D70F},w}\circ \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}[p^{m}]:({\mathcal{O}}/p^{m}{\mathcal{O}})\otimes _{{\mathcal{O}}}{\mathcal{O}}_{\mathfrak{ X}_{1}(p^{2m})(v)}\overset{{\sim}}{\longrightarrow }{\mathcal{F}}_{\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}_{K}}{\mathcal{O}}_{K}/p^{w},\end{eqnarray}$$

$$\begin{eqnarray}\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}\circ \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}:({\mathcal{O}}/p^{m}{\mathcal{O}})\otimes _{{\mathcal{O}}}{\mathcal{O}}_{\mathfrak{ X}_{1}(p^{2m})(v)}{\hookrightarrow}{\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}_{K}}{\mathcal{O}}_{K}/p^{w},\end{eqnarray}$$

and denote by ${\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{\text{can}}$ the image of the second morphism. It is a locally direct factor of ${\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes {\mathcal{O}}_{K}/p^{w}$ , and, passing through the quotient, we get a map,

$$\begin{eqnarray}\overline{\operatorname{HT}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}}\circ \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70F}}[p^{m}]:({\mathcal{O}}/p^{m}{\mathcal{O}})\otimes _{{\mathcal{O}}}{\mathcal{O}}_{\mathfrak{ X}_{1}(p^{m})(v)}\overset{{\sim}}{\longrightarrow }({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes _{{\mathcal{O}}_{K}}{\mathcal{O}}_{K}/p^{w})/({\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F},w}^{\text{can}}).\end{eqnarray}$$

Using the construction of torsors of the previous section, we get a chain of maps,

$$\begin{eqnarray}\mathfrak{I}\mathfrak{W}_{w}^{+}\overset{\unicode[STIX]{x1D70B}_{1}}{\longrightarrow }\mathfrak{I}\mathfrak{W}_{w}\overset{\unicode[STIX]{x1D70B}_{2}}{\longrightarrow }\mathfrak{X}_{1}(p^{2m})(v)\overset{\unicode[STIX]{x1D70B}_{3}}{\longrightarrow }\mathfrak{X}(v).\end{eqnarray}$$

Moreover, $\unicode[STIX]{x1D70B}_{1}$ is a torsor over the formal torus $\mathfrak{T}_{w}$ , $\unicode[STIX]{x1D70B}_{2}$ is affine, and we have an action of ${\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}$ and $B_{n}$ on $\mathfrak{X}_{1}(p^{2m})$ over $\mathfrak{X}(v)$ . Denote by $B$ the Borel subgroup of $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ , $\mathfrak{B}$ its formal completion along its special fiber, and $\mathfrak{B}_{w}$ ,

$$\begin{eqnarray}\mathfrak{B}_{w}(A)=\operatorname{Ker}(\mathfrak{B}(A)\longrightarrow \mathfrak{B}(A/p^{w}A)).\end{eqnarray}$$

We can embed $\mathfrak{T}$ in $\mathfrak{B}$ (which induce an embedding $\mathfrak{T}_{w}\subset \mathfrak{B}_{w}$ ) and ${\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}$ in $\mathfrak{T}$ , via

$$\begin{eqnarray}(a,b)\in {\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}\longmapsto \left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(b) & & \\ & \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(a) & \\ & & \unicode[STIX]{x1D70F}(a)\end{array}\right)\in \mathfrak{T},\end{eqnarray}$$

such that the action of ${\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}$ on $X_{1}(p^{2m})(v)$ and via $\mathfrak{T}$ on ${\mathcal{G}}r^{+}$ preserves $\mathfrak{I}\mathfrak{W}_{w}^{+}$ (over $\mathfrak{X}(v)$ ). More generally, the action of $B_{\infty }(\mathbb{Z}_{p})$ on $X_{1}(p^{2m})(v)$ (and thus $\mathfrak{X}_{1}(p^{2m})(v)$ ) and via $\mathfrak{B}$ on ${\mathcal{G}}r^{+}$ preserves $\mathfrak{I}\mathfrak{W}_{w}^{+}$ .

Let $\unicode[STIX]{x1D705}\in {\mathcal{W}}_{w}(L)$ . The character $\unicode[STIX]{x1D705}:{\mathcal{O}}^{\times }\times {\mathcal{O}}^{1}\longrightarrow {\mathcal{O}}_{L}^{\times }$ extends to a character,

$$\begin{eqnarray}\unicode[STIX]{x1D705}:({\mathcal{O}}^{\times }\times {\mathcal{O}}^{1})\mathfrak{T}_{w}\longrightarrow \widehat{\mathbb{G}_{m}},\end{eqnarray}$$

which can be extended as a character of

$$\begin{eqnarray}\unicode[STIX]{x1D705}:({\mathcal{O}}^{\times }\times {\mathcal{O}}^{1})\mathfrak{B}_{w}\longrightarrow \widehat{\mathbb{G}_{m}},\end{eqnarray}$$

where $\mathfrak{U}_{w}\subset \mathfrak{B}_{w}$ acts trivially, and even as a character,

$$\begin{eqnarray}\unicode[STIX]{x1D705}:B(\mathbb{Z}_{p})\mathfrak{B}_{w}\longrightarrow \widehat{\mathbb{G}_{m}},\end{eqnarray}$$

where $U(\mathbb{Z}_{p})\mathfrak{U}_{w}$ acts trivially. Let us write $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{3}\circ \unicode[STIX]{x1D70B}_{2}\circ \unicode[STIX]{x1D70B}_{1}$ .

Definition 7.6. We call $\mathfrak{w}_{w}^{\unicode[STIX]{x1D705}\dagger }:=\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{\mathfrak{I}\mathfrak{W}_{w}^{+}}[\unicode[STIX]{x1D705}]$ the sheaf of $v$ -overconvergent $w$ -analytic modular forms of weight $\unicode[STIX]{x1D705}$ . The space of integral $v$ -overconvergent, $w$ -analytic modular forms of weight $\unicode[STIX]{x1D705}$ and level (outside $p$ ) $K^{p}$ , for the group $G$ is

$$\begin{eqnarray}M_{w}^{\unicode[STIX]{x1D705}\dagger }(\mathfrak{X}(v))=H^{0}(\mathfrak{X}(v),\mathfrak{w}_{w}^{\unicode[STIX]{x1D705}\dagger }).\end{eqnarray}$$

7.2 Changing the analytic radius

Let $m-((p^{2m}-1)/(p^{2}-1))v>w^{\prime }>w$ and $\unicode[STIX]{x1D705}\in \mathfrak{W}_{w}(L)$ , and thus $\unicode[STIX]{x1D705}\in \mathfrak{W}_{w^{\prime }}(L)$ . There is a natural inclusion,

$$\begin{eqnarray}\mathfrak{I}\mathfrak{W}_{w^{\prime }}^{+}{\hookrightarrow}\mathfrak{I}\mathfrak{W}_{w}^{+},\end{eqnarray}$$

compatible with the action of $({\mathcal{O}}^{\times }\times {\mathcal{O}}^{1})\mathfrak{B}_{w}$ . This induces a map $\mathfrak{w}_{w}^{\unicode[STIX]{x1D705}\dagger }\longrightarrow \mathfrak{w}_{w^{\prime }}^{\unicode[STIX]{x1D705}\dagger }$ and thus a map,

$$\begin{eqnarray}M_{w}^{\dagger \unicode[STIX]{x1D705}}(\mathfrak{X}(v))\longrightarrow M_{w^{\prime }}^{\dagger \unicode[STIX]{x1D705}}(\mathfrak{X}(v)).\end{eqnarray}$$

Definition 7.8. The space of integral overconvergent locally analytic Picard modular forms of weight $\unicode[STIX]{x1D705}$ , and level (outside $p$ ) $K^{p}$ , is

$$\begin{eqnarray}M_{\unicode[STIX]{x1D705}}^{\dagger }(\mathfrak{X})=\mathop{\varinjlim }\nolimits_{v\rightarrow 0,w\rightarrow \infty }M_{w}^{\unicode[STIX]{x1D705}\dagger }(\mathfrak{X}(v)).\end{eqnarray}$$

7.3 Classical and overconvergent forms in rigid fiber

Denote by $\mathfrak{X}_{Iw^{+}(p^{2m})}(v)$ the quotient of $\mathfrak{X}_{1}(p^{2m})(v)$ by $\tilde{U_{m}}\subset B_{m}$ , which is isomorphic to

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}1 & {\mathcal{O}}/p^{m}{\mathcal{O}}\\ & 1+p^{m}{\mathcal{O}}/p^{2m}{\mathcal{O}}\end{array}\right)\subset \left(\begin{array}{@{}cc@{}}({\mathcal{O}}/p^{m}{\mathcal{O}})^{\times } & {\mathcal{O}}/p^{m}{\mathcal{O}}\\ & ({\mathcal{O}}/p^{2m}{\mathcal{O}})^{\times }\end{array}\right).\end{eqnarray}$$

Let us also denote by $X_{Iw^{+}(p^{2m})}(v)$ the corresponding rigid space. Over the scheme $X$ , we have the locally free sheaf $\unicode[STIX]{x1D714}_{A}=\unicode[STIX]{x1D714}_{A,\unicode[STIX]{x1D70F}}\oplus \unicode[STIX]{x1D714}_{A,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ , which is locally isomorphic to ${\mathcal{O}}_{X}\oplus {\mathcal{O}}_{X}^{2}$ , with the corresponding action of ${\mathcal{O}}$ . Denote by ${\mathcal{T}}$ the scheme $\operatorname{Hom}_{X,{\mathcal{O}}}({\mathcal{O}}_{X}^{2}\oplus {\mathcal{O}}_{X},\unicode[STIX]{x1D714}_{G})$ of trivialization of $\unicode[STIX]{x1D714}_{G}$ as a ${\mathcal{O}}_{X}\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}={\mathcal{O}}_{X}\,\oplus \,{\mathcal{O}}_{X}$ -sheaf, and denote by ${\mathcal{T}}^{\times }$ its subsheaf of isomorphisms; it is a $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ -torsor over $X$ , where $g\in \operatorname{GL}_{2}\times \operatorname{GL}_{1}$ acts on ${\mathcal{T}}^{\times }$ by $g\cdot \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}\circ g^{-1}$ . For $\unicode[STIX]{x1D705}\in X^{+}(T)$ a classical weight, denote by $\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}$ the sheaf $\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{T}}^{\times }}[\unicode[STIX]{x1D705}^{\prime }]$ , where $\unicode[STIX]{x1D70B}:{\mathcal{T}}^{\times }\longrightarrow X$ is the projection and $\unicode[STIX]{x1D705}\longrightarrow \unicode[STIX]{x1D705}^{\prime }$ the involution on classical weights. In down-to-earth terms, $\unicode[STIX]{x1D705}=(k_{1},k_{2},l)$ where $k_{1}\geqslant k_{2}$ and

$$\begin{eqnarray}\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}=\operatorname{Sym}^{k_{1}-k_{2}}\unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\otimes (\det \unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}})^{\otimes k_{2}}\otimes (\det \unicode[STIX]{x1D714}_{G,\unicode[STIX]{x1D70F}})^{\otimes l}.\end{eqnarray}$$

We have defined $X(v)$ , which is the rigid fiber of $\mathfrak{X}(v)$ . Denote by ${\mathcal{T}}_{\text{an}},{\mathcal{T}}_{\text{an}}^{\times },(\operatorname{GL}_{2}\times \operatorname{GL}_{1})_{\text{an}}$ the analytification of the schemes ${\mathcal{T}},{\mathcal{T}}^{\times },\operatorname{GL}_{g}$ , and ${\mathcal{T}}_{\text{rig}},{\mathcal{T}}_{\text{rig}}^{\times },(\operatorname{GL}_{2}\times \operatorname{GL}_{1})_{\text{rig}}$ Raynaud’s rigid fiber of the completion along the special fibers of the same schemes. As ${\mathcal{T}}^{\times }/B$ is complete, ${\mathcal{T}}_{\text{an}}^{\times }/B_{\text{an}}={\mathcal{T}}_{\text{rig}}^{\times }/B_{\text{rig}}$ , over which there is the diagram,

where $f$ is a torsor over $U_{\text{rig}}/B_{\text{rig}}=T_{\text{rig}}$ (the torus, not to be mistaken with ${\mathcal{T}}_{\text{rig}}$ ) and $g$ a torsor over $T_{\text{an}}$ (same remark).

Definition 7.9. Let $\unicode[STIX]{x1D705}\in {\mathcal{W}}_{w}(K)$ . We denote by $\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}\dagger }$ the rigid fiber of $\mathfrak{w}_{w}^{\unicode[STIX]{x1D705}\dagger }$ on $X(v)$ . It exists by [Reference Andreatta, Iovita and PilloniAIP15, Proposition A.2.2.4]. It is called the sheaf of $w$ -analytic overconvergent modular forms of weight $\unicode[STIX]{x1D705}$ . The space of $v$ -overconvergent, $w$ -analytic modular forms of weight $\unicode[STIX]{x1D705}$ is the space,

$$\begin{eqnarray}H^{0}(X(v),\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}\dagger }).\end{eqnarray}$$

The space of locally analytic overconvergent Picard modular forms of weight $\unicode[STIX]{x1D705}$ (and level $K^{p}$ ) is the space,

$$\begin{eqnarray}M_{\unicode[STIX]{x1D705}}^{\dagger }(X)=\mathop{\varinjlim }\nolimits_{v\rightarrow 0,w\rightarrow \infty }H^{0}(X(v),\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}\dagger }).\end{eqnarray}$$

The injection of ${\mathcal{O}}_{\mathfrak{X}(v)}$ -modules ${\mathcal{F}}_{\unicode[STIX]{x1D70F}}\oplus {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\subset \unicode[STIX]{x1D714}_{A}=\unicode[STIX]{x1D714}_{A,\unicode[STIX]{x1D70F}}\oplus \unicode[STIX]{x1D714}_{A,\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ is an isomorphism in generic fiber, and this induces an open immersion,

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{W}}_{w}{\hookrightarrow}{\mathcal{T}}_{\text{rig}}^{\times }/B_{\text{rig}}\times _{X(v)}X_{1}(p^{2m})(v).\end{eqnarray}$$

We also have an open immersion,

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{W}}_{w}^{+}{\hookrightarrow}{\mathcal{T}}_{\text{an}}^{\times }/U_{\text{an}}\times _{X(v)}X_{1}(p^{2m})(v).\end{eqnarray}$$

The action of $B_{n}$ on $\mathfrak{X}_{1}(p^{2m})(v)$ (or $X_{1}(p^{2m})(v)$ ) lifts to an action on $\mathfrak{I}\mathfrak{W}_{w}$ (or ${\mathcal{I}}{\mathcal{W}}_{w}$ ) because being $w$ -compatible for $\operatorname{Fil}^{1}{\mathcal{F}}_{\unicode[STIX]{x1D70F}}$ it only depends on the trivialization of $K_{n}^{D}$ modulo $B_{n}$ . Similarly the action of $\tilde{U_{n}}$ lifts to $\mathfrak{I}\mathfrak{W}_{w}^{+}$ and ${\mathcal{I}}{\mathcal{W}}_{w}^{+}$ . We can thus define ${\mathcal{I}}{\mathcal{W}}_{w}^{0}$ and ${\mathcal{I}}{\mathcal{W}}_{w}^{+,0}$ the respective quotients of ${\mathcal{I}}{\mathcal{W}}_{w}$ and ${\mathcal{I}}{\mathcal{W}}_{w}^{+}$ by $B_{n}$ and $\tilde{U_{n}}$ , which induces open immersions,

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{W}}_{w}^{0}{\hookrightarrow}{\mathcal{T}}_{\text{rig}}^{\times }/B_{\text{rig}}\times _{X(v)}X(v)\quad \text{and}\quad {\mathcal{I}}{\mathcal{W}}_{w}^{+,0}{\hookrightarrow}{\mathcal{T}}_{\text{an}}^{\times }/U_{\text{an}}\times _{X(v)}X_{Iw^{+}(p^{2m})}(v).\end{eqnarray}$$

Proposition 7.10. Suppose $w>m-1+(p+v)/(p^{2}-1)$ . Then there are embeddings

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{W}}_{w}^{0}\subset ({\mathcal{T}}^{\text{an}}/B)_{X(v)}\quad \text{and}\quad h:{\mathcal{I}}{\mathcal{W}}_{w}^{0,+}\subset ({\mathcal{T}}^{\text{an}}/U)_{X(v)}.\end{eqnarray}$$

Proof. Let $S$ be a set of representatives in $I_{\infty }\simeq \Bigl(\begin{smallmatrix}{\mathcal{O}}^{1} & {\mathcal{O}}\\ p{\mathcal{O}} & {\mathcal{O}}^{\times }\end{smallmatrix}\!\Bigr)$ of $I_{n}/\widetilde{U_{n}}$ which we can suppose of the form,

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}[b] & \\ p[c] & [a]\end{array}\right),\quad a\in ({\mathcal{O}}/p^{m})^{\times },b\in ({\mathcal{O}}^{1}/p^{m}),c\in {\mathcal{O}}/p^{m-1}.\end{eqnarray}$$

Here, $[.]$ denotes any lift. Then $h$ is locally (over $X(v)$ ) isomorphic to

$$\begin{eqnarray}h:\coprod _{\unicode[STIX]{x1D6FE}\in S}\left(\begin{array}{@{}ccc@{}}1+p^{w}B(0,1) & & \\ p^{w}B(0,1) & 1+p^{w}B(0,1) & \\ & & 1+p^{w}B(0,1)\end{array}\right)M\widetilde{\unicode[STIX]{x1D6FE}}\longrightarrow (\operatorname{GL}_{2}\times \operatorname{GL}_{1}/U)_{\text{an}},\end{eqnarray}$$

where $M$ is the matrix which is locally given by the Hodge–Tate map, and thus corresponds to the inclusion ${\mathcal{F}}_{\unicode[STIX]{x1D70F}}\oplus {\mathcal{F}}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}\subset \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70F}}\oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}}$ , and if $\unicode[STIX]{x1D6FE}\in S$ , then $\widetilde{\unicode[STIX]{x1D6FE}}$ is given by

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FE}}=\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(b) & & \\ p\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(c) & \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(a) & \\ & & \unicode[STIX]{x1D70F}(a)\end{array}\right)\quad \text{if}\quad \unicode[STIX]{x1D6FE}=\left(\begin{array}{@{}cc@{}}b & \\ pc & a\end{array}\right).\end{eqnarray}$$

But there exists $M^{\prime }$ with integral coefficients such that

$$\begin{eqnarray}M^{\prime }M=\left(\begin{array}{@{}cc@{}}p^{(p+v)/(p^{2}-1)}I_{2} & \\ & p^{v/(p^{2}-1)}\end{array}\right),\end{eqnarray}$$

and it is easily checked that $M^{\prime }\circ h$ is then injective if $w>m-1+(p+v)/(p^{2}-1)$ . The proof for the other embedding is similar (and easier).◻

From now on, we suppose that we have fixed $m$ , and we suppose that $m-1+(p+v)/(p^{2}-1)<w<m-((p^{2m}-1)/(p^{2}-1))v$ .

We could have defined $\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}\dagger }$ directly, by $g_{\ast }{\mathcal{O}}_{{\mathcal{I}}{\mathcal{W}}_{w}^{0,+}}[\unicode[STIX]{x1D705}]$ where $g$ is the composite,

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{W}}_{w}^{0,+}\longrightarrow {\mathcal{I}}{\mathcal{W}}_{w}^{0}\longrightarrow X(v)\end{eqnarray}$$

as shown by the next proposition. Remark that $X(v)\subset X_{Iw(p)}(v)$ via the canonical filtration of level 1.

Proposition 7.13. For $\unicode[STIX]{x1D705}\in X_{+}(T^{0})$ and $\unicode[STIX]{x1D714}>0$ , there is a restriction map,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{X(v)}^{\unicode[STIX]{x1D705}}{\hookrightarrow}\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}^{\prime }\dagger },\end{eqnarray}$$

induced by the inclusion ${\mathcal{I}}{\mathcal{W}}_{w}^{0,+}\subset ({\mathcal{T}}^{\text{an}}/U)$ . Moreover, locally for the étale topology, this inclusion is isomorphic to the following composition,

$$\begin{eqnarray}V_{\unicode[STIX]{x1D705}^{\prime }}{\hookrightarrow}V_{\unicode[STIX]{x1D705}^{\prime }}^{w\text{-an}}\overset{\text{res}_{0}}{\longrightarrow }V_{0,\unicode[STIX]{x1D705}^{\prime }}^{w\text{-an}}.\end{eqnarray}$$

Proof. Locally for the étale topology, $\unicode[STIX]{x1D714}^{\unicode[STIX]{x1D705}}$ is identified with algebraic functions on $\operatorname{GL}_{2}\times \operatorname{GL}_{1}$ which are invariant by $U$ and varies as $\unicode[STIX]{x1D705}^{\prime }$ under the action of $T$ , i.e. to $V_{\unicode[STIX]{x1D705}^{\prime }}$ . But a function $f\in \unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}^{\prime }\dagger }$ is locally identified with a function,

$$\begin{eqnarray}f:\left\{\!\!\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D70F}(a)(1+p^{w}B(0,1)) & & \\ p^{w}B(0,1) & \unicode[STIX]{x1D70F}(b)(1+p^{w}B(0,1)) & \\ & & \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}(b)(1+p^{w}B(0,1))\end{array}\!\right)\!\!,a\in {\mathcal{O}}^{1},b\in {\mathcal{O}}^{\times }\!\right\}\!\longrightarrow L,\end{eqnarray}$$

which varies as $\unicode[STIX]{x1D705}^{\prime }$ under the action on the right of $T(\mathbb{Z}_{p})\mathfrak{T}_{w}$ . As $\unicode[STIX]{x1D705}^{\prime }=(k_{1},k_{2},k_{3})\in X_{+}(T)$ we can extend $f$ to a $\unicode[STIX]{x1D705}^{\prime }$ -varying function on

$$\begin{eqnarray}I_{p^{w}}=\left\{\!\left(\begin{array}{@{}ccc@{}}\mathbb{G}_{m} & B(0,1) & \\ p^{w}B(0,1) & \mathbb{G}_{m} & \\ & & \mathbb{G}_{m}\end{array}\right)\right\},\end{eqnarray}$$

extending it ‘trivially’; i.e.

$$\begin{eqnarray}\displaystyle f\left(\left(\begin{array}{@{}ccc@{}}x & u & \\ p^{w}z & y & \\ & & t\end{array}\right)\right) & =f\left(\left(\begin{array}{@{}ccc@{}}1 & 0 & \\ p^{w}zx^{-1} & 1 & \\ & & 1\end{array}\right)\left(\begin{array}{@{}ccc@{}}x & u & \\ 0 & y-p^{w}zu & \\ & & t\end{array}\right)\right) & \displaystyle \nonumber\\ \displaystyle & =x^{k_{1}}(y-p^{w}zu)^{k_{2}}t^{k_{3}}f\left(\left(\begin{array}{@{}ccc@{}}1 & 0 & \\ p^{w}zx^{-1} & 1 & \\ & & 1\end{array}\right)\right). & \displaystyle \nonumber\end{eqnarray}$$

Under this identification, locally for the étale topology $\unicode[STIX]{x1D714}_{w}^{\unicode[STIX]{x1D705}^{\prime }\dagger }$ is identified with $V_{0,\unicode[STIX]{x1D705}^{\prime }}^{w\text{-an}}$ .◻

8.1 Hecke operators outside $p$

These operators have been defined already in [Reference BrascaBra16, § 4]. We explain their definition quickly, and refer to [Reference BrascaBra16] (see also [Reference Andreatta, Iovita and PilloniAIP15, § 6.1]) for the details. Let $\ell \neq p$ be an integer, and suppose $\ell \!\not |~N$ , the set of places where $K_{v}$ is not maximal. Let $\unicode[STIX]{x1D6FE}\in G(\mathbb{Q}_{\ell })\cap \operatorname{End}{\mathcal{O}}_{E,\ell }({\mathcal{O}}_{E,\ell }^{3})\times \mathbb{Q}_{\ell }^{\times }$ , and consider

$$\begin{eqnarray}C_{\unicode[STIX]{x1D6FE}}\rightrightarrows {\mathcal{Y}}_{Iw(p)},\end{eqnarray}$$

the moduli space of isogeny $f:A_{1}\longrightarrow A_{2}$ such that:

1. (i) $f$ is ${\mathcal{O}}_{E}$ -linear, and of degree a power of $\ell$ ;

2. (ii) $f$ is compatible with polarizations, i.e. $f^{\ast }\unicode[STIX]{x1D706}_{2}$ is a multiple of $\unicode[STIX]{x1D706}_{1}$ ;

3. (iii) $f$ is compatible with the $K^{p}$ -level structure (at places that divides $N$ ) (we remark that $f$ is an isomorphism on $T_{q}(A_{i})$ when $q\neq \ell$ is a prime);

4. (iv) $f$ is compatible with the filtration given by the Iwahori structure at $p$ ;

5. (v) the type of $f$ is given by the double class $G(\mathbb{Z}_{\ell })\unicode[STIX]{x1D6FE}G(\mathbb{Z}_{\ell })$ .

Denote by $p_{i}:C_{\unicode[STIX]{x1D6FE}}\longrightarrow {\mathcal{Y}}$ the two (finite) maps that sends $f$ to $A_{i}$ . Denote by $C_{\unicode[STIX]{x1D6FE}}(p^{n})$ the fiber product with $p_{1}$ of $C_{\unicode[STIX]{x1D6FE}}$ with ${\mathcal{Y}}_{1}(p^{n})(v)\longrightarrow {\mathcal{Y}}(v)\subset _{s}{\mathcal{Y}}_{Iw(p)}(v)$ , where $s$ is the canonical filtration of $A[p]$ . Denote by $f$ the universal isogeny over $G_{\unicode[STIX]{x1D6FE}}(p^{n})$ . It induces an ${\mathcal{O}}$ -linear isomorphism,