w-ordinary abelian varieties and their moduli

(This post describes ongoing joint work with Przemyslaw Chojecki and Christian Johansson.)

In this post I want to describe a new gauge for the ordinarity of abelian varieties over p-adic fields. The definition is rather simple; the part which actually requires work is showing it cuts out a well-behaved locus in the relevant Siegel moduli spaces. This locus is most naturally defined at infinite level, where it carries some remarkable extra structures (most notably, the functions {\mathfrak{z}_{ij}} defined below). These observations lead naturally to an alternate construction of sheaves of overconvergent modular forms which absorbs any explicit use of difficult (i.e. torsion) p-adic Hodge theory into Scholze’s construction of his infinite-level Shimura varieties. This approach is the nonarchimedean analogue of the classical “{cz+d}” approach to holomorphic modular forms.

Let {C} be a complete algebraically closed extension of {\mathbf{Q}_{p}}, with ring of integers {\mathcal{O}_{C}}. Given any smooth proper rigid space {X/C}, we have a general Hodge-Tate spectral sequence

\displaystyle E_{2}^{i,j}=H^{i}(X,\Omega_{X}^{j})(-j)\Rightarrow H_{\acute{\mathrm{e}}\mathrm{t}}^{i+j}(X,\mathbf{Q}_{p})\otimes C.

Taking {X=A} an abelian variety and recalling the identification {H_{\mathrm{\acute{e}t}}^{1}(A,\mathbf{Q}_{p})=T_{p}A^{\ast}\otimes_{\mathbf{Z}_{p}}\mathbf{Q}_{p}(-1)}, the edge map {H^{1}\rightarrow E_{2}^{0,1}} gives (after twisting and replacing {A} by {A^{\ast}}) a natural Hodge-Tate map

\displaystyle \mathrm{HT}_{A}:T_{p}A\rightarrow\omega_{A^{\ast}}.

Let {\omega_{A^{\ast}}^{+}} denote the {\mathcal{O}_{C}}-span of the image of the Hodge-Tate map. For any rational {w>0} and any {n\geq w}, we write {\mathrm{HT}_{A,w}} for the induced map {A[p^{n}](C)\rightarrow\omega_{A^{\ast}}^{+}/p^{w}}. For later use, we note that the invariant differentials on the Néron model of {A^{\ast}} define a canonical {\mathcal{O}_{C}}-lattice {\omega_{A^{\ast}}^{\circ}\subset\omega_{A^{\ast}}}, and work of Faltings and Fargues implies the inclusions {p^{\frac{1}{p-1}}\omega_{A^{\ast}}^{\circ}\subset\omega_{A^{\ast}}^{+}\subset\omega_{A^{\ast}}^{\circ}}.

Definition.

i. A {g}-dimensional abelian variety {A/C} is w-ordinary if {\ker\mathrm{HT}_{A,w}\simeq(\mathbf{Z}/p^{n})^{g}\subset A(C)[p^{n}]}, where {n=\left\lceil w\right\rceil }.

ii. Given a rigid space {X} and a family of abelian varieties {A\rightarrow X}, {A} is {w}-ordinary if {A_{x}} is w-ordinary at all geometric points {x\in X(C)}.

Note that {A/C} is classically ordinary if and only if it is {\infty}-ordinary.

Given a {w}-ordinary abelian variety {A} and any {0<v\leq\min(w,1)}, the kernel {\mathcal{C}} of the map {\mathrm{HT}_{A,v}:A(C)[p]\rightarrow\omega_{A^{\ast}}^{+}/p^{v}} is an étale group scheme isomorphic to {(\mathbf{Z}/p)^{g}}, which we term the pseudocanonical subgroup. We say a trivialization {\alpha_{n}:(\mathbf{Z}/p^{n})^{2g}\overset{\sim}{\rightarrow}A[p^{n}](C)} is strict if the induced map {\alpha_{n}|_{(p^{n-1}\mathbf{Z}/p^{n})^{g}\oplus0}:(\mathbf{Z}/p)^{g}\rightarrow A(C)[p]} trivializes the pseudocanonical subgroup.

Let {G=\mathrm{GSp}_{2g}} be the symplectic group for the usual matrix {J=\left(\begin{array}{cc} & I_{g}\\ -I_{g}\end{array}\right)}, with {g=\left(\begin{array}{cc} A & B\\ C & D\end{array}\right)} a typical element. Let {g\mapsto g^{\sigma}} be the involution of {G} given by {g^{\sigma}=\left(\begin{array}{cc} D & -C\\ -B & A\end{array}\right)}. Let {Q} be the Siegel parabolic defined by {C=0}, {U} its unipotent radical, {H=\mathrm{GL}_{g}} embedded by {h\mapsto\left(\begin{array}{cc} h\\ & h^{-t}\end{array}\right)}, so {Q=U\rtimes(H\times\mathbf{G}_{m})}. Inside {G(\mathbf{Z}_{p})} we have the usual open subgroups {\Gamma(p^{n})\subset\Gamma_{1}(p^{n})\subset\Gamma_{0}(p^{n})}. Let {K(N)\subset G(\widehat{\mathbf{Z}})} be the principal congruence subgroup of level {N}, and let {K\subset G(\mathbf{A}_{f})} be any open subgroup contained in {K(N)} for some {N\geq3}. Let {X_{K}} denote the Siegel modular variety over {\mathrm{Spec}\mathbf{Q}_{p}} parametrizing principally polarized abelian varieties with {K}-level structure. We shall consider levels of the form {K=K^{p}K_{p}} for some fixed open {K^{p}\subset G(\mathbf{A}_{f}^{p})}; we write {X_{K_{p}}} for {X_{K^{p}K_{p}}} and {X_{n}} for {X_{K^{p}\Gamma(p^{n})}}. Let {X_{K_{p}}^{\ast}} denote the minimal compactification, and write {\mathcal{X}_{K_{p}}}, {\mathcal{X}_{K_{p}}^{\ast}}, {\mathcal{X}_{n}}, etc. for the associated rigid analytic (equivalently, locally tft adic) spaces over {\mathbf{Q}_{p}}, so the {C}-points of {X_{n}} are in bijection with isomorphism classes of quadruples {(A,i,\alpha^{p},\alpha_{n})}, where {A/C} is a {g}-dimensional abelian variety, {i:A\overset{\sim}{\rightarrow}A^{\ast}} is a principal polarization, {\alpha^{p}} is a {K^{p}}-level structure, and {\alpha_{n}:(\mathbf{Z}/p^{n})^{2g}\overset{\sim}{\rightarrow}A[p^{n}](C)} is a symplectic trivialization. In what follows I’ll suppress {i} and {\alpha^{p}}, conflating a point {x\in X_{n}(C)=\mathcal{X}_{n}(C,\mathcal{O}_{C})} with the associated pair {(A/C,\alpha_{n})}. We have an action of {G(\mathbf{Z}/p^{n})} on {X_{n}} given by {(A,\alpha_{n})\mapsto(A,g\cdot\alpha_{n})} where

\displaystyle (g\cdot\alpha_{n})(e_{i})=\sum_{j=1}^{2g}g_{ij}^{\sigma}\alpha_{n}(e_{j}).

Let {\mathcal{X}_{\infty}^{\ast}} denote the perfectoid space constructed by Scholze in Chapter III of his torsion paper, so we have a {G(\mathbf{Z}_{p})}-equivariant identification

\displaystyle \mathcal{X}_{\infty}^{\ast}\sim\lim_{n}\mathcal{X}_{n}^{\ast}

and a {G(\mathbf{Q}_{p})}-equivariant Hodge-Tate period map

\displaystyle \pi_{\mathrm{HT}}:\mathcal{X}_{\infty}^{\ast}\rightarrow\mathcal{F}l=G/Q,

whose definition we recall below.

Theorem A.

i. There is a canonical open rigid subspace {\mathcal{X}_{w}\subset\mathcal{X}=\mathcal{X}_{G(\mathbf{Z}_{p})}} parametrizing principally polarized {w}-ordinary abelian varieties with {K^{p}}-level structure, and this inclusion factors through the composite of a natural open immersion {\mathcal{X}_{w}\subset\mathcal{X}_{\Gamma_{0}(p)}} with the natural morphism {\mathcal{X}_{\Gamma_{0}(p)}\rightarrow\mathcal{X}}.

ii. For any open {K_{p}\subset\Gamma_{0}(p)}, the space {\mathcal{X}_{K_{p},w}=\mathcal{X}_{w}\times_{\mathcal{X}_{\Gamma_{0}(p)}}\mathcal{X}_{K_{p}}} parametrizes principally polarized {w}-ordinary abelian varieties equipped with a {K^{p}}-level structure and a strict {K_{p}}-level structure. The spaces {\mathcal{X}_{K_{p},w}} admit canonical affinoid minimal compactifications {\mathcal{X}_{K_{p},w}^{\ast}\subset\mathcal{X}_{K_{p}}^{\ast}}, and for {K_{p}'\subset K_{p}} the natural morphism {\mathcal{X}_{K_{p}',w}\rightarrow\mathcal{X}_{K_{p},w}} (resp. {\mathcal{X}_{K_{p}',w}^{\ast}\rightarrow\mathcal{X}_{K_{p},w}^{\ast}}) is finite etale (resp. finite).

iii. In the Shimura variety {\mathcal{X}_{\infty}^{\ast}} of infinite level we have {\Gamma_{0}(p)}-stable open subspaces {\mathcal{X}_{\infty,w}} and {\mathcal{X}_{\infty,w}^{\ast}} such that

\displaystyle \mathcal{X}_{\infty,w}\sim\lim_{K_{p}}\mathcal{X}_{K_{p},w}

and

\displaystyle \mathcal{X}_{\infty,w}^{\ast}\sim\lim_{K_{p}}\mathcal{X}_{K_{p},w}^{\ast}.

The space {\mathcal{X}_{\infty,w}^{\ast}} is affinoid perfectoid, and there is a natural {\Gamma_{0}(p)}-stable affinoid subdomain {\mathcal{F}l_{w}\subset\mathcal{F}l} such that {\mathcal{X}_{\infty,w}^{\ast}=\pi_{\mathrm{HT}}^{-1}(\mathcal{F}l_{w})}.

I’ve stated the results in this theorem in the opposite order from their proofs: the simple description of {\mathcal{X}_{\infty,w}^{\ast}} in terms of the Hodge-Tate period map is the key ingredient in our comprehension of these {w}-ordinary loci. To explain this, let {\Lambda=\mathbf{Z}_{p}^{2g}} with basis {e_{1},\dots,e_{2g}} and the usual symplectic form {\psi}, and let {\mathcal{F}l=G/Q} be the flag variety with {\mathcal{F}l(C)} parametrizing maximal isotropic flags {M\subset\Lambda\otimes_{\mathbf{Z}_{p}}C}. Given a point {x=(A,\alpha:\Lambda\overset{\sim}{\rightarrow}T_{p}A)\in\mathcal{X}_{\infty}(C,\mathcal{O}_{C})}, the Hodge-Tate map for {A} sits in a short exact sequence

\displaystyle 0\rightarrow(\mathrm{Lie}A)(1)\rightarrow T_{p}A\otimes_{\mathbf{Z}_{p}}C\rightarrow\omega_{A^{\ast}}\rightarrow0,

and Scholze’s Hodge-Tate period map sends {x} to the flag

\displaystyle (\alpha\otimes1)^{-1}\left(\mathrm{Lie}A\right)\subset\Lambda\otimes_{\mathbf{Z}_{p}}C.

Inside {\mathcal{F}l} we have the (Zariski-open and dense) big cell {\mathcal{F}l^{+}} where for some (any) basis {v_{1},\dots,v_{g}} of {M} the matrix {\psi(v_{j},e_{i+g})_{1\leq i,j\leq g}} is invertible. Over {\mathcal{F}l^{+}} we have canonical global sections {u_{ij}\in\mathcal{O}(\mathcal{F}l^{+})} ({1\leq i,j\leq g}) characterized by the fact that for any point {x\in\mathcal{F}l^{+}(C)}, the vectors {v_{j}=e_{j}+\sum_{i=1}^{g}u_{ij}(x)e_{i+g}} form the unique basis of {M_{x}\subset\Lambda\otimes_{\mathbf{Z}_{p}}C} with {\psi(v_{j},e_{i+g})=\delta_{ij}} (the canonical basis of {M_{x}}). The matrix {u=(u_{ij})\in M_{g}\left(\mathcal{O}(\mathcal{F}l^{+})\right)} is symmetric. For {w>0} rational, let {\mathcal{F}l_{w}} be the subspace of {\mathcal{F}l^{+}} cut out by the condition

\displaystyle \mathrm{sup}_{i,j}\mathrm{inf}_{a\in p\mathbf{Z}_{p}}|u_{ij}-a|\leq|p|^{w}.

This is naturally an affinoid, with {u_{ij}\in\mathcal{O}(\mathcal{F}l_{w})^{+}}, and it’s not hard to check that {\mathcal{F}l_{w}} is {\Gamma_{0}(p)}-stable – in fact, we have the explicit formula

\displaystyle \gamma^{\ast}u=\left(C+Du\right)\left(A+Bu\right)^{-1}

for {\gamma\in\Gamma_{0}(p)}.

Definition. We set {\mathcal{X}_{\infty,w}^{\ast}=\pi_{\mathrm{HT}}^{-1}(\mathcal{F}l_{w})}. The fundamental periods are the global sections of the structure sheaf of {\mathcal{X}_{\infty,w}^{\ast}} defined by {\mathfrak{z}_{ij}=\pi_{\mathrm{HT}}^{\ast}u_{ij}\in\mathcal{O}(\mathcal{X}_{\infty,w}^{\ast})^{+}}.

The matrix {\mathfrak{z}=(\mathfrak{z}_{ij})\in M_{g}\left(\mathcal{O}(\mathcal{X}_{\infty,w}^{\ast})^{+}\right)} is the nonarchimedean version of the usual coordinates {Z=(Z_{ij})\in M_{g}(\mathbf{C})} on Siegel upper half-space. The functions {\mathfrak{z}_{ij}} aren’t exactly coordinates anymore (since {\pi_{\mathrm{HT}}} isn’t injective), but they still carry an enormous amount of information.

The theorem above follows from the following two propositions.

Proposition B. Given a point {x\in\mathcal{X}_{\Gamma_{0}(p)}(C)} with associated abelian variety {A} and {\Gamma_{0}(p)}-subgroup {\mathcal{C}\subset A[p]}, then {A} is {w}-ordinary and {\mathcal{C}} is pseudocanonical if and only if some (any) point {\tilde{x}\in\mathcal{X}_{\infty}} lifting {x} is contained in {\mathcal{X}_{\infty,w}^{\ast}}.

Proof. Let {\alpha:\Lambda\overset{\sim}{\rightarrow}T_{p}A} be the trivialization associated with {\tilde{x}}, and set {\omega_{i}=\mathrm{HT}_{A}(\alpha(e_{i}))\in\omega_{A^{\ast}}^{+}}. If {\tilde{x}\in\mathcal{X}_{\infty,w}^{\ast}}, then {\omega_{j}=-\sum_{i=1}^{g}\mathfrak{z}_{ij}(\tilde{x})\omega_{i+g}} for {1\leq j\leq g} with {\mathfrak{z}_{ij}(\tilde{x})\in p\mathbf{Z}_{p}+p^{w}\mathcal{O}_{C}}, and after replacing {\tilde{x}} by a {\Gamma_{0}(p)}-translate we may arrange so that {\mathfrak{z}_{ij}(\tilde{x})\in p^{w}\mathcal{O}_{C}}. Now each {\omega_{j}} has trivial image in {\omega_{A^{\ast}}^{+}/p^{w}} for {1\leq j\leq g}, so the image of each {\alpha(e_{j})} in {A(C)[p^{n}]} for {1\leq j\leq g} (with {n=\left\lceil w\right\rceil }) lies in the kernel of {\mathrm{HT}_{A,w}}, and therefore {A} is {w}-ordinary. By definition the images of these {\alpha(e_{j})}‘s in {A[p]} trivialize the {\Gamma_{0}(p)}-subgroup {\mathcal{C}}, but the former clearly generate the pseudocanonical subgroup.

The converse direction is similar. {\square}

Proposition C. The space {\mathcal{X}_{\infty,w}^{\ast}} is affinoid perfectoid, and comes by pullback from an affinoid {\mathcal{X}_{\Gamma_{0}(p),w}^{\ast}\subset\mathcal{X}_{\Gamma_{0}(p)}^{\ast}}.

Proof. We briefly recall the Plucker cells of {\mathcal{F}l}. For {J\subset\{1,\dots,2g\}} a subset of cardinality {g}, let {e_{J}\in\wedge^{g}\Lambda} denote the wedge of the {e_{i}}‘s for {i\in J} (in the natural ordering). Given {x\in\mathcal{F}l(C)} with associated {M\subset\Lambda\otimes_{\mathbf{Z}}C}, then choosing any basis {v_{1},\dots,v_{g}} of {M} and writing

\displaystyle v_{1}\wedge\cdots\wedge v_{g}=\sum_{J'}s_{J'}(x)e_{J'},

{x} defines a point of {\mathcal{F}l_{J}} if and only if {|s_{J'}(x)|\leq|s_{J}(x)|} for all {J'\neq J}. (Alternately, each {s_{J'}/s_{J}} is a well-defined meromorphic function on {\mathcal{F}l}, and {\mathcal{F}l_{J}} is cut out by the condition {\mathrm{sup}_{J'}|s_{J'}/s_{J}|\leq1}.)

Now setting {J_{0}=\{1,\dots,g\}} and letting {v_{1},\dots,v_{g}} be the canonical basis of the flag associated with a point {x\in\mathcal{F}l_{w}}, the following things are clear:

{s_{J_{0}}(x)=1},

{u_{ij}(x)=\pm s_{J(ij)}(x)} where {J(ij)=\{1,\dots,i-1,\hat{i},i+1,\dots,g\}\cup\{g+j\}},

-each {s_{J}(x)} is a homogeneous polynomial in the {u_{ij}(x)}‘s of degree {J\cap\{g+1,\dots,2g\}}.

In particular, {\mathcal{F}l_{w}\subset\mathcal{F}l_{J_{0}}} and each {u_{ij}} extends to a section of {\mathcal{O}(\mathcal{F}l_{J_{0}})^{+}}. Now the following are some of the main results in Scholze’s torsion paper:

i. the subspaces {\mathcal{X}_{\infty,J}^{\ast}=\pi_{\mathrm{HT}}^{-1}(\mathcal{F}l_{J})} are affinoid perfectoid,

ii. we may find some {n_{0}} such that {\mathcal{X}_{\infty,J}^{\ast}} is the preimage of an affinoid {\mathcal{X}_{n,J}^{\ast}\subset\mathcal{X}_{n}^{\ast}} for all {n\geq n_{0}},

iii. writing {\mathcal{X}_{n,J}^{\ast}=\mathrm{Spa}(R_{n,J},R_{n,J}^{+})} and defining {R_{\infty,J}^{+}} as the {p}-adic completion of {\lim_{n\rightarrow\infty}R_{n,J}^{+}}, then {\mathcal{X}_{\infty,J}^{\ast}=\mathrm{Spa}(R_{\infty,J}^{+}[\frac{1}{p}],R_{\infty,J}^{+})}.

Taking {J=J_{0}}, we see that each {\mathfrak{z}_{ij}} canonically extends to a section of {\mathcal{O}(\mathcal{X}_{\infty,J_{0}})^{+}=R_{\infty,J_{0}}^{+}}, and the condition

\displaystyle \mathrm{sup}_{i,j}\mathrm{inf}_{a\in p\mathbf{Z}_{p}}|\mathfrak{z}_{ij}-a|\leq|p|^{w}

cuts out {\mathcal{X}_{\infty,w}^{\ast}} as a rational subdomain of {\mathcal{X}_{\infty,J_{0}}^{\ast}}. One of the important basic results on perfectoid spaces is that rational subdomains of affinoid perfectoids are themselves affinoid perfectoid, so this proves the first clause of the proposition. Now for some {N} sufficiently large depending on {w}, we may choose some large {n} and some {\tilde{\mathfrak{z}}_{ij}\in R_{n,J_{0}}^{+}} such that {\mathfrak{z}_{ij}-\tilde{\mathfrak{z}}_{ij}\in p^{N}R_{\infty,J_{0}}^{+}}, and then the condition

\displaystyle \mathrm{sup}_{i,j}\mathrm{inf}_{a\in p\mathbf{Z}_{p}}|\tilde{\mathfrak{z}}_{ij}-a|\leq|p|^{w}

again cuts out {\mathcal{X}_{\infty,w}^{\ast}} (cf. Remark 2.8 in “Perfectoid spaces”) and visibly descends to a condition cutting out a rational subdomain {\mathcal{X}_{n,w}^{\ast}\subset\mathcal{X}_{n,J_{0}}^{\ast}} whose preimage at infinite level is {\mathcal{X}_{\infty,w}^{\ast}}. Since {\mathcal{X}_{\infty,w}^{\ast}} is {\Gamma_{0}(p)}-stable, {\mathcal{X}_{n,w}^{\ast}} is as well.

It remains to descend {\mathcal{X}_{n,w}^{\ast}} down to level {\Gamma_{0}(p)}. Given a finite group {G} acting on a rigid space {Y}, we say a finite morphism {f:Y\rightarrow X} is a {G}covering if {f} is {G}-equivariant for the trivial action on {X}, {f} is generically of degree {|G|}, and {G} acts transitively on the fibers of {f.} (Is there a simpler definition? I don’t want to assume that {f} is etale.)

Lemma. If {f:Y\rightarrow X} is a {G}-covering with {X} normal, there is a functorial bijection between affinoid subdomains {U\subset X} and {G}-stable affinoid subdomains {U'\subset Y} realized explicitly by the functors

\displaystyle U\rightsquigarrow U\otimes G=\mathrm{Sp}(f_{\ast}\mathcal{O}_{Y})(U)

and

\displaystyle U'\rightsquigarrow U'^{G}=\mathrm{Sp}\mathcal{O}_{Y}(U')^{G}.

Proof. By the discussion in 6.3.3 of BGR, {\mathcal{O}_{Y}(U')^{G}} is affinoid and {\mathcal{O}_{Y}(U')} is a module-finite and integral {\mathcal{O}_{Y}(U')^{G}}-algebra. By Corollary 9.4.4/2 in BGR,

\displaystyle (f_{\ast}\mathcal{O}_{Y})(U)=\mathcal{O}_{Y}(f^{-1}(U))=\mathcal{O}(U\otimes G)

is affinoid and module-finite over {\mathcal{O}_{X}(U)}. To see that these functors are mutually inverse, note that {(U\otimes G)^{G}=\mathrm{Sp}\left(\mathcal{O}_{Y}(f^{-1}U)^{G}\right)\rightarrow U} is finite and birational, and therefore an isomorphism by the normality of {X}. The identification {U'=U'^{G}\otimes G} is an easy consequence of the set-theoretic surjectivity of the natural map {U'\rightarrow U'^{G}}. {\square}

Applying this lemma with {G=\Gamma_{0}(p)/\Gamma(p^{n})}, {X=\mathcal{X}_{\Gamma_{0}(p)}^{\ast}}, {Y=\mathcal{X}_{n}^{\ast}}, {f} the obvious map, and {U'=\mathcal{X}_{n,w}^{\ast}}, we obtain {\mathcal{X}_{\Gamma_{0}(p),w}^{\ast}} as {U'^{G}}. (The finiteness of {f} and the normality of {X} are basic properties of the minimal compactifications.) Now the rest is easy: we define {\mathcal{X}_{K_{p},w}^{\ast}} as the preimage of {\mathcal{X}_{\Gamma_{0}(p),w}^{\ast}}, and then {\mathcal{X}_{K_{p},w}=\mathcal{X}_{K_{p},w}^{\ast}\cap\mathcal{X}_{K_{p}}}. The twiddly relation in part iii. of the theorem follows immediately from a result in Scholze-Weinstein.

It’s natural to wonder how {w}-ordinarity compares with the usual measure of ordinarity, namely the size of the Hasse invariant. For {v\geq0}, let {\mathcal{X}_{K_{p}}^{\ast}(v)} denote the locus where {\mathrm{Ha}(A_{x})\leq v}, so {\mathcal{X}_{K_{p}}^{\ast}(0)} is the ordinary locus. Since {\mathcal{X}_{K_{p}}^{\ast}(v),v\rightarrow0} and {\mathcal{X}_{K_{p},w}^{\ast},w\rightarrow\infty} both give a sequence of shrinking affinoid neighborhoods of {\mathcal{X}_{K_{p}}^{\ast}(0)}, they must intertwine “for compacity reasons”. Here’s a more concrete statement.

Theorem D. If {p\geq5} and {\mathrm{Ha}(A)<\frac{1}{2p^{n-1}}} for some integer {n\geq1}, then {A} is {w}-ordinary for

\displaystyle w=n-\frac{p^{n}}{p-1}\mathrm{Ha}(A)-\frac{1}{p-1}\approx n-1/2.

In particular, for {v<\frac{1}{2p^{n-1}}} we have {\mathcal{X}_{K_{p}}^{\ast}(v)\subset\mathcal{X}_{K_{p},w}^{\ast}} for the same {w}.

This crucially uses results of Fargues on the canonical subgroup (plus some fiddling to deal with the boundary). There’s a similar statement for {p=2,3}.

Let me end with a teaser, which I’ll discuss more carefully in a followup post. Set {g=1}, so we’re in the realm of modular curves (and I’ll replace {A,...,D} by {a,...,d} as is customary). Let {L} be a finite extension of {\mathbf{Q}_{p}}, and choose a continuous character {\lambda:\mathbf{Z}_{p}^{\times}\rightarrow L^{\times}}. Let {\eta:\mathcal{X}_{\infty,w}^{\ast}\rightarrow\mathcal{X}_{\Gamma_{0}(p),w}^{\ast}=:X_{w}} be the natural map, so for any affinoid {U\subset X_{w}} the preimage {\eta^{-1}U} is a perfectoid space, and I’ll write {\mathcal{O}(\eta^{-1}U)} for the global sections of its structure sheaf. (Note: {\eta^{-1}U} might not be affinoid perfectoid! But if {U} is a rational subdomain then it will be, so by Gerritzen-Grauert {\eta^{-1}U} has a finite covering by affinoid perfectoids.) There’s a left action of {\Gamma_{0}(p)} on {\eta^{-1}U}, inducing a right action {f\rightarrow\gamma^{\ast}f} on {\mathcal{O}(\eta^{-1}U)}. Now there’s a unique fundamental period {\mathfrak{z}=\mathfrak{z}_{11}}, and there’s a rational {w_{\lambda}} depending on {\lambda} such that for any {w\geq w_{\lambda}}, the expression {\lambda(a+b\mathfrak{z})} gives a well-defined element of {\mathcal{O}(\mathcal{X}_{\infty,w}^{\ast})\otimes_{\mathbf{Q}_{p}}L} for any {a\in\mathbf{Z}_{p}^{\times}}, {b\in\mathbf{Z}_{p}}.

Definition. We define a sheaf {\omega_{\lambda}^{\dagger}} on {X_{w}} by the rule

\displaystyle \omega_{\lambda}^{\dagger}(U)=\left\{ f\in\mathcal{O}(\eta^{-1}U)\otimes L\mid f=\lambda(a+b\mathfrak{z})\cdot\gamma^{\ast}f\,\forall\gamma\in\Gamma_{0}(p)\right\}

for any rational subdomain {U}.

Theorem E. The sheaf {\omega_{\lambda}^{\dagger}} is a line bundle on {X_{w}\times_{\mathrm{Sp}\mathbf{Q}_{p}}\mathrm{Sp}L}, and coincides with the sheaf of overconvergent forms of weight {\lambda} defined in Pilloni’s Annales de l’Institut Fourier paper.

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