# Spherical varieties and $p$-adic families of cohomology classes

David Loeffler, Rob Rockwood and Sarah Livia Zerbes

###### Abstract.

We prove a “twist-compatibility” result for $p$-adic families of cohomology
classes associated to symmetric spaces. This shows that a single family of
classes (lying in a finitely-generated Iwasawa module) interpolates classical
cohomology classes of many different weights, including twists by
Grössencharacters of possibly non-trivial infinity-type. This subsumes and
generalises a number of prior results relating to Euler systems and $p$-adic
$L$-functions, and we conclude with some novel applications to Euler systems
for $\operatorname{GSp}_{4}$,
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$, and
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}$.

We gratefully acknowledge support from the following research grants: EPSRC
Standard Grant EP/S020977/1 and ERC Consolidator Grant 101001051 “ShimBSD”
(Loeffler); Royal Society URF Enhancement Award RGF/EA/180212 (Rockwood).

_This paper is dedicated to Massimo Bertolini, to mark the occasion of his
60th birthday. Massimo’s beautiful work has been an inspiration to us
throughout our careers and we wish him all the best for the future._

##  1\. Introduction

This article is a sequel to the article [Loe21b] of the first author. In the
previous article, we constructed norm-compatible families of cohomology
classes for Shimura varieties and other symmetric spaces, associated to pairs
of reductive groups satisfying an “open orbit” condition. This very general
construction subsumes a number of well-known results in Iwasawa theory,
including the norm-compatibility of modular symbols and Heegner points for
$\operatorname{GL}_{2}$, as well as more recent constructions such as the
Euler system for $\operatorname{GSp}_{4}$ constructed in [LSZ21a].

In this paper, we consider varying the algebraic weight defining the
coefficient system for our cohomology. Our main result, theorem 5.2.3, shows
that the norm-compatible families constructed in [Loe21b] interpolate
classical cohomology classes of many different weights (including twists by
Grössencharacters of possibly non-trivial infinity-type). Several specific
instances of this result are already known – see remark 5.2.6 below – and form
an important ingredient in the proofs of “explicit reciprocity laws” for Euler
systems and $p$-adic $L$-functions. The general theorem proved here subsumes
all of these earlier results, and we conclude with several new applications of
this theory to Euler systems for $\operatorname{GSp}_{4}$,
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$, and
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}$.

##  2\. Control theorems for ordinary cohomology

The aim of this section is to prove a general control result (theorem 2.6.2)
for “$Q$-ordinary” cohomology groups associated to a general parabolic
subgroup $Q$. Our formulation is heavily influenced by the work of Urban
[Urb05] in the setting of Coleman theory; however, unlike Urban we obtain
integral results, without inverting $p$ in the coefficients, as a consequence
of imposing the stronger condition of $Q$-ordinarity rather than just
$Q$-finite-slope. These results are adapted from the second author’s 2022
University of Warwick PhD thesis.

###  2.1. Setup

Let $p$ be a prime. We consider the following situation:

  * •

$\mathcal{G}$ is a connected reductive group over $\mathbf{Q}$, satisfying
Milne’s axiom (SV5), i.e. its centre contains no torus which is
$\mathbf{R}$-split but not $\mathbf{Q}$-split.

  * •

$G$ is an unramified $\mathbf{Z}_{p}$-model of $\mathcal{G}$, i.e. a reductive
group scheme over $\mathbf{Z}_{p}$ whose base extension to $\mathbf{Q}_{p}$
coincides with that of $\mathcal{G}$.

  * •

$Q$ is a parabolic subgroup of $G$, and $\bar{Q}$ a choice of opposite
parabolic; thus $L=Q\cap\bar{Q}$ is a Levi subgroup of $Q$, and the “big cell”
$\overline{N}\times L\times N$ is an open subscheme of $G$ over
$\mathbf{Z}_{p}$, where $N$ is the unipotent radical of $Q$ (and similarly
$\overline{N}$). Note that $Q$ and $\bar{Q}$ are local at $p$; we do not
assume that they come from parabolic subgroups of $\mathcal{G}$ defined over
$\mathbf{Q}$.

  * •

$S$ denotes the torus $L/L^{\mathrm{der}}$, and $X^{\bullet}(S)$ its character
lattice. We let $X_{+}^{\bullet}(S)$ be the cone of dominant weights (relative
to $Q$).

  * •

$A$ is the maximal $\mathbf{Q}_{p}$-split subtorus in the centre of $L$.

We choose a torus $S^{0}\subset S$, and let $L^{0}$ and $Q^{0}$ be its
preimages in $L$ and $Q$ respectively (so $Q^{0}=L^{0}\ltimes N$). We write
$\mathfrak{S}$ for the profinite abelian group
$L(\mathbf{Z}_{p})/L^{0}(\mathbf{Z}_{p})\subseteq(S/S^{0})(\mathbf{Z}_{p})$.
The base ring for our constructions will be the Iwasawa algebra of
$\mathfrak{S}$. Note that $\mathfrak{S}$ is $p$-torsion-free (since $S/S^{0}$
splits over an unramified extension).

######  Remark 2.1.1.

The assumption (SV5) can be weakened, but at the cost of more complicated
statements: very roughly, if $Z(\mathcal{G})$ contains a torus which is
$\mathbf{R}$-split but not $\mathbf{Q}$-split, we need to assume that this
torus is contained in $S^{0}$. An important example where (SV5) does not hold
is the group
$\mathcal{G}=\operatorname{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$ for $F$ a
totally real field; this case will be treated in detail in forthcoming work of
Arshay Sheth.

###  2.2. Algebraic representations

Let us fix a maximal torus $T\subset L$, and a finite extension
$K/\mathbf{Q}_{p}$, with ring of integers $\mathcal{O}$, such that $T$ splits
over $K$. We can and do assume that $K$ is unramified (since $G$ has a smooth
integral model).

For $\lambda\in X^{\bullet}_{+}(S)$, there is a unique isomorphism class of
irreducible representations $(\rho_{\lambda},V_{\lambda})$ of $G_{/K}$ of
highest weight $\lambda$ (with respect to any choice of Borel contained in
$Q$). A representative of this isomorphism class can be constructed using the
Borel–Weil–Bott theorem, as the space of all polynomials

$\\{f\in K[G]:f(\overline{n}\ell
g)=\lambda(\ell)f(g)\quad\forall\overline{n}\in\overline{N},\ell\in L,g\in
G\\},$

with $G$ acting by right-translation. In particular, the highest weight space
of $V_{\lambda}$ has a unique basis vector $f_{\lambda}^{\mathrm{hw}}$ whose
restriction to the big Bruhat cell $\overline{N}LN$ is given by
$\overline{n}\ell n\mapsto\lambda(\ell)$. This is also the highest _relative_
weight space of $V_{\lambda}$, i.e. the highest-weight eigenspace for the
action of $A$.

###  2.3. Integral lattices

Let $\lambda\in X^{\bullet}_{+}(S)$.

######  Definition 2.3.1.

An _admissible lattice_ in $V_{\lambda}$ is an $\mathcal{O}$-lattice
$\mathcal{L}\subset V_{\lambda}$ which is invariant under $G_{/\mathcal{O}}$
(i.e. $G(R)$ preserves $\mathcal{L}\otimes R$ for every $\mathcal{O}$-algebra
$R$), and whose intersection with the highest-weight subspace is
$\mathcal{O}\cdot f_{\lambda}^{\mathrm{hw}}$.

We recall the following standard facts, which can be found in [Lin92] for
example:

  1. (a)

There are only finitely many admissible lattices in $V_{\lambda}$.

  2. (b)

Every admissible lattice is the direct sum of its intersections with the
weight spaces of $V^{G}_{\lambda}$.

  3. (c)

There are unique maximal and minimal admissible lattices $\mathcal{L}^{\max}$
and $\mathcal{L}^{\min}$ such that every admissible $\mathcal{L}$ satisfies
$\mathcal{L}^{\max}\supseteq\mathcal{L}\supseteq\mathcal{L}^{\min}$.

  4. (d)

If we realise $V_{\lambda}$ via the Borel–Weil construction as above, then the
maximal admissible lattice $\mathcal{L}^{\max}$ is given by the intersection
of $V_{\lambda}$ with the coordinate ring of $G_{/\mathcal{O}}$.

  5. (e)

The dual of an admissible lattice in $V_{\lambda}$ is an admissible lattice in
$(V_{\lambda})^{\vee}$. In particular, the dual of the maximal lattice is the
minimal lattice and vice versa.

We shall henceforth always write $V_{\lambda,\mathcal{O}}$ for the _maximal_
admissible lattice, realised via the Borel–Weil construction as above, and
$V^{\min}_{\lambda,\mathcal{O}}$ for the minimal lattice.

######  Remark 2.3.2.

Note that if $G=\operatorname{GL}_{2}$ and
$V_{\lambda}=\operatorname{Sym}^{n}(K^{2})$ is the $n$-th symmetric power of
the standard representation, then $\operatorname{Sym}^{n}(\mathcal{O}^{2})$ is
the maximal lattice, and the module $\operatorname{TSym}^{n}(\mathcal{O}^{2})$
of symmetric tensors (which features prominently in [KLZ17]) is the minimal
lattice.

######  Definition 2.3.3.

Let $\Sigma^{+}$ denote the monoid of cocharacters $\eta\in X_{\bullet}(A)$
which are dominant with respect to $Q$, so that $\langle\eta,\phi\rangle\geq
0$ for every relative root $\phi$. We let $\Sigma^{++}\subset\Sigma^{+}$ be
the ideal of elements that are strictly dominant, so
$\langle\eta,\phi\rangle>0$ for all $\phi$.

######  Lemma 2.3.4.

If $\eta\in\Sigma^{+}$, then the endomorphism
$p^{\langle\eta,\lambda\rangle}\rho_{\lambda}\left(\eta(p)^{-1}\right)$ of
$V_{\lambda}$ acts on every weight space as a non-negative power of $p$, and
hence restricts to an endomorphism of $\mathcal{L}$ for any admissible lattice
$\mathcal{L}$. If $\eta\in\Sigma^{++}$ then this endomorphism acts as a
strictly positive power of $p$ on every weight space except the highest one.

###### Proof.

This is an easy explicit check. ∎

###  2.4. Cohomology of symmetric spaces

Let us fix a prime-to-$p$ level group $K^{p}$ (which we shall suppose to be
neat). We define $H^{*}(U,\mathcal{O})$, for $U\subset G(\mathbf{Z}_{p})$ open
compact, to mean _one_ of the following:

  * •

Betti cohomology of the locally symmetric space $Y_{G}(K^{p}U)$, viewed as a
real manifold, with coefficients in $\mathcal{O}$;

  * •

étale cohomology of the Shimura variety of that level, viewed as an algebraic
variety over $\overline{\mathbf{Q}}$, and supposing $\mathcal{G}$ to be
equipped with a choice of Shimura datum;

  * •

étale cohomology of the canonical integral model of the Shimura variety over
$\mathcal{O}_{E}[1/S]$, where $E$ is the reflex field, and $S$ is a
sufficiently large finite set of primes containing all of those dividing $p$.
Here we suppose the Shimura datum to be of Hodge type. In this case, we may
also allow arbitrary cyclotomic twists of the coefficients.

One can also consider cohomology with compact support (in any of the above
settings). More generally, since we may regard an algebraic representation of
$G$ as a coefficient sheaf (for any of the above cohomology theories), we can
make sense of $H^{*}(U,V_{\lambda,\mathcal{O}})$ for any $U\subseteq
G(\mathbf{Z}_{p})$.

When considering étale cohomology of canonical models, we shall suppose that
either $p>2$ or $E$ has no real places. Then in all the above cases the
cohomology groups $H^{i}(U,-)$ are finitely-generated over $\mathcal{O}$, and
zero for $i$ exceeding some bound independent of $U$.

###  2.5. Hecke actions

For general $\lambda$, the integral étale cohomology groups
$H^{*}(-,V_{\lambda,\mathcal{O}})$ are _Cartesian cohomology functors_ for
$G(\mathbf{Z}_{p})$ in the sense of [Loe21b]. However, if the coefficients are
non-trivial they are typically not cohomology functors for the whole of
$G(\mathbf{Q}_{p})$ since this group does not act on the lattice
$V_{\lambda,\mathcal{O}}$. We shall work around this by renormalising our
Hecke operators using lemma 2.3.4.

######  Definition 2.5.1.

Suppose $U\subseteq G(\mathbf{Z}_{p})$. Then, for any $\lambda\in
X^{\bullet}_{+}(S)$ and $\eta\in\Sigma^{+}$, we can define a normalised Hecke
operator acting on $H^{*}(U,V_{\lambda,\mathcal{O}})$ by

$\mathcal{T}^{\prime}_{\eta}=p^{h}[U\eta(p^{-1})U],\qquad
h={\langle\lambda,\eta\rangle}.$

######  Remark 2.5.2.

In the special cases of this theory we have previously worked out, the scalar
factor $p^{h}$ appearing above is hidden from view, since we had always chosen
$\eta$ and $\lambda$ such that $\langle\eta,\lambda\rangle=0$.

######  Definition 2.5.3.

We say an open compact subgroup $U\subset G(\mathbf{Z}_{p})$ has an _Iwahori
decomposition_ (with respect to $Q$) if we have $U=\bar{N}_{U}L_{U}N_{U}$,
where $N_{U}=U\cap N(\mathbf{Z}_{p})$ etc, and $N_{U}$ and $\bar{N}_{U}$ are
the products of their intersections with the relative root spaces for $A$.

The prototypical examples are the parahoric subgroups $J=\\{g\in
G(\mathbf{Z}_{p}):g\bmod p\in Q(\mathbf{F}_{p})\\}$ and its opposite
$\bar{J}$.

######  Proposition 2.5.4.

If $U$ has an Iwahori decomposition, then the operators
$\mathcal{T}^{\prime}_{\eta}$ for varying $\eta$ define an action of
$\Sigma^{+}$ on $H^{*}(U,V_{\lambda,\mathcal{O}})$; and if we choose an
$\eta_{0}$ which is strictly dominant, then the _anti-ordinary idempotent_

$e_{Q}^{\prime}=\lim_{k\to\infty}(\mathcal{T}^{\prime}_{\eta_{0}})^{k}$

is independent of the choice of $\eta_{0}$, and cuts out the maximal direct
summand on which the operators $\mathcal{T}_{\eta}^{\prime}$ are all
invertible.

We will show momentarily that $e^{\prime}_{Q}H^{*}(U,V_{\lambda,\mathcal{O}})$
in fact depends only on $U\cap Q(\mathbf{Z}_{p})$.

######  Proposition 2.5.5.

For any $U$ with an Iwahori decomposition, the natural map

$\mathop{e^{\prime}_{Q}}H^{i}(U,\mathcal{V}_{\lambda,\mathcal{O}}^{\mathrm{min}})\to
e^{\prime}_{Q}H^{i}(U,\mathcal{V}_{\lambda,\mathcal{O}}),$

induced by the inclusion
$\mathcal{V}_{\lambda,\mathcal{O}}^{\mathrm{min}}\hookrightarrow\mathcal{V}_{\lambda,\mathcal{O}}$,
is an isomorphism.

###### Proof.

Both $V_{\lambda,\mathcal{O}}^{\min}$ and $V_{\lambda,\mathcal{O}}$ are direct
sums of their weight spaces for $A$. The inclusion map is an isomorphism on
the highest relative weight space (by definition). On each lower weight space,
$p^{\langle\lambda,\eta_{0}\rangle}\rho_{\lambda}(\eta_{0}(p)^{-1})$ acts as a
positive power of $p$, so some power of this map will annihilate the quotient
of the weight spaces in the two lattices. Thus, if $r\gg 0$ (depending on
$\lambda$), the operator $(\mathcal{T}^{\prime}_{\eta_{0}})^{r}$ annihilates
the cohomology of the quotient. ∎

###  2.6. Iwasawa cohomology and control theorems

######  Definition 2.6.1.

For $i\geq 0$, and $\lambda\in X^{\bullet}_{+}(S)$, we set

$H^{i}_{\mathrm{Iw}}(Q^{0},V_{\lambda,\mathcal{O}})=\varprojlim_{U}H^{i}(U,\mathcal{V}_{\lambda,\mathcal{O}}),$

where $U$ varies over open compact subgroups of $G(\mathbf{Z}_{p})$ containing
$Q^{0}(\mathbf{Z}_{p})$, and the inverse limit is taken relative to the
pushforward maps.

One checks that the levels $U$ having an Iwahori decomposition are cofinal in
the above inverse system, and the operators
$\\{\mathcal{T}^{\prime}_{\tau}:\tau\in\Sigma^{+}\\}$ are compatible with the
pushforward maps and hence act on the inverse limit.

In this section and the next, we prove the following result, generalising
results due to Ohta [Oht99, Oht00] for Betti or étale cohomology of modular
curves to arbitrary reductive groups:

######  Theorem 2.6.2.

  1. (i)

The modules $e^{\prime}_{Q}H^{i}_{\mathrm{Iw}}(Q^{0},\mathcal{O})$ are
finitely-generated over $\mathcal{O}[[\mathfrak{S}]]$, and vanish for $i\gg
0$.

  2. (ii)

Let $U$ be any open compact subgroup of $G(\mathbf{Z}_{p})$ such that
$U\supseteq Q^{0}(\mathbf{Z}_{p})$ and $U$ admits an Iwahori decomposition.
For any $\lambda\in X^{\bullet}_{+}(S/S_{0})$, there is a convergent spectral
sequence (supported in the second quadrant $i\leq 0,j\geq 0$):

$E_{2}^{ij}=\operatorname{Tor}_{-i}^{\mathcal{O}[[\mathfrak{S}_{U}]]}\Big{(}e^{\prime}_{Q}H^{j}_{\mathrm{Iw}}(Q^{0},\mathcal{O}),\mathcal{O}[-\lambda]\Big{)}\Rightarrow
e^{\prime}_{Q}H^{i+j}(U,V_{\lambda,\mathcal{O}}),$

where $\mathfrak{S}_{U}$ is the image of $U\cap Q(\mathbf{Z}_{p})$ in
$\mathfrak{S}$, and $\mathcal{O}[-\lambda]$ denotes $\mathcal{O}$ with
$\mathfrak{S}_{U}$ acting via the algebraic character $\lambda$. This spectral
sequence is compatible with the action of Hecke operators away from $p$, and
with the operators $\\{\mathcal{T}^{\prime}_{\tau}:\tau\in\Sigma^{+}\\}$.

We shall give the proof below assuming that $H^{*}(-)$ denotes Betti
cohomology. The case of étale cohomology follows from this; one may define all
the spaces and maps for a finite truncation of the coefficients, using general
functoriality properties of étale cohomology, and then the remaining
assertions – that the inverse limits are finitely-generated, and that the
moment map in (ii) is an isomorphism on the ordinary part – can be checked
using the comparison between Betti cohomology and étale cohomology over
$\overline{\mathbf{Q}}$, and the Hochschild–Serre spectral sequence relating
étale cohomology over number fields to Galois cohomology of étale cohomology
over $\overline{\mathbf{Q}}$.

######  Remark 2.6.3.

In the above formulation, we have assumed that $\lambda$ factors through the
quotient $S/S_{0}$, and in particular is trivial on $L^{\mathrm{der}}$. More
generally, we can fix any $\nu\in X^{\bullet}_{+}(T)$ (not necessarily trivial
on $L^{\mathrm{der}}$) and study the cohomology groups
$e_{Q}^{\prime}\cdot\varprojlim_{U}H^{i}(U,V_{\nu,\mathcal{O}})$; one can show
that these are finite-type $\mathcal{O}[[\mathfrak{S}]]$-modules which
interpolate the cohomology groups $e_{Q}^{\prime}\cdot
H^{i}(U,V_{\lambda+\nu,\mathcal{O}})$ as $\lambda$ varies over characters of
$S/S_{0}$ (so $\lambda+\nu$ varies over a coset of $X^{\bullet}(S/S_{0})$ in
$X^{\bullet}(T)$). We shall stick to the case $\nu=0$ here for ease of
notation, but the proof in the general case is very similar.

###  2.7. Proof of the control theorem

For brevity we set $R\coloneqq\mathcal{O}[[\mathfrak{S}]]$.

Fix a choice of $\eta\in\Sigma^{++}$ and set $\tau:=\eta(p)$. We recall the
following notation of [Loe21b], setting for $r\geq 0$:

$\displaystyle N_{r}$
$\displaystyle:=\tau^{r}N(\mathbf{Z}_{p})\tau^{-r},\bar{N}_{r}:=\tau^{-r}\bar{N}(\mathbf{Z}_{p})\tau^{r}$
$\displaystyle L_{r}$ $\displaystyle:=\\{\ell\in L(\mathbf{Z}_{p}):\ell\in
L^{0}(\mathbf{Z}_{p})\ \mathrm{mod}\ p^{r}\\}$ $\displaystyle\text{and for
$r\geq 1$},\quad V_{r}$ $\displaystyle:=\bar{N}_{r}L_{r}N_{0}.$

Note that $\bar{N}_{0}L_{0}N_{0}$ is not a group in general, and in _op.cit._
we defined $V_{0}=G(\mathbf{Z}_{p})$, but we shall only consider $V_{r}$ for
$r\geq 1$ here. We shall also use the group

$V_{0}^{\prime}=\bar{N}_{1}L_{0}N_{0}=\tau^{-1}G(\mathbf{Z}_{p})\tau\cap
G(\mathbf{Z}_{p}).$

######  Remark 2.7.1.

Note that $V_{0}^{\prime}\subseteq J$, where $J$ is the parahoric subgroup
associated to $Q(\mathbf{F}_{p})$; but this inclusion is in general strict (it
is an equality iff $\eta$ is a minuscule cocharacter).

Let $Q^{0}(\mathbf{Z}_{p})\subset U\subset G(\mathbf{Z}_{p})$ be an open
compact subgroup admitting an Iwahori decomposition

$U=\bar{N}_{U}\times L_{U}\times N_{0},$

and for $r\geq 0$ set

$U^{(r)}=\tau^{-r}U\tau^{r}\cap U.$

#### Betti cohomology complexes

For an open compact subgroup $K=K^{p}U\subset G(\mathbf{A}_{f})$ the symmetric
space $Y(K)$ is given by a finite disjoint union

$Y(K)=\sqcup_{i}\mathcal{X}/\Gamma_{i}.$

where $\Gamma_{i}\subset G(\mathbf{Q})^{+}$ are arithmetic subgroups and
$\mathcal{X}=G(\mathbf{R})^{+}/K_{\infty}Z(\mathbf{R})\cap G^{+}(\mathbf{R})$
with $K_{\infty}$ a maximal compact subgroup of $G(\mathbf{R})$. If $K$ is
neat then $\mathcal{X}/\Gamma_{i}$ has the structure of a smooth manifold.
This space satisfies $\pi_{1}\left(\mathcal{X}/\Gamma_{i}\right)=\Gamma_{i}$
and thus

$H^{i}(K,V_{\lambda})=\bigoplus_{j}H^{i}(\Gamma_{i},V_{\lambda}).$

Let $M$ be a $\mathbf{Z}[K]$-module with $K$ acting through its projection to
$U$. By the theory of Borel–Serre the cohomology groups $H^{i}(\Gamma_{i},M)$
vanish for $i\gg 0$ and are computed by a complex
$\mathcal{C}^{\bullet}(\Gamma_{i},M)$ of $\mathbf{Z}[\Gamma_{i}]$-modules such
that each term is isomorphic to a finite direct sum of copies of $M$. We write

$\mathcal{C}^{\bullet}(K,M):=\bigoplus_{i}\mathcal{C}^{\bullet}(\Gamma_{i},M).$

This complex computes the Betti cohomology of $Y(K)$. If $M$ has an action of
$\tau^{-1}$ compatible with the $K$-action then we can define an action of
$\mathcal{T}^{\prime}_{\eta}$ on $\mathcal{C}^{\bullet}(K,M)$ lifting the
action on cohomology. There are many such liftings but these differ by a chain
homotopy and thus induce the same morphism in the derived category.

#### Modules of continuous functions

For $\lambda\in X_{+}^{\bullet}(S)$ we define $V_{0}^{\prime}$-modules

$\displaystyle C_{\lambda,U}$
$\displaystyle=\\{f:V_{0}^{\prime}\to\mathcal{O}:f\ \text{continuous},\ f(\ell
ng)=\lambda^{-1}(\ell)f(g)\ \forall\ell n\in L_{U}N_{0},g\in
V^{\prime}_{0}\\}$ $\displaystyle C_{\mathrm{univ}}$
$\displaystyle=\\{f:V^{\prime}_{0}\to\mathcal{O}:f\ \text{continuous},\ f(\ell
ng)=f(g)\ \forall\ell n\in Q^{0}(\mathbf{Z}_{p}),g\in V^{\prime}_{0}\\}$

with $V_{0}^{\prime}$ acting by right translation. (We shall only use this
when $\lambda$ is trivial on $S_{0}$, so that $\lambda(\ell)=1$ for all $\ell
n\in Q^{0}(\mathbf{Z}_{p})$). We endow these spaces with an action of $\tau$
given by

$\left(\tau\cdot f\right)(n\ell\bar{n})=f(n\ell\tau^{-1}\bar{n}\tau).$

There are natural inclusions

$V_{\lambda^{\vee},\mathcal{O}}^{\mathrm{max}}\to C_{\lambda,U}\to
C_{\mathrm{univ}}$

intertwining the $V_{0}^{\prime}$ and $\tau$ actions, where
$\lambda^{\vee}=-w_{0}\lambda$ and $w_{0}$ is the long Weyl element.

#### Modules of measures

We define modules of bounded distributions

$\displaystyle D_{\lambda,U}$
$\displaystyle=\mathrm{Hom}_{\mathrm{cont}}\left(C_{\lambda,U},\mathcal{O}\right)$
$\displaystyle D_{\mathrm{univ}}$
$\displaystyle=\mathrm{Hom}_{\mathrm{cont}}\left(C_{\mathrm{univ}},\mathcal{O}\right),$

which inherit actions of $V_{0}^{\prime}$ and $\tau^{-1}$ by duality. By
dualising the above sequence, these spaces admit the following maps

$D_{\mathrm{univ}}\to D_{\lambda,U}\to
V_{\lambda,\mathcal{O}}^{\mathrm{min}},$

which are equivariant for the actions of $V_{0}^{\prime}$ and $\tau^{-1}$.
Writing $\mathfrak{S}_{U}$ for the image of $U$ in $\mathfrak{S}$, these
groups are naturally modules over $R=\mathcal{O}[[\mathfrak{S}]]$ and this
structure is given explicitly by the isomorphisms:

$\displaystyle D_{\lambda,U}$
$\displaystyle\cong\left(\mathcal{O}[-\lambda]\otimes_{\mathcal{O}[[\mathfrak{S}_{U}]]}R\right)\hat{\otimes}_{\mathcal{O}}\mathcal{O}[[\bar{N}_{0}]],$
$\displaystyle D_{\mathrm{univ}}$ $\displaystyle\cong
R\hat{\otimes}_{\mathcal{O}}\mathcal{O}[[\bar{N}_{0}]],$

where $\mathcal{O}[-\lambda]$ denotes $\mathcal{O}$ regarded as an
$\mathfrak{S}$ (and accordingly $\mathcal{O}[[\mathfrak{S}]]$) module via the
inverse of $\lambda$. In particular

$D_{\mathrm{univ}}\otimes_{R}\mathcal{O}[-\lambda]\cong D_{\lambda,U}$

as $R$-modules. (In this case the completed tensor product and the abstract
tensor product agree, since both modules are complete and
$\mathcal{O}[-\lambda]$ is finitely-generated.)

######  Lemma 2.7.2.

The $R$-module $D_{\mathrm{univ}}$ is flat.

###### Proof.

We have shown that $D_{\mathrm{univ}}$ is isomorphic as an $R$-module to a
power series ring in finitely many variables over $R$, which is in particular
flat. ∎

######  Proposition 2.7.3.

There is an idempotent $e_{Q}^{\prime}$ acting on
$\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$ such that

  * •

For all $i$ and all $f\in\mathcal{C}^{i}(U,D_{\mathrm{univ}})$ we have
$e^{\prime}_{Q}f=\lim_{n\to\infty}\left(\mathcal{T}^{\prime}_{\eta}\right)^{n!}f$.

  * •

There is a decomposition

$\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})=e^{\prime}_{Q}\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})\oplus\left(1-e_{Q}^{\prime}\right)\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$

such that $\mathcal{T}^{\prime}_{\eta}$ acts invertibly on the first component
and is topologically nilpotent on the second.

Moreover, the complex $e_{Q}^{\prime}\mathcal{C}(U,D_{\mathrm{univ}})$
consists of flat $R$-modules.

###### Proof.

Note that $D_{\mathrm{univ}}$ is a profinite $R$-module: there is an
isomorphism

$D_{\mathrm{univ}}\cong\mathcal{O}[[Q(\mathbf{Z}_{p})]]\cong\varprojlim_{r}\left(\mathcal{O}/p^{r}\right)[[Q(\mathbf{Z}/p^{r}\mathbf{Z})]].$

If we write $W_{r}$ for the kernel of the map
$D_{\mathrm{univ}}\to(\mathcal{O}/p^{r})[[Q(\mathbf{Z}/p^{r}\mathbf{Z})]]$
then this is a basis of open neighbourhoods for the profinite topology on
$D_{\mathrm{univ}}$ and each $W_{r}$ is preserved by the action of
$\tau^{-1}$. The complex $\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$ has
profinite components and the $W_{r}$ induce subcomplexes $W^{\bullet}_{r}$
whose components are bases of of open neighbourhoods for the components of
$\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$. The action of
$\mathcal{T}^{\prime}_{\eta}$ on $\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$
preserves $W^{\bullet}_{r}$ and thus descends to an action on
$\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})/W_{r}^{\bullet}$ for every $r$.
This complex consists of modules of finite cardinality. We can thus use the
ideas developed in [Pil20, Section 2.1] to deduce the existence of
$e^{\prime}_{Q}$.

Since each component of $\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$ is a
direct sum of copies of $D_{\mathrm{univ}}$, this is a complex of flat
$R$-modules by the previous lemma. Since
$e_{Q}^{\prime}\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$ is a direct summand
of $\mathcal{C}^{\bullet}(U,D_{\mathrm{univ}})$ we conclude that it is also a
complex of flat $R$-modules. ∎

The key input into the proof of the control theorem is the following lemma.

######  Lemma 2.7.4.

Let $M$ be a $\mathbf{Z}[U]$-module with a compatible action of $\tau^{-1}$.
The following diagram commutes on cohomology

${\mathcal{C}^{\bullet}(U^{(s)},M)}$${\mathcal{C}^{\bullet}(U,M)}$${\mathcal{C}^{\bullet}(U^{(s)},M)}$${\mathcal{C}^{\bullet}(U,M)}$$\scriptstyle{\mathcal{T}_{\eta}^{s}}$$\scriptstyle{\mathrm{cores}}$$\scriptstyle{{[\tau^{s}]}_{*}}$$\scriptstyle{\mathcal{T}_{\eta}^{s}}$$\scriptstyle{\mathrm{cores}}$

###### Proof.

This is a standard argument in Hida theory, compare e.g. [Hid95, Proposition
4.1] or [TU99, Lemma 3.1]. ∎

######  Corollary 2.7.5.

When $\mathcal{C}^{\bullet}(U,M)$ in addition admits an ordinary idempotent
$e$ in the sense of Proposition 2.7.3, the corestriction maps induce
isomorphisms

$eH^{i}(U^{(s)},M)\cong eH^{i}(U,M),$

for any $s\geq 0$.

######  Proposition 2.7.6.

There is an isomorphism

$e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\lambda,U})\cong
e^{\prime}_{Q}H^{i}(U,V_{\lambda})$

###### Proof.

We sketch the proof. The ordinary idempotent $e^{\prime}_{Q}$ contracts the
$\bar{N}_{0}$ part of $D_{\lambda,U}$ and combining this with Lemma 2.7.4
yields the isomorphism

$e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\lambda,U}/p^{s})\cong
e^{\prime}_{Q}H^{i}(V_{0}^{(s)},\mathcal{O}/(p^{s})[-\lambda]\otimes_{\mathcal{O}[\mathfrak{S}_{U}]}\mathcal{O}/(p^{s})[\mathfrak{S}]),$

where the modules occurring in the tensor product are polynomial rather than
power series because $\mathfrak{S}_{U}\backslash\mathfrak{S}$ is finite. Since
for $s\gg 0$ we have $\mathfrak{S}_{U}\backslash\mathfrak{S}\cong U\cap
V_{0}^{(s)}\backslash V_{0}^{(s)}$, Shapiro’s lemma gives an isomorphism

$\displaystyle e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\lambda,U}/p^{s})$
$\displaystyle\cong e^{\prime}_{Q}H^{i}(U\cap
V_{0}^{(s)},\mathcal{O}/(p^{s})[-\lambda])$ $\displaystyle\cong
e^{\prime}_{Q}H^{i}(U,\mathcal{O}/(p^{s})[-\lambda])$ $\displaystyle\cong
e^{\prime}_{Q}H^{i}(U,V_{\lambda}/(p^{s}))$

where the second isomorphism is given by the previous lemma and the third is
given by the fact that the ordinary idempotent contracts the representation
$V_{\lambda}$ onto its highest weight space. Then

$\displaystyle e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\lambda,U})$
$\displaystyle\cong\varprojlim_{s}e^{\prime}_{Q}H^{i}(U,V_{\lambda}/(p^{s}))$
$\displaystyle=e^{\prime}_{Q}H^{i}(U,V_{\lambda}).$

∎

######  Corollary 2.7.7.

We have an isomorphism

$e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\mathrm{univ}})\cong
e^{\prime}_{Q}H^{i}_{\mathrm{Iw}}(Q^{0},\mathcal{O})$

###### Proof.

We have

$\displaystyle e^{\prime}_{Q}H^{i}(V_{0}^{\prime},D_{\mathrm{univ}})$
$\displaystyle\cong\varprojlim_{Q^{0}\subset
U,s}e^{\prime}_{Q}H^{i}(V^{\prime}_{0},D_{1,U}/(p^{s}))$
$\displaystyle\cong\varprojlim_{Q^{0}\subset
U,s}e^{\prime}_{Q}H^{i}(U,\mathcal{O}/(p^{s}))$ $\displaystyle\cong
e^{\prime}_{Q}H^{i}_{\mathrm{Iw}}(Q^{0},\mathcal{O}).\qed$

Define $M^{\bullet}$ to be the image of
$e_{Q}^{\prime}\mathcal{C}^{\bullet}(V_{0}^{\prime},D_{\mathrm{univ}})$ in the
subcategory $\mathcal{D}^{\mathrm{flat}}(R)$ of the derived category of
$R$-modules generated by flat objects.

######  Proposition 2.7.8.

Suppose
$e_{Q}^{\prime}\mathcal{C}^{\bullet}(V_{0}^{\prime},D_{\mathrm{univ}})$ is
concentrated in degrees $[a,b]$. Then $M^{\bullet}$ is represented by a
perfect complex concentrated in degrees $[a,b]$.

###### Proof.

The cohomology groups

$H^{i}(M^{\bullet}\otimes_{R}R/J(R))=e^{\prime}_{Q}H^{i}(V_{1},\mathcal{O}/p)$

are finite and thus we can use the same method as [Pil20, Proposition 2.2.1]
to obtain the result. (The lemma as stated is for complete Noetherian _local_
rings, and our ring $R$ is only semi-local, but we can obtain the result for
$R$ by applying the lemma to the localisations of our complex at each of the
finitely many maximal ideals of $R$.) ∎

######  Theorem 2.7.9.

For all $r\geq 1$ and $\lambda\in X_{+}^{\bullet}(S)$ there is a quasi-
isomorphism

$M^{\bullet}\otimes^{\mathbb{L}}_{\mathcal{O}[[\mathfrak{S}_{U}]]}\mathcal{O}[-\lambda]\cong
e^{\prime}_{Q}\mathcal{C}^{\bullet}(U,V_{\lambda})$

###### Proof.

Since $M^{\bullet}$ is represented by the flat complex
$e^{\prime}_{Q}\mathcal{C}^{\bullet}(V_{0}^{\prime},D_{\mathrm{univ}})$, we
can compute the derived tensor product as

$e^{\prime}_{Q}\mathcal{C}^{\bullet}(V_{0}^{\prime},D_{\mathrm{univ}})\otimes_{\mathcal{O}[[\mathfrak{S}_{U}]]}\mathcal{O}[-\lambda]=e^{\prime}_{Q}\mathcal{C}^{\bullet}(V_{0}^{\prime},D_{\mathrm{univ}}\otimes_{\mathcal{O}[[\mathfrak{S}_{U}]]}\mathcal{O}[-\lambda])$

and by the previous lemmas this complex is quasi-isomorphic to
$e^{\prime}_{Q}\mathcal{C}^{\bullet}(U,V_{\lambda})$. ∎

######  Corollary 2.7.10.

There is a spectral sequence

(1)
$E_{2}^{i,j}:\mathrm{Tor}^{\mathcal{O}[[\mathfrak{S}_{U}]]}_{-i}(e_{Q}^{\prime}H^{j}_{\mathrm{Iw}}(Q^{0},\mathcal{O}),\mathcal{O}[-\lambda])\implies
e_{Q}^{\prime}H^{i+j}(U,V_{\lambda})$

supported in the second quadrant.

###### Proof.

This follows from the previous theorem using the spectral sequence for the Tor
functor (cf. Stacks Project tag 061Y). ∎

######  Definition 2.7.11.

We denote by

$\operatorname{mom}^{\lambda}_{U}:e^{\prime}_{Q}H^{j}_{\mathrm{Iw}}(Q^{0},\mathcal{O})\otimes_{\mathcal{O}[[\mathfrak{S}_{U}]]}\mathcal{O}[-\lambda]\longrightarrow
e^{\prime}_{Q}H^{i}(U,V_{\lambda,\mathcal{O}})$

the edge map of the spectral sequence (1).

We will need below an explicit description of this map as an inverse limit.
The natural edge map

$e_{Q}^{\prime}H^{i}(V_{0}^{\prime},D_{\mathrm{univ}})\otimes_{\mathcal{O}[[\mathfrak{S}_{U}]]}\mathcal{O}[-\lambda]\to
e_{Q}^{\prime}H^{i}(U,V_{\lambda})$

comes from the map

$H^{i}_{\mathrm{Iw}}(Q^{0},\mathcal{O})\cong
H^{i}(V_{0}^{\prime},D_{\mathrm{univ}})\to
H^{i}(V_{0}^{\prime},D_{\mathrm{univ}}\otimes V_{\lambda})\cong
H^{i}_{\mathrm{Iw}}(Q^{0},V_{\lambda})$

induced by the map on coefficients given by $\mu\mapsto\mu\otimes
f_{\lambda}^{\mathrm{hw}}$ and projection to level $U$. Since modulo $p^{s}$
we have $f^{\mathrm{hw}}_{\lambda}\in H^{0}(V_{0}^{(s)},V_{\lambda}/(p^{s}))$
it’s easy to see that the above map is given by

$\displaystyle\mathrm{mom}^{\lambda}:H^{i}_{\mathrm{Iw}}(Q^{0},\mathcal{O})\to
H^{i}_{\mathrm{Iw}}(Q^{0},V_{\lambda})$ $\displaystyle
x\mapsto\varprojlim\left(x\cup f_{\lambda}^{\mathrm{hw}}\ \mathrm{mod}\
p^{s}\right).$

##  3\. Branching laws for algebraic representations

We recall the setup considered in [Loe21b], where we consider pushforward of
“Eisenstein-type” cohomology classes from a group $H$ to a larger group $G$.
We shall fix an embedding $\mathcal{H}\hookrightarrow\mathcal{G}$ of reductive
groups over $\mathbf{Q}$, extending to an embedding $H\hookrightarrow G$ of
reductive group schemes over $\mathbf{Z}_{p}$.

###  3.1. Flag varieties

We fix choices of subgroups $Q,S$ etc as in §2.1, both for $G$ and for $H$;
and we distinguish between them using subscripts, so $Q_{G}$ and $Q_{H}$ are
parabolics in $G$ and $H$ respectively. Note that we do not assume any direct
compatibility between these (e.g. we do not suppose that $Q_{H}=Q_{G}\cap H$
etc). We are interested in the action of $H$, and its subgroup $Q_{H}^{0}$, on
the flag variety $\mathcal{F}=G/\overline{Q}_{G}$. We assume that there exists
$u\in\mathcal{F}(\mathbf{Z}_{p})$ such that

  1. (A)

the $Q_{H}^{0}$-orbit of $u$ is Zariski-open in $\mathcal{F}$.

  2. (B)

the image of $\overline{Q}_{G}\cap u^{-1}Q_{H}^{0}u$ under the projection
$\overline{Q}_{G}\twoheadrightarrow L_{G}\twoheadrightarrow S_{G}$ is
contained in the subtorus $S_{G}^{0}$.

######  Remark 3.1.1.

Note that the validity of (B) is independent of the choice of representative
$u$ in (A), since $\overline{Q}_{G}\cap u^{-1}Q_{H}^{0}u$ is well-defined up
to conjugation (and $S_{G}$ is commutative).

###  3.2. Highest-weight representations

######  Proposition 3.2.1.

For any $\lambda\in X^{\bullet}_{+}(S_{G}/S_{G}^{0})$, we have
$\dim(V^{G}_{\lambda})^{Q_{H}^{0}}\leq 1$. If this dimension is 1, then
$(V_{\lambda}^{G})^{Q_{H}^{0}}$ is spanned by the unique polynomial
$\operatorname{br}_{\lambda}\in K[G]$ (a “branching polynomial”) such that
$\operatorname{br}_{\lambda}\left(\overline{q}u^{-1}q^{\prime}\right)=\lambda(\overline{q})$
for $\overline{q}\in\overline{Q}_{G}$ and $q^{\prime}\in Q_{H}^{0}$.

###### Proof.

Suppose $f\in(V_{\lambda}^{G})^{Q_{H}^{0}}$. Then the restriction of $f$ to
the subvariety $\overline{Q}_{G}u^{-1}Q_{H}^{0}$ of $G$ is uniquely determined
by $f(u^{-1})$. Since $\overline{Q}_{G}u^{-1}Q_{H}^{0}$ is open (and $G$ is
connected), this implies that $f$ itself is uniquely determined, as an element
of $K[G]$, by $f(u^{-1})$. In particular, if $f$ is non-zero, then we can
scale it so that $f(u^{-1})=1$. ∎

######  Remark 3.2.2.

Conversely, if $S_{G}^{0}$ is exactly the image of $\overline{Q}_{G}\cap
u^{-1}Q_{H}^{0}u$ in $S_{G}$, then any weight $\lambda\in
X^{\bullet}_{+}(S_{G})$ such that $(V^{G}_{\lambda})^{Q_{H}^{0}}\neq 0$ must
lie in $X^{\bullet}_{+}(S_{G}/S_{G}^{0})$. However, it is convenient not to
require $S_{G}^{0}$ to have this minimality property.

######  Definition 3.2.3.

We shall say that a weight $\lambda\in X^{\bullet}(S_{G}/S_{G}^{0})$ is
_$Q_{H}^{0}$ -admissible_ if $(V_{\lambda}^{G})^{Q_{H}^{0}}\neq 0$.

If this holds, then $Q_{H}/Q_{H}^{0}$ must act on
$(V_{\lambda}^{G})^{Q_{H}^{0}}$ via a weight $\mu\in
X^{\bullet}(S_{H}/S^{0}_{H})$, and this weight must be dominant (since it is
the highest weight of the $H$-representation spanned by
$(V_{\lambda}^{G})^{Q_{H}^{0}}$, which is finite-dimensional). We denote this
weight by $\mu(\lambda)$, or just $\mu$ if $\lambda$ is clear from context.

######  Remark 3.2.4.

The natural map

$\frac{Q_{H}\cap u\overline{Q}_{G}u^{-1}}{Q_{H}^{0}\cap
u\overline{Q}_{G}u^{-1}}\to Q_{H}/Q_{H}^{0}$

is injective, and for dimension reasons, it must in fact be an isomorphism.
The left-hand side maps naturally to $S_{G}/S_{G}^{0}$; so we can characterise
$\mu(\lambda)$ as the pullback of $\lambda$ to this quotient.

######  Proposition 3.2.5.

If $\lambda$ is $Q_{H}^{0}$-admissible, then there exists a unique
homomorphism

$V^{H}_{\mu}\to\left(V^{G}_{\lambda}\middle)\right|_{H},$

where $\mu\in X^{\bullet}(L_{H}/L^{0}_{H})$ is the character by which $Q_{H}$
acts on $\left(V^{G}_{\lambda}\right)^{Q_{H}^{0}}$. This is characterised by
mapping the highest-weight vector $f^{\mathrm{hw}}_{\mu}$ of $V^{H}_{\mu}$ to
$\operatorname{br}_{\lambda}$.

###### Proof.

This follows from Frobenius reciprocity: $V^{H}_{\mu}$ is isomorphic to the
co-induced module $\operatorname{Coind}_{Q_{H}}^{H}(\mu)$ (which is left
adjoint to restriction of representations, while the usual induction is right
adjoint). ∎

######  Proposition 3.2.6.

If $\lambda$ is a $Q_{H}^{0}$-admissible weight, then
$\operatorname{br}_{\lambda}\in V^{G}_{\lambda,\mathcal{O}}$.

###### Proof.

Since $G_{/\mathcal{O}}$ is smooth over $\mathcal{O}$, it is a normal scheme,
so it suffices to check that $\operatorname{br}_{\lambda}$ is regular away
from a subvariety of codimension $\geq 2$ (Hartogs’ lemma). By definition, it
is regular in characteristic 0, and also on an open dense subvariety of
$G_{/k}$, where $k=\mathcal{O}/p\mathcal{O}$; so we are done. ∎

######  Corollary 3.2.7.

The branching map of proposition 3.2.5 restricts to a map of
$\mathcal{O}$-modules

$V^{H,\min}_{\mu,\mathcal{O}}\to V^{G}_{\lambda,\mathcal{O}}.$

######  Remark 3.2.8.

Compare [LSZ21a, Proposition 4.3.5]. (We do not know whether the minimal
lattice maps to the minimal lattice, nor whether the maximal one maps to the
maximal one.)

###  3.3. Projection to the highest-weight subspace

If $\lambda$ is an $Q_{H}^{0}$-admissible weight, we thus have two canonical
vectors in $V^{G}_{\lambda}$: namely, the highest-weight vector
$f_{\lambda}^{\mathrm{hw}}$, and the $Q_{H}^{0}$-invariant vector
$\operatorname{br}_{\lambda}$. We also have a canonical linear functional
$\psi_{\lambda}$ on $V_{\lambda}$, given by evaluation at $\mathrm{id}_{G}$,
which factors through projection from $V_{\lambda}$ to its highest-weight
space (i.e. it is a lowest-weight vector of weight $-\lambda$ for the
contragredient representation $(V_{\lambda}^{G})^{\vee}$).

######  Proposition 3.3.1.

We have
$\psi_{\lambda}(f_{\lambda}^{\mathrm{hw}})=\psi_{\lambda}(u^{-1}\cdot\operatorname{br}_{\lambda})$.

###### Proof.

Both sides are, by definition, equal to 1. ∎

######  Corollary 3.3.2.

Let $\eta\in\Sigma^{++}$. If $\lambda$ is $Q_{H}^{0}$-admissible, then for any
$r\geq 1$ we have

$p^{hr}\rho_{\lambda}(\eta(p)^{-r})\cdot(f_{\lambda}^{\mathrm{hw}}-u^{-1}\cdot\operatorname{br}_{\lambda})\in
p^{r}V_{\lambda,\mathcal{O}}.$

###### Proof.

Since $V_{\lambda,\mathcal{O}}$ is stable under the action of $G$, it is
certainly stable under the action of $\mathbf{G}_{m,/\mathbf{Z}_{p}}$ defined
by $\eta$, and hence is equal to the direct sum of its eigenspaces for the
powers of $\eta$. So to prove that $p^{h}\rho_{\lambda}(\eta(p)^{-1})$ maps
$V_{\lambda,\mathcal{O}}$ to itself, it suffices to show that the eigenvalues
of $\rho_{\lambda}(\eta(p)^{-1})$ on $V_{\lambda}$ are in
$p^{-h}\mathbf{Z}_{p}$. Clearly, these eigenvalues are given by
$p^{-\langle\eta,\mu\rangle}$ where $\mu$ varies over the weights
$V_{\lambda}$. However, since $\lambda$ is the highest weight of
$V_{\lambda}$, and $\eta$ is dominant, all such weights satisfy
$\langle\eta,\mu\rangle\leq\langle\eta,\lambda\rangle=h$.

To prove the second statement, it suffices to take $r=1$. By proposition
3.3.1, $f_{\lambda}^{\mathrm{hw}}-u^{-1}\cdot\operatorname{br}_{\lambda}$ has
zero projection to the highest relative weight space. So it is a sum of
vectors lying in relative weight spaces of strictly smaller weights, and hence
$p^{h}\rho_{\lambda}(\eta(p)^{-1})$ acts on these with eigenvalues in
$p\mathbf{Z}_{p}$. ∎

##  4\. Norm-compatible families

###  4.1. Subgroups

Let us choose $\eta\in\Sigma^{++}$ (which will remain fixed throughout the
construction), and set $\tau=\eta(p)$. For $r\geq 0$, let us write

$N_{r}=\tau^{r}N_{G}(\mathbf{Z}_{p})\tau^{-r},\qquad\overline{N}_{r}=\tau^{-r}\overline{N}_{G}(\mathbf{Z}_{p})\tau^{r}.$

Note that $N_{r}$ and $\overline{N}_{r}$ are contained (usually strictly) in
the kernel of reduction modulo $p^{r}$. We set $L_{r}=\\{\ell\in
L_{G}(\mathbf{Z}_{p}):\ell\bmod{p^{r}}\in L_{G}^{0}\\}$. Then the “norm-
compatibility machine” of [Loe21b] uses the open compact subgroups $U_{r}$ and
$V_{r}$ of $G(\mathbf{Z}_{p})$ defined for $r\geq 1$ by

$U_{r}=\overline{N}_{0}L_{r}N_{r},\qquad V_{r}=\overline{N}_{r}L_{r}N_{0}.$

Note that $(U_{r})_{r\geq 1}$ is a decreasing sequence of open neighbourhoods
of $\overline{Q}_{G}^{0}(\mathbf{Z}_{p})$, and $(V_{r})_{r\geq 1}$ similarly
of $Q^{0}_{G}(\mathbf{Z}_{p})$.

###  4.2. Hecke actions

For $r\geq 1$, the groups $U_{r}$ and $V_{r}$ have Iwahori decompositions, so
we have an integrally normalised Hecke operator $\mathcal{T}^{\prime}_{\eta}$
on cohomology at these levels, defined in weight $\lambda$ using the
normalisation factor $p^{h}$ where $h=\langle\eta,\lambda\rangle$. Similarly,
we obtain an integrally normalised pushforward map

$[\tau^{r}]_{*}:H^{i}(U_{r},\mathcal{V}_{\lambda,\mathcal{O}})\to
H^{i}(V_{r},\mathcal{V}_{\lambda,\mathcal{O}}),$

whose effect on the coefficient sheaves is given by
$p^{hr}\rho_{\lambda}(\tau^{-r})$. Note that this map commutes with the
actions of $\mathcal{T}^{\prime}_{\eta}$.

###  4.3. Pushforward from $H$

######  Definition 4.3.1.

Let $c$ be the integer

$\dim_{\mathbf{R}}(Y_{\mathcal{G}})-\dim_{\mathbf{R}}(Y_{\mathcal{H}})-\operatorname{rk}_{\mathbf{R}}\left(\tfrac{Z_{H}}{Z_{G}\cap
H}\right)\geq 0,$

where $\operatorname{rk}_{\mathbf{R}}$ denotes the split rank over
$\mathbf{R}$. (Note that this is simply
$2\left(\dim_{\mathbf{C}}(Y_{\mathcal{G}})-\dim_{\mathbf{C}}Y_{\mathcal{H}}\right)$,
if $\mathcal{H}$ and $\mathcal{G}$ admit compatible Shimura data.)

For any open compacts $U_{G}\subseteq G(\mathbf{Z}_{p})$ and $U_{H}\subset
H(\mathbf{Z}_{p})$ with $U_{H}\subset H\cap U_{G}$, and any $H$-admissible
weight $\lambda$, the maps of corollary 3.2.7 give us morphisms

$\iota_{\star}^{[\lambda]}:H^{i}\left(U_{H},\mathcal{V}^{H,\min}_{\mu,\mathcal{O}}\right)\to
H^{i+c}\left(U_{G},\mathcal{V}^{G}_{\lambda,\mathcal{O}}\right).$

(As usual, we work with some fixed tame levels $K_{\mathcal{G}}^{p}$ and
$K_{\mathcal{H}}^{p}$ away from $p$; we shall suppose
$K_{\mathcal{H}}^{p}=K_{\mathcal{G}}^{p}\cap\mathcal{H}(\mathbf{A}_{f}^{p})$.)

######  Remark 4.3.2.

In the Shimura-variety setting, $c=2d$ is always even, and the pushforward map
for étale cohomology takes the form

$H^{i}_{\text{{\'{e}t}}}\left(U_{H},\mathcal{V}^{H,\min}_{\mu,\mathcal{O}}(j)\middle)\to
H^{i+2d}_{\text{{\'{e}t}}}\middle(U,\mathcal{V}^{G}_{\lambda,\mathcal{O}}(j+d)\right)$

for any $i,j$, where $(j)$ denotes a cyclotomic twist.

Now let $\lambda$ be a $Q_{H}^{0}$-admissible weight, and $\mu$ the
corresponding weight for $H$. We suppose we are given a compatible family of
classes (for some given $i\geq 0$),

$z_{H}^{[\mu]}\in
H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0},\mathcal{V}_{\mu,\mathcal{O}}^{H,\min}\right).$

######  Remark 4.3.3.

We remind the reader that the special case when $Q_{H}^{0}=H$, $\mu=0$, and
$z_{H}^{[\mu]}$ is the identity class in
$H^{0}(H(\mathbf{Z}_{p}),\mathcal{O})$ is permitted (and not at all trivial).

######  Definition 4.3.4.

We let $z_{G,r}^{[\lambda]}\in
H^{i+c}\left(U_{r},\mathcal{V}_{\lambda,\mathcal{O}}^{G}\right)$ be the
following class: it is the image of $z_{H}^{[\mu]}$ under the composite map

$\displaystyle
H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0},\mathcal{V}_{\mu,\mathcal{O}}^{H,\min}\right)$
$\displaystyle\xrightarrow{\operatorname{pr}^{\star}}H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0}\cap
uU_{r}u^{-1},\mathcal{V}_{\mu,\mathcal{O}}^{H,\min}\right)$
$\displaystyle\xrightarrow{\iota_{\star}^{[\lambda]}}H^{i+c}\left(uU_{r}u^{-1},\mathcal{V}_{\lambda,\mathcal{O}}^{G}\right)$
$\displaystyle\xrightarrow{[u]_{*}}H^{i+c}\left(U_{r},\mathcal{V}_{\lambda,\mathcal{O}}^{G}\right).$

We set

$\xi_{G,r}^{[\lambda]}=[\tau^{r}]_{*}(z_{G,r}^{[\lambda]})\in
H^{i+c}\left(V_{r},\mathcal{V}_{\lambda,\mathcal{O}}^{G}\right).$

(Here $\operatorname{pr}^{\star}$ is the natural pullback map, and
$[\tau]_{*}$ the integrally normalised pushforward as above.)

The main theorem of [Loe21b] is the following:

######  Theorem 4.3.5.

If $\operatorname{pr}_{V_{r+1},V_{r},\star}$ denotes pushforward along the map
$Y_{\mathcal{G}}(V_{r+1})\to Y_{\mathcal{G}}(V_{r})$, then we have

$\operatorname{pr}_{V_{r+1},V_{r},\star}\left(\xi^{[\lambda]}_{G,r+1}\right)=\mathcal{T}_{\eta}^{\prime}\cdot\xi^{[\lambda]}_{G,r},$

for all $r\geq 1$.

######  Definition 4.3.6.

We define

$\xi_{G,\infty}^{[\lambda]}\coloneqq\left(\mathop{(\mathcal{T}_{\eta}^{\prime})^{-r}e^{\prime}_{Q_{G}}}\xi_{G,r}^{[\lambda]}\right)_{r\geq
1}\in\mathop{e^{\prime}_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(V_{\infty},\mathcal{V}^{G}_{\lambda,\mathcal{O}}\right),$

and we let

$\iota_{\infty}^{[\lambda]}:H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0},\mathcal{V}_{\mu,\mathcal{O}}^{H,\min}\right)\longrightarrow\mathop{e^{\prime}_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{V}^{G}_{\lambda,\mathcal{O}}\right)$

to be the map sending $z_{H}^{[\mu]}$ to $\xi_{G,\infty}^{[\lambda]}$.

Note that $\lambda$ (and hence $\mu$) are fixed in the above construction. The
goal of this paper is to compare the classes $\xi_{G,\infty}^{[\lambda]}$ for
different values of $\lambda$.

##  5\. Moment maps

###  5.1. Definitions

Let $r\geq 1$, and let $\lambda\in X^{\bullet}_{+}(S_{G}/S_{G}^{0})$. (For the
moment we do not assume $\lambda$ is $H$-admissible.)

Since $f_{\lambda}^{\mathrm{hw}}$ is a highest-weight vector, and its weight
is trivial on $L_{G}^{0}$, the mod $p^{r}$ reduction of this vector is
invariant under the group $Q_{G}^{0}(\mathbf{Z}/p^{r})$, which is the image of
$V_{r}$ in $G(\mathbf{Z}/p^{r})$. Thus the reduction of
$f_{\lambda}^{\mathrm{hw}}$ defines a class

$f_{\lambda,r}^{\mathrm{hw}}\in H^{0}(V_{r},\mathcal{V}_{\lambda,r}),$

and these have the property

(2) $f_{\lambda,{r+1}}^{\mathrm{hw}}\bmod
p^{r}=\operatorname{pr}_{V_{r+1},V_{r}}^{\star}\left(f_{\lambda,r}^{\mathrm{hw}}\right)\quad\in
H^{0}(V_{r+1},\mathcal{V}_{\lambda,r}).$

######  Definition 5.1.1.

The _moment map_ of weight $\lambda$ and level $r$ is the map

$\operatorname{mom}^{\lambda}_{G}\ \ :\ \
H^{c}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{O}\right)\longrightarrow
H^{c}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{V}_{\lambda,\mathcal{O}}\right)$

defined as follows: if $\underline{x}=(x_{s})_{s\geq
1}\in\varprojlim_{s}H^{c}\left(V_{s},\mathcal{O}\right)$, then
$\operatorname{mom}^{\lambda}(\underline{x})$ is given by

$\left((x_{s}\bmod p^{s})\cup f_{\lambda,s}^{\mathrm{hw}}\right)_{s\geq
1}\in\varprojlim_{s}H^{c}\left(V_{s},\mathcal{V}_{\lambda,s}\right)\cong
H^{c}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{V}_{\lambda,\mathcal{O}}\right).$

which is well-defined by (2).

By construction this map is compatible with the action of the Hecke operator
$\mathcal{T}^{\prime}_{\eta}$, and hence with the ordinary projector
$e_{Q_{G}}$. (It is _not_ compatible with the action of the abelian group
$S_{G}(\mathbf{Z}_{p})$.) Moreover, we can make the same definitions with the
lattice $V_{\lambda,\mathcal{O}}$ replaced with
$V_{\lambda,\mathcal{O}}^{\mathrm{min}}$ (and the moment maps for the two
lattices are compatible).

###  5.2. Twist-compatibility

We now bring the group $H$ back into the picture. Suppose $\lambda\in
X^{\bullet}_{+}(S_{G})$ is $Q_{H}^{0}$-admissible, and $\mu=\mu(\lambda)$ as
above. We consider the following diagram of maps:

${H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0},\mathcal{O}\right)}$${\mathop{e_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{O}\right)}$${H^{i}_{\mathrm{Iw}}\left(Q_{H}^{0},\mathcal{V}_{\mu,\mathcal{O}}^{H,\min}\right)}$${\mathop{e_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{V}^{G}_{\lambda,\mathcal{O}}\right).}$$\scriptstyle{\iota_{\infty}^{[0]}}$$\scriptstyle{\operatorname{mom}^{\mu}_{H}}$$\scriptstyle{\operatorname{mom}_{G}^{\lambda}}$$\scriptstyle{\iota_{\infty}^{[\lambda]}}$

The following is the key technical result of the present paper:

######  Theorem 5.2.1.

The above diagram is commutative.

######  Remark 5.2.2.

As in remark 2.6.3 above, this can be generalised so that instead of comparing
trivial coefficients with coefficients in
$\mathcal{V}^{G}_{\lambda,\mathcal{O}}$, we compare coefficients
$\mathcal{V}_{\nu,\mathcal{O}}$ and $\mathcal{V}_{\lambda+\nu,\mathcal{O}}$,
where $\nu$ is not necessarily trivial on $L^{\mathrm{der}}$. We leave the
details to the interested reader.

###### Proof.

From the definition of the moment maps, it suffices to prove the following
claim: suppose we define
$z_{H}^{[\mu]}=\operatorname{mom}_{H}^{\mu}(z_{H}^{[0]})$ and apply the
constructions above in weight 0 and in weight $\lambda$. Then for each $r\geq
1$ we have the following equality of elements of
$H^{c}(V_{r},\mathcal{V}_{\lambda,r})$:

$\xi_{G,r}^{[\lambda]}\bmod p^{r}=(\xi_{G,r}^{[0]}\bmod p^{r})\cup
f_{\lambda,r}^{\mathrm{hw}}.$

Recall that $V_{r}=\tau^{-r}U_{r}\tau^{r}$. We thus have an isomorphism
$\tau^{r}:Y_{G}(U_{r})\to Y_{G}(V_{r})$, and our (integrally normalised)
pushforward map $H^{c}(Y_{G}(U_{r}),\mathcal{V}_{\lambda,r})\to
H^{c}(Y_{G}(V_{r}),\mathcal{V}_{\lambda,r})$ can be written more explicitly as
the composite

$H^{c}(Y_{G}(U_{r}),\mathcal{V}_{\lambda,r})\longrightarrow
H^{c}(Y_{G}(U_{r}),(\tau_{r})^{*}\mathcal{V}_{\lambda,r})\xrightarrow[\cong]{\
(\tau_{r})_{\star}\ }H^{c}(Y_{G}(V_{r}),\mathcal{V}_{\lambda,r}),$

where the first map is given by the morphism of sheaves
$\mathcal{V}_{\lambda,r}\to(\tau_{r})^{*}\mathcal{V}_{\lambda,r}$
corresponding to the endomorphism $A=p^{hr}\rho_{\lambda}(\tau^{-r})$ of
$V_{\lambda,\mathcal{O}}$.

Since $V_{\lambda,\mathcal{O}}$ is an admissible lattice, it is the direct sum
of its intersections with the weight spaces of $V_{\lambda}$ for the action of
$S$. On the highest-weight space, $A$ acts as the identity, and on all other
weight spaces, it acts as multiplication by $p^{nr}$ for some $n\geq 1$.
Hence, modulo $p^{r}$, the map $A$ factors through projection to the highest
weight space.

Since $A$ acts trivially on the highest-weight space, we have
$A(f^{\mathrm{hw}}_{\lambda,r})=(\tau^{r})^{\star}(f^{\mathrm{hw}}_{\lambda,r})$.
So it suffices to prove that

$z_{G,r}^{[\lambda]}\bmod p^{r}=\Big{(}(z_{G,r}^{[0]}\bmod p^{r})\cup
f_{\lambda,r}^{\mathrm{hw}}\Big{)}\pmod{\ker A},$

as elements of $H^{i+c}(U_{r},\mathcal{V}_{\lambda,r})$. Via adjunction
between pushforward and pullback, we are reduced to showing that the classes
$u^{-1}f_{\lambda,r}^{H}$ and $f_{\lambda,r}^{\mathrm{hw}}$, viewed as vectors
in $V_{\lambda,r}$ invariant under $u^{-1}Hu\cap U_{r}$, agree modulo
$\ker(A)$; and this is precisely the statement of corollary 3.3.2. ∎

Applying the maps in the diagram to the class $z_{H}^{[0]}$, and defining
$z_{H}^{[\mu]}$ using the moment maps as above, we have proved the following:

######  Theorem 5.2.3.

Let
$\xi_{G,\infty}\coloneqq\xi_{G,\infty}^{[0]}\in\mathop{e_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{O}\right)$.
Then, for all $Q_{H}^{0}$-admissible $\lambda$ and all $r\geq 1$, we have

$\operatorname{mom}_{G,V_{r}}^{[\lambda]}\left(\xi_{G,\infty}\right)=(\mathcal{T}_{\eta}^{\prime})^{-r}e^{\prime}_{Q_{G}}\cdot\xi_{G,r}^{[\lambda]}.$

(We shall give a formula in the next section for the moments of
$\xi_{G,\infty}$ at parahoric level, which may be more convenient in
applications than the level groups $V_{r}$ used above.)

######  Remark 5.2.4.

Thus the single Iwasawa-comology class $\xi_{G,\infty}$ “knows” the classes
$\xi_{G,r}^{[\lambda]}$ (or at least their projections to the ordinary part)
for all $r$ and all $\lambda$. As $\xi_{G,\infty}$ lies in the module
$\mathop{e_{Q_{G}}}H^{(i+c)}_{\mathrm{Iw}}\left(Q_{G}^{0},\mathcal{O}\right)$,
which is finitely-generated over $\mathcal{O}[[\mathfrak{S}]]$ by theorem
2.6.2, this forces very strong compatibilities between the
$\xi_{G,r}^{[\lambda]}$ for varying $\lambda$, which is the crucial input
needed to prove explicit reciprocity laws.

######  Corollary 5.2.5.

Suppose that our cohomology is étale cohomology, and for all $r\geq 1$ and all
$Q_{H}^{0}$-admissible $\lambda$, the class $z_{H,r}^{[\mu(\lambda)]}$ is the
image of a motivic cohomology class. Then
$\operatorname{mom}_{G,V_{r}}^{\lambda}(\xi_{G,\infty})$ is in the image of
motivic cohomology for all $\lambda$, where we set
$\xi_{G,\infty}=\xi_{G,\infty}^{[0]}$.

Note that the “motivicity” assumption on the classes $z_{H,r}^{[\mu]}$ is
automatic if $Q_{H}^{0}=H$ (which implies $\mu(\lambda)=0$ for all $\lambda$)
and $z_{H,r}^{[0]}$ is the identity class in $H^{0}$; this is the “algebraic
cycle” setting. Crucially for many applications, the $\operatorname{GL}_{2}$
Eisenstein classes also satisfy this motivicity condition, by the main result
of [Kin15].

######  Remark 5.2.6.

Special cases of this computation appear in many of our previous works, such
as:

  * •

Theorem 6.3.4 of [KLZ17] is the case
$\mathcal{G}=\operatorname{GL}_{2}\times\operatorname{GL}_{2}$,
$\mathcal{H}=\operatorname{GL}_{2}$, with $Q_{H}$ and $Q_{G}$ taken to be
Borel subgroups.

  * •

The case $\mathcal{G}=\operatorname{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$
and $\mathcal{H}=\operatorname{GL}_{2}$, with $F$ a quadratic field, is
considered in [LLZ18, Theorem 8.2.3] (for $F$ real quadratic, using étale
cohomology) and [LW20, Proposition 5.6] (for $F$ imaginary quadratic, using
Betti cohomology). Again, we take $Q_{H}$ and $Q_{G}$ here to be Borel
subgroups.

  * •

Theorem 9.6.4 of [LSZ21a] is an analogous result for
$\mathcal{G}=\operatorname{GSp}_{4}$ and
$\mathcal{H}=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$,
taking $Q_{G}$ to be the Siegel parabolic subgroup. In this case we did not
assume $L_{G}/L_{G}^{0}$ to be commutative.

  * •

Theorem 11.2.1 of [LSZ21b] is the case $G=\operatorname{GU}(2,1)$,
$H=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{Res}_{E/\mathbf{Q}}\operatorname{GL}_{1}$,
for $E$ an imaginary quadratic field.

In some of these works there are various correction terms appearing (such as
the factor $(-2)^{-q}$ which appears in several places in [LSZ21a]) which are,
in effect, a consequence of using the “wrong” normalisations for the branching
maps. In the present work, we have avoided these correction terms, by
normalising the branching maps for algebraic representations using the same
element $u$ used to define the pushforward maps $\iota_{\infty}^{[0]}$.

We briefly mention some related works by other authors in the literature. A
result roughly equivalent to the above theorem in the case of
Gross–Kudla–Schoen diagonal cycles (associated to
$G=(\operatorname{GL}_{2})^{3}$, $H=\operatorname{GL}_{2}$ embedded
diagonally) has recently been proved in [DR22] and independently in [BSV22].
The methods of these works (in particular the former) appear to be closely
related to the present work. The case of $G=\operatorname{GL}_{2}$ and $H$ a
non-split torus (the “Heegner point” setting) has also been explored,
initially by Castella [Cas13] using very different methods based on $p$-adic
$L$-functions, and in a more general setting by Disegni [Dis22] using methods
rather closer to those of this paper.

##  6\. Interpolating properties

###  6.1. Parahoric-level classes

It will be convenient to work with a slightly different family of subgroups.
Let $J_{G}=\\{g\in G(\mathbf{Z}_{p}):g\bmod p\in Q_{G}\\}$, and similarly
$\bar{J}_{G}$. On the other hand, let $J_{H}^{0}=\\{h:h\bmod p\in
Q_{H}^{0}\\}$. Then the quotient $J_{H}^{0}\backslash
G(\mathbf{Z}_{p})/\bar{J}_{G}$ is identified with the
$Q_{H}^{0}(\mathbf{F}_{p})$-orbits on $\mathcal{F}(\mathbf{F}_{p})$, and the
double coset of $u$ represents the unique open orbit. Thus we have a
pushforward map

$u_{\star}\circ\iota_{\star}:H^{i}(J_{H}^{0},V^{H,\min}_{\mu,\mathcal{O}})\to
H^{i+c}(\bar{J}_{G},V^{G}_{\lambda,\mathcal{O}}),$

and this map is independent of the choice of $u$ in its orbit (except via the
normalisation of the branching map).

######  Proposition 6.1.1.

The map $\mathcal{B}:H^{*}(J_{G},V_{\lambda,\mathcal{O}})\to
H^{*}(\bar{J}_{G},V_{\lambda,\mathcal{O}})$ given by pullback to level
$\bar{J}_{G}\cap J_{G}$ composed with the trace map, commutes with the action
of the operators $\mathcal{T}_{\eta}^{\prime}$, and induces an isomorphism on
the ordinary part for these operators.

###### Proof.

Let $\mathcal{A}$ be the map $H^{*}(\bar{J}_{G},V_{\lambda,\mathcal{O}})\to
H^{*}(J_{G},V_{\lambda,\mathcal{O}})$, given by the double coset
$[J_{G}\eta(p)^{-1}\bar{J}_{G}]$ (normalised as in definition 2.5.1). Then one
checks that the composite of $\mathcal{A}$ and $\mathcal{B}$, either way
around, is $\mathcal{T}^{\prime}_{\eta}$; and the result follows. ∎

For $\lambda$ an $H$-admissible weight, and $\mu$ the corresponding weight of
$H$, we have a class

$\xi_{J_{G}}^{[\lambda]}=\mathcal{B}^{-1}\left(e^{\prime}_{Q}\cdot\left(u_{\star}\circ\iota_{\star}\right)(z_{J_{H}^{0}}^{[\mu]})\right)\in
H^{i+c}(J_{G},V_{\lambda,\mathcal{O}}),$

where $z_{J_{H}^{0}}^{[\mu]}$ is the image of $z_{H}^{[\mu]}$ at level
$J_{H}^{0}$, pulled back to $J_{H}^{0}\cap u\bar{J}_{G}u^{-1}$.

######  Proposition 6.1.2.

For each $Q_{H}^{0}$-admissible $\lambda$, we have

$\xi_{J_{G}}^{[\lambda]}=\operatorname{mom}_{G,J_{G}}^{[\lambda]}\left(\xi_{G,\infty}\right).$

###### Proof.

This follows by an argument using Cartesian diagrams which is very similar to
the proof of the norm-compatibility relations; we leave the details to the
reader. ∎

There is an analogous map
$\mathcal{B}^{*}:H^{*}(\bar{J}_{G},V_{\lambda,\mathcal{O}}^{\vee})\to
H^{*}(J_{G},V_{\lambda,\mathcal{O}}^{\vee})$ defined by interchanging the
roles of $Q_{G}$ and $\bar{Q}_{G}$, and one checks that $\mathcal{B}^{*}$ is
the transpose of $\mathcal{B}$ with respect to Poincaré duality. This commutes
with the “dominant” Hecke operator $\mathcal{T}_{\eta}$ defined by the double
coset of $\eta(p)$ (the transpose of $\mathcal{T}^{\prime}_{\eta}$). So, if we
are given some $v\in H^{*}_{c}(J_{G},V_{\lambda,\mathcal{O}}^{\vee})$ which is
an ordinary eigenvector for $\mathcal{T}_{\eta}$, in the cohomological degree
complementary to $i+c$, we can define $\bar{v}=(\mathcal{B}^{*})^{-1}(v)$,
which is a $\mathcal{T}$-eigenvector at level $\bar{J}_{G}$; and we compute
that

$\left\langle\xi^{[\lambda]}_{J_{G}},v\right\rangle_{J_{G}}=\left\langle\left(u_{\star}\circ\iota_{\star}\right)(z_{J_{H}}^{[\mu]}),\bar{v}\right\rangle_{\bar{J}_{G}}=\left\langle
z_{J_{H}^{0}}^{[\mu]},\iota^{*}u^{*}(\bar{v})\right\rangle_{J_{H}^{0}\cap
u\overline{J}_{G}u^{-1}}.$

######  Remark 6.1.3.

It is natural to expect that the latter expression – an “open-orbit
cohomological period” at parahoric level – should be related to cohomological
periods at prime-to-$p$ level, via an appropriate Euler factor. For examples
of statements of this kind for specific groups $G$ and $H$, see e.g. [KLZ17,
Theorem 5.7.6] for
$\operatorname{GL}_{2}\subset\operatorname{GL}_{2}\times\operatorname{GL}_{2}$,
and the results for $\operatorname{GSp}_{4}$ proved in [Loe21a] and recalled
in the next section. It would be interesting to attempt to find a general
recipe for this Euler factor; at least in the special case when $Q_{H}^{0}=H$
and $Q_{G}$ is a Borel, this may be possible using the results of
Sakellaridis, in particular [Sak13, Theorem 1.3.1].

##  7\. Applications to $\operatorname{GSp}_{4}$

We now consider applications of the above theory to Euler systems for
$\operatorname{GSp}_{4}$.

###  7.1. A variant of the setting of [LSZ21a]

Let $0\leq q\leq a,0\leq r\leq b$ be integers; and let
$G=\operatorname{GSp}_{4}$. We choose a set of primes $S\ni p$, and auxiliary
data $K_{S},\underline{\phi}_{S},W,c_{1},c_{2}$, as in §8.4.6 of [LSZ21a];
these will remain fixed throughout the following discussion. For simplicity,
we also assume that the function $\underline{\phi}_{S}$ has the form
$\phi_{1,S}\otimes\phi_{2,S}$ where each $\phi_{i,S}$ vanishes at $(0,0)$ and
transforms under $\mathbf{Z}_{S}^{\times}$ by (the adelic character attached
to) $\chi_{i}^{-1}$, where $\chi_{i}$ is a Dirichlet character unramified
outside $S$. This allows us to define classes

${}_{c_{1},c_{2}}z_{\text{{\'{e}t}}}^{[a,b,q,r]}\in
H^{4}_{\text{{\'{e}t}}}\left(G(\mathbf{Z}_{p}),\mathscr{D}^{a,b}_{\mathbf{Z}_{p}}(3-q)\right),$

which are given by the same construction as _loc.cit._ (with $M=1$) but taking
the test data at $p$ to be the spherical test data.

We now consider the group

$e^{\prime}_{B_{G}}H^{4}_{\text{{\'{e}t}},\mathrm{Iw}}\Big{(}N_{G}(\mathbf{Z}_{p}),\mathbf{Z}_{p}(3)\Big{)},$

where $B_{G}$ is the upper-triangular Borel and $N_{G}$ its unipotent radical.
This is a finitely-generated module over the Iwasawa algebra of
$T_{G}(\mathbf{Z}_{p})$, and we can identify this group with
$(\mathbf{Z}_{p}^{\times})^{3}$ in such a way that the character $(a,b,q)$
corresponds to the highest weight of $\mathscr{D}^{a,b}(-q)$. Thus the fibre
of the Iwasawa cohomology at $(a,b,q)$ maps canonically (via the edge map of
eq. 1) to the ordinary part of the cohomology of $\mathscr{D}^{a,b}(3-q)$ at
Iwahori level.

Given an automorphic representation $\Pi$ of $G$ as in §10 of _op.cit._ , of
weight $(a,b)$, good ordinary at $p$, we can identify the $\Pi^{*}$-eigenspace
in the ordinary cohomology at Iwahori level with the $\Pi^{*}$-eigenspace at
prime-to-$p$ level, via a suitable Hecke operator (compare Note 17.1.6 of
[LZ20a] for Siegel-parahoric level). We label the Hecke parameters of $\Pi$ as
$(\alpha,\beta,\gamma,\delta)$ in increasing order of valuation, so
$v_{p}(\alpha)=0$ and $v_{p}(\beta)=a+1$.

######  Theorem 7.1.1.

There exists a class

${}_{c_{1},c_{2}}z_{\text{{\'{e}t}}}\in
H^{4}_{\text{{\'{e}t}},\mathrm{Iw}}\Big{(}N_{G}(\mathbf{Z}_{p}),\mathbf{Z}_{p}(3)\Big{)}\mathop{\hat{\otimes}}\mathbf{Z}_{p}[[\mathbf{Z}_{p}^{\times}]]$

such that for all $a,b,q,r$ as above, and any cuspidal automorphic
representation $\Pi$ of weight $(a,b)$ ordinary at $p$, the image of
${}_{c_{1},c_{2}}z_{\text{{\'{e}t}}}$ under evaluation at $(a,b,q,r)$ and
projection into $H^{1}(\mathbf{Q},W_{\Pi_{f}}^{*}(-q))$ is
$(-2)^{-q}\mathcal{E}\cdot\operatorname{pr}_{\Pi}\left({}_{c_{1},c_{2}}z_{\text{{\'{e}t}}}^{[a,b,q,r]}\right)$,
where $\mathcal{E}$ is the Euler factor

$\left(1-\tfrac{p^{q}}{\alpha}\right)\left(1-\tfrac{\beta}{p^{1+q}}\right)\left(1-\tfrac{\gamma}{p^{1+q}}\right)\left(1-\tfrac{\delta}{p^{1+q}}\right)\\\
\times\left(1-\tfrac{p^{a+1+r}\chi_{2}(p)}{\alpha}\right)\left(1-\tfrac{p^{a+1+r}\chi_{2}(p)}{\beta}\right)\left(1-\tfrac{\gamma}{p^{a+2+r}\chi_{2}(p)}\right)\left(1-\tfrac{\delta}{p^{a+2+r}\chi_{2}(p)}\right).$

######  Remark 7.1.2.

This refines a result in [LZ20a, §17] in which $(a,q)$ were varying and
$(b,r)$ were fixed (but only Siegel-ordinarity was assumed, rather than full
Borel-ordinarity as here). In the above result, the parameter space is four-
dimensional and all four parameters $(a,b,q,r)$ are varying independently.

###### Proof.

We consider the groups
$\widetilde{G}=\operatorname{GSp}_{4}\times\operatorname{GL}_{1}$, and
$\widetilde{H}=(\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2})\times\operatorname{GL}_{1}$.
There is an obvious embedding $\widetilde{H}\hookrightarrow\widetilde{G}$,
extending the embedding $H\hookrightarrow G$ used in [LSZ21a]. We can also
view $\widetilde{H}$ as a double cover of
$\operatorname{GL}_{2}\times\operatorname{GL}_{2}$ via
$(h_{1},h_{2},z)\mapsto(h_{1},zh_{2})$. All of these maps are compatible with
the standard Shimura data, if we take the Shimura cocharacter to project
trivially into the $\operatorname{GL}_{1}$ factors of $\widetilde{G}$ and
$\widetilde{H}$. The cohomology group
$H^{4}_{\text{{\'{e}t}}}\Big{(}N_{G}(\mathbf{Z}_{p}),\mathbf{Z}_{p}(3)\Big{)}\mathop{\hat{\otimes}}\mathbf{Z}_{p}[[\mathbf{Z}_{p}^{\times}]]$
then has a natural interpretation as étale cohomology for $\widetilde{G}$ with
level $N_{\widetilde{G}}(\mathbf{Z}_{p})$.

We apply the machinery of [Loe21b] and the present paper to the groups
$\widetilde{G}$ and $\widetilde{H}$, taking the input class
$z_{\widetilde{H}}^{[0]}$ to be the pullback of the Eisenstein–Iwasawa class
for $\operatorname{GL}_{2}\times\operatorname{GL}_{2}$; the associated
mirabolic subgroup $Q_{\widetilde{H}}^{0}$ is then given by the preimage of
the usual mirabolic of $\operatorname{GL}_{2}\times\operatorname{GL}_{2}$,
namely

$\\{\left(\left(\begin{smallmatrix}x&\star\\\
0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}xy&\star\\\
0&y^{-1}\end{smallmatrix}\right),y\right):x,y\in\mathbf{G}_{m}\\}.$

This group has an open orbit on the Borel flag variety of $\widetilde{G}$,
with trivial stabiliser. We can now obtain a family of classes in the Iwahori
level tower for $\widetilde{G}$. Comparing these with their analogues at
prime-to-$p$ level is done via an Euler-factor computation, for which we refer
to [Loe21a].∎

###  7.2. The setting of [HJS20]

We now let
$\mathcal{G}=\operatorname{GSp}_{4}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$.
As in _op.cit._ we can construct classes

${}_{e}z^{[a,b,c,q]}_{\text{{\'{e}t}}}\in
H^{5}_{\text{{\'{e}t}}}\left(G(\mathbf{Z}_{p}),\mathscr{W}^{a,b,c,*}_{\mathbf{Z}_{p}}(3-q)\right)$

where $\mathscr{W}^{a,b,c}$ is a certain representation of $G$ for $a,b,c\geq
0$, and $\max(a,c)\leq q\leq\min(a+b,a+c)$. As in the previous section, we are
fixing some arbitrary test data away from $p$, including the smoothing
parameter $e>1$; and we are taking the test data at $p$ to be the spherical
data. We have also relabelled the parameters slightly, so $q$ is $a+r$ in the
notation of _op.cit._.

######  Theorem 7.2.1.

There exists a class ${}_{e}z_{\mathrm{Iw}}\in
e^{\prime}_{B_{G}}H^{5}_{\text{{\'{e}t}},\mathrm{Iw}}\left(N_{G}(\mathbf{Z}_{p}),\mathbf{Z}_{p}(3)\right)$
whose specialisation at an automorphic representation $\Pi\times\Sigma$ of
weight $(a,b,c)$ twisted by $\det^{q}$, with $\Pi$ and $\Sigma$ good ordinary
at $p$, is given by
$\mathcal{E}\operatorname{pr}_{\Pi\times\Sigma}\left({}_{e}z^{[a,b,c,q]}_{\text{{\'{e}t}}}\right)$,
where $\mathcal{E}$ is the Euler factor

$\prod\left\\{\left(1-\tfrac{\lambda}{p^{2+q}}\right):\lambda=\beta\mathfrak{b},\gamma\mathfrak{a},\gamma\mathfrak{b},\delta\mathfrak{a},\delta\mathfrak{b}\right\\}\cdot\prod\left\\{\left(1-\tfrac{p^{1+q}}{\lambda}\right):\lambda=\alpha\mathfrak{a},\alpha\mathfrak{b},\beta\mathfrak{a}\right\\}$

Here $\mathfrak{a}$ and $\mathfrak{b}$ are the Hecke parameters of $\Sigma$
(with $v_{p}(\mathfrak{a})=0$, $v_{p}(\mathfrak{b})=c+1$).

###### Proof.

We apply the machinery of the previous sections with
$\mathcal{H}=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$
embedded in the usual way, and an Eisenstein class on the first factor of
$\mathcal{H}$ alone. As before, we obtain a family of classes at Iwahori
level; and these are related to their analogues at prime-to-$p$ level via an
Euler-factor computation, which is again carried out in [Loe21a]. ∎

This result is an important ingredient in the proof of new cases of the
Bloch–Kato conjecture for $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$
in [LZ21].

###  7.3. The setting of [LZ20b]

Finally, we consider the triple product of cusp forms, as in [LZ20b]. This
requires some care owing to the necessity of choosing a self-dual twist
(compare §3.3 of op.cit.).

Let
$\mathcal{G}=\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}\times\operatorname{GL}_{1}$,
and let
$\mathcal{H}=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$
embedded via

$(h_{1},h_{2})\mapsto(h_{1}\boxplus h_{2},h_{1},h_{2},\det h_{1}^{-1}).$

In this setting, we let $Q_{H}^{0}=H$, and $Q_{G}$ the upper-triangular Borel
of $G$. Then one computes that $H\cap u\bar{Q}_{G}u^{-1}$ is a copy of
$\mathbf{G}_{m}$, embedded as $z\mapsto(z,z,z,z^{-2})$. We take $S_{G}^{0}$ to
be the subgroup

$\left\\{\left(\left(\begin{smallmatrix}\star\\\ &\star\\\ &&z\\\
&&&z\end{smallmatrix}\right),\left(\begin{smallmatrix}\star&\\\
&z\end{smallmatrix}\right)\left(\begin{smallmatrix}\star&\\\
&z\end{smallmatrix}\right),z^{-2}\right):z\in\mathbf{G}_{m}\right\\}.$

With these choices, the group $\mathfrak{S}$ is once more isomorphic to
$(\mathbf{Z}_{p}^{\times})^{4}$; and a $p$-adic modular form for
$\mathcal{G}$, of level $Q_{G}^{0}(\mathbf{Z}_{p})$, is identified with a
triple of $p$-adic modular forms for the groups $\operatorname{GSp}_{4}$,
$\operatorname{GL}_{2}$ and $\operatorname{GL}_{2}$ such that the product of
their weight-characters is a square, together with a choice of square root of
this character. Applying the machinery of the previous sections, combined with
yet another zeta-integral computation from [Loe21a], gives Theorem 8.2.5 of
[LZ20b], showing that diagonal-cycle cohomology classes for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}$
vary in $p$-adic families.

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