#  $p$-adic Asai $L$-functions for quadratic Hilbert eigenforms

Giada Grossi  CNRS, Institut Galilée, Université Sorbonne Paris Nord, 93430
Villetaneuse, FRANCE<EMAIL_ADDRESS>,  David Loeffler
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
<EMAIL_ADDRESS>and  Sarah Livia Zerbes  Department of Mathematics,
ETH Zürich, Rämistrasse 101, 8092 Zürich, SWITZERLAND
<EMAIL_ADDRESS>
# Asai–Flach classes and $p$-adic L-functions

Giada Grossi  CNRS, Institut Galilée, Université Sorbonne Paris Nord, 93430
Villetaneuse, FRANCE<EMAIL_ADDRESS>,  David Loeffler
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
<EMAIL_ADDRESS>and  Sarah Livia Zerbes  Department of Mathematics,
ETH Zürich, Rämistrasse 101, 8092 Zürich, SWITZERLAND
<EMAIL_ADDRESS>
###### Abstract.

We prove a formula for the Bloch–Kato logarithm of the bottom class in the
Asai–Flach Euler system associated to a quadratic Hilbert modular form. We
show that this can be expressed as a value, outside the interpolation range,
of the $p$-adic Asai $L$-function constructed in the prequel paper [GLZ23].

###### Key words and phrases:

Hilbert modular varieties, $p$-adic modular forms, higher Hida theory,
coherent cohomology of Shimura varieties

###### 2020 Mathematics Subject Classification:

11F41, 11F33, 11G18, 14G35

Supported by European Research Council Consolidator Grant No. 101001051
(ShimBSD) (Loeffler) and US National Science Foundation Grant No. DMS-1928930
(all authors)

###### Contents

  1. 1 Introduction
  2. 2 Conventions
  3. 3 The Asai motive and its realisations
  4. 4 Definition of the $p$-adic regulator
  5. 5 Lifting $\nu$ to the $\mathfrak{p}_{1}$-ordinary locus
  6. 6 Restricting to the fully-ordinary locus
  7. 7 Expression via coherent cohomology

##  1\. Introduction

The theory of Euler systems – norm-compatible families of cohomology classes
for global Galois representations – has played an important role in many
discoveries in number theory. A key aspect of this theory is _explicit
reciprocity laws_ , which relate Euler systems to the special values of
$L$-functions; this can be used as a criterion for the non-vanishing of Euler
systems, a crucial input in applications to the Bloch–Kato conjecture and
related problems.

One strategy for proving such explicit reciprocity laws, introduced in the
seminal paper [BDP13] in the context of Heegner points, proceeds as follows.
Supposing we already have an Euler system for some Galois representation $V$,
we aim to carry out the following steps:

  1. (1)

Construct a $p$-adic $L$-function, for some $p$-adic family of Galois
representations specialising to $V$, whose values in a suitable interpolation
range are critical values of the corresponding complex $L$-functions.

  2. (2)

Relate values of this $p$-adic $L$-function at suitable points _outside_ the
range of interpolation to Bloch–Kato logarithms of Euler system classes.

  3. (3)

Via analytic continuation, deduce a relation between Euler systems and
$p$-adic $L$-values at points _inside_ the interpolation range, where the
$p$-adic $L$-function specialises to the complex $L$-value.

In the present paper we focus on the _Euler system of Asai–Flach classes_ ,
introduced in [LLZ18], which is an Euler system for the Asai, or twisted
tensor, Galois representation associated to a Hilbert modular form for a real
quadratic field. In this setting, step (1) of the above strategy was carried
out in the prequel paper [GLZ23]. The goal of the present paper is to carry
out step (2): the main result (Theorem 7.7.6) expresses the logarithms of
Euler system classes as non-critical values of the $p$-adic $L$-function
constructed in _op.cit._. The remaining step (3), with applications to the
Bloch–Kato conjecture and the Iwasawa main conjecture for Asai $L$-functions,
will be carried out in a third paper in the series, currently in preparation.

This strategy was successfully carried out for the Euler system of
Beilinson–Flach elements (attached to a Rankin–Selberg convolution of modular
forms) in [BDR15, KLZ17]; and more recently in the $\operatorname{GSp}_{4}$
setting in [LPSZ21, LZ20]. The Asai representation can be seen as a “twisted
version” of the Rankin–Selberg convolution, and the construction of the Euler
system classes is recognisably a direct generalisation of the Beilinson–Flach
case; but the techniques used to construct $p$-adic $L$-functions and prove
the reciprocity law for Rankin–Selberg convolutions do not straightforwardly
generalise, since they rely on the decomposition of the Shimura variety for
$\operatorname{GL}_{2}\times\operatorname{GL}_{2}$ as a product of two
factors, and new techniques are required. Hence the present paper follows a
strategy rather closer to the $\operatorname{GSp}_{4}$ case, relying on the
use of _higher Hida theory_ , which has been developed for Hilbert modular
varieties at split primes by the first author [Gro21]; and its
“overconvergent” version, _higher Coleman theory_. The higher Coleman theory
we use is based on the general results of [BP20], with some slight
modifications to allow $p$-adic interpolation at only one of the primes above
$p$, for which we refer to [LZ22].

(We remark that a related problem is also considered in the paper [LSZ20],
where Skinner and two of the present authors computed a pairing which also
involves the Bloch–Kato logarithm of the Asai–Flach class, but projected into
a different piece of the de Rham cohomology which can be computed using only
conventional, rather than higher, Coleman theory. The earlier paper gives an
explicitly computable formula in terms of $p$-adic Hilbert modular forms, but
one which does not seem to have any straightforward relationship to $p$-adic
$L$-values, in contrast to the formula of Theorem 7.7.6.)

##  2\. Conventions

Throughout this paper $F$ denotes a real quadratic field, and we fix an
enumeration of the embeddings $F\hookrightarrow\mathbf{R}$ as
$\sigma_{1},\sigma_{2}$.

###  2.1. Algebraic groups

As in [LLZ18], define
$G=\operatorname{Res}_{F/\mathbf{Q}}(\operatorname{GL}_{2})$, and
$H=\operatorname{GL}_{2}$, with $\iota:H\hookrightarrow G$ the natural
embedding. We write $B_{H}$ and $B_{G}$ for the upper-triangular Borel
subgroups.

Recall that if $V$ is an algebraic representation of $B_{H}$, then $V$ gives
rise to a vector bundle on the Shimura variety $Y_{H}$ (for any sufficiently
small level group), endowed with an action of Hecke correspondences. There are
two possible normalisations for this functor, and we normalise so that the
defining 2-dimensional representation of $H$ (restricted to $B_{H}$) maps to
the relative de Rham _cohomology_ sheaf of the universal elliptic curve
$\mathcal{E}\to Y_{H}$ (rather than the relative homology, which is the other
convention in use). There is an analogous construction for $G$ as long as we
restrict to representations trivial on the norm-one subgroup
$\\{\left(\begin{smallmatrix}x&\\\
&x\end{smallmatrix}\right):N_{F/\mathbf{Q}}(x)=1\\}$ of $Z_{G}$, cf. [LLZ18,
§3.2c].

###  2.2. Algebraic representations and the Clebsch–Gordan map

For the group $G$, we shall always work with representations over fields
containing $F$. Given two representations $V,V^{\prime}$ of
$\operatorname{GL}_{2}$, we write $V\boxtimes V^{\prime}$ for the tensor
product of $V,V^{\prime}$, endowed with a $G$-action as the tensor product of
the actions via $\sigma_{1}$ on $V$ and via $\sigma_{2}$ on $V^{\prime}$.

Given integers $k_{1},k_{2}\geqslant 0$, and $0\leqslant
j\leqslant\min\\{k_{1},k_{2}\\}$, let

$V_{G}\coloneqq\operatorname{Sym}^{k_{1}}W_{G}\boxtimes\operatorname{Sym}^{k_{2}}W_{G}\qquad\text{and}\qquad
V_{H}=\operatorname{Sym}^{t}W_{H},\quad t=k_{1}+k_{2}-2j,$

where $W_{H}$ and $W_{G}$ denote the defining 2-dimensional representations of
$\operatorname{GL}_{2}/\mathbf{Q}$ and $\operatorname{GL}_{2}/F$ respectively.

The representation $V_{H}$ has a canonical basis $(v^{a}w^{t-a})_{0\leqslant
a\leqslant t}$, where $v,w$ are the two standard basis vectors of $W_{H}$; and
is equipped with a decreasing $B_{H}$-stable weight filtration
$\operatorname{Fil}^{n}V_{H}=\langle\\{v^{a}w^{t-a}:a\geqslant n\\}\rangle$.
Similarly, $V_{G}$ is equipped with a bi-filtration (a decreasing filtration
indexed by $\mathbf{Z}^{2}$) arising from the weight vector filtrations on
each factor.

There is a non-zero morphism of $H$-representations (the _Clebsch–Gordan
map_), unique up to scalars,

$\operatorname{CG}^{[k_{1},k_{2},j]}:V_{H}\rightarrow
V_{G}\otimes\det{}^{-j}.$

See (7.1) below for explicit formulae. If we equip the one-dimensional
representation $\det^{-j}$ with a filtration concentrated in degree $-j$, then
this map respects the filtrations, and hence induces a map on the graded
pieces.

##  3\. The Asai motive and its realisations

###  3.1. Automorphic representations of $G$

######  Definition 3.1.1.

Let $\Pi$ be the unitary cuspidal automorphic representation of
$G(\mathbf{A})$ generated by a holomorphic Hilbert modular newform of weight
$(k_{1}+2,k_{2}+2)$, for some integers $k_{1},k_{2}\geqslant 0$, and some
level $\mathcal{N}_{\Pi}\trianglelefteqslant\mathcal{O}_{F}$.

We shall suppose throughout this work that the following conditions hold:

  * •

$\Pi$ is not of CM type,

  * •

the central character of $\Pi$ is trivial (and hence $k_{i}$ are both even).

Both of these could almost certainly be relaxed with a little extra work, but
we shall not pursue this here. The representation $\Pi$ (and hence the
integers $(k_{1},k_{2})$) will remain fixed throughout the remainder of this
paper.

We define $a_{\mathfrak{n}}^{\circ}(\Pi)$, for each ideal
$\mathfrak{n}\trianglelefteqslant\mathcal{O}_{F}$, to be the Hecke eigenvalues
of the new vector of $\Pi$, normalised in the analytic fashion, so that
$|a_{\mathfrak{p}}^{\circ}(\Pi)|\leqslant 2$ for primes
$\mathfrak{p}\nmid\mathcal{N}_{\Pi}$. For each prime
$\mathfrak{p}\nmid\mathcal{N}_{\Pi}$ we let $\alpha_{\mathfrak{p}}^{\circ}$
and $\beta_{\mathfrak{p}}^{\circ}$ be the Satake parameters of
$\Pi_{\mathfrak{p}}$, normalised so that
$|\alpha_{\mathfrak{p}}^{\circ}|=|\beta_{\mathfrak{p}}^{\circ}|=1$ and
$\alpha_{\mathfrak{p}}^{\circ}+\beta_{\mathfrak{p}}^{\circ}=a_{\mathfrak{p}}^{\circ}(\Pi)$.

By a rationality theorem due to Shimura [Shi86, Proposition 1.2], there exists
a number field $L\subset\mathbf{C}$ (depending on $\Pi$) such that the
quantities $N(\mathfrak{n})^{1/2}a_{\mathfrak{n}}^{\circ}$ lie in $L$, for all
$\mathfrak{n}\trianglelefteqslant\mathcal{O}_{F}$. Extending $L$ if necessary,
we may assume that $L$ also contains the images of the embeddings
$\sigma_{i}:F\hookrightarrow\mathbf{R}$ (this is automatic if $k_{1}\neq
k_{2}$).

######  Definition 3.1.2.

For $n\in\mathbf{Z}_{\geqslant 1}$, we define $a_{n}(\Pi)$ by

$a_{n}(\Pi)=n^{(k_{1}+k_{2}+2)/2}a_{n}^{\circ}(\Pi).$

One checks that $a_{n}(\Pi)\in\mathcal{O}_{L}$ for all $n$. (However, unless
we make the stronger assumption that $k_{1}=k_{2}\bmod 4$, we cannot extend
this normalisation of the Hecke eigenvalues to include ideals $\mathfrak{n}$
which do not come from $\mathbf{Q}$, without introducing infinitely many
square roots to $L$.)

###  3.2. The Asai $L$-function

For each rational prime $\ell$, let us set
$\Pi_{\ell}=\bigotimes_{v\mid\ell}\Pi_{v}$, considered as a representation of
$G(\mathbf{Q}_{\ell})=\prod_{v\mid\ell}\operatorname{GL}_{2}(F_{v})$.
Associated to $\Pi_{\ell}$ we have an _Asai $L$-factor_111This can be defined
as the “common denominator” of a family of zeta-integrals, or alternatively
via the local Langlands correspondence, but both definitions give the same
$L$-factor. For $\ell$ split this is one of the defining properties of the
local Langlands correspondence; for $\ell$ non-split, see [Mat10, Theorem
3.3]., $L_{\operatorname{As}}(\Pi_{\ell},s)$. We write
$L_{\operatorname{As}}(\Pi,s)=\prod_{\text{$\ell$
prime}}L_{\operatorname{As}}(\Pi_{\ell},s)$ (without any Archimedean factors)
for the Asai $L$-function; note that this Dirichlet series has coefficients in
$L$.

Since the Euler factors of $L_{\operatorname{As}}$ at primes dividing
$\mathcal{N}_{\Pi}$ are hard to describe explicitly, we shall also consider
the “imprimitive” Asai $L$-function

$L_{\operatorname{As}}^{\mathrm{imp}}\left(\Pi,s\right)=\zeta_{(N)}(2s)\sum_{n\geqslant
1}a_{n}^{\circ}(\Pi)n^{-s},$

where the notation $(N)$ signifies removing the Euler factors at the primes
dividing $N=\mathcal{N}_{\Pi}\cap\mathbf{Z}$. Note that this Dirichlet series
has coefficients in $L$. The following result is standard:

######  Lemma 3.2.1.

The function
$L_{\operatorname{As}}^{\mathrm{imp}}\left(\Pi,s\right)/L_{\operatorname{As}}\left(\Pi,s\right)$
is a product of polynomials in $\ell^{-s}$ for $\ell\mid N$. All of its zeroes
have real part in $\\{0,-\tfrac{1}{2},-1\\}$. ∎

###  3.3. De Rham and coherent cohomology groups

We let $U_{1}(\mathcal{N}_{\Pi})$ denote the open compact subgroup $\\{g\in
G(\mathbf{A}_{\mathrm{f}}):g=\left(\begin{smallmatrix}*&*\\\
0&1\end{smallmatrix}\right)\bmod\mathcal{N}_{\Pi}\\}$, so that the space of
$U_{1}(\mathcal{N}_{\Pi})$-invariants of $\Pi_{\mathrm{f}}$ (the new subspace
of $\Pi_{\mathrm{f}}$) is one-dimensional. We assume, for simplicity222The
excluded cases may be dealt with via the usual trick of introducing full level
$\mathfrak{h}$ structure, for an auxiliary ideal $\mathfrak{h}$, and then
taking invariants under $\operatorname{GL}_{2}(\mathcal{O}/\mathfrak{h})$; we
leave the details to the interested reader. that $\mathcal{N}_{\Pi}$ does not
divide $6\operatorname{disc}_{F/\mathbf{Q}}$, so that
$U_{1}(\mathcal{N}_{\Pi})$ is _sufficiently small_ in the sense of [LLZ18,
Definition 2.2.1]; thus the Hilbert modular variety $Y_{G}$ of level
$U_{1}(\mathcal{N}_{\Pi})$ is a smooth quasiprojective variety defined over
$\mathbf{Q}$.

######  Definition 3.3.1.

Let $\mathcal{V}_{G}$ denote the vector bundle with connection on $Y_{G}$
corresponding to the algebraic representation333More precisely, we need to
twist by an appropriate character to make the action of norm-one units
trivial, but the resulting vector bundle is independent of the choice up to a
canonical isomorphism. $V_{G}$; and set

$D_{L}^{\operatorname{As}}(\Pi)=H^{2}_{\mathrm{dR},c}\Big{(}Y_{G},\mathcal{V}_{G}\Big{)}[\Pi_{\mathrm{f}}],$

which is 4-dimensional over $L$.

The de Rham cohomology can be computed via the coherent cohomology of a smooth
toroidal compactification $X_{G}$ of $Y_{G}$. We can then compute de Rham
cohomology using the logarithmic de Rham complex
$\mathcal{V}_{G}\otimes\Omega^{\bullet}_{X_{G}}\langle D\rangle$, where
$D=X_{G}-Y_{G}$ is the boundary divisor; and we can compute compactly-
supported de Rham cohomology using the “minus-log” complex
$\mathcal{V}_{G}\otimes\Omega^{\bullet}_{X_{G}}\langle-D\rangle$, where
$\Omega^{\bullet}\langle-D\rangle\coloneqq\Omega^{\bullet}\langle
D\rangle(-D)$. We denote the corresponding cohomologies by

$\displaystyle R\Gamma_{\mathrm{dR}}\left(X_{G},\mathcal{V}_{G}\langle
D\rangle\right)$ $\displaystyle\coloneqq
R\Gamma\left(X_{G},\mathcal{V}\otimes\Omega^{\bullet}_{X_{G}}\langle
D\rangle\right)\cong R\Gamma_{\mathrm{dR}}(Y_{G},\mathcal{V}_{G}),$
$\displaystyle R\Gamma_{\mathrm{dR}}(X_{G},\mathcal{V}_{G}\langle-D\rangle)$
$\displaystyle\coloneqq
R\Gamma(X_{G},\mathcal{V}\otimes\Omega^{\bullet}_{X_{G}}\langle-D\rangle)\cong
R\Gamma_{\mathrm{dR},c}(Y_{G},\mathcal{V}_{G}).$

Rather than explicitly working with the full de Rham complex, it is more
convenient to work with the _dual BGG complex_

$\operatorname{BGG}^{\bullet}=\left[\omega^{(-k_{1},-k_{2})}\longrightarrow\omega^{(-k_{1},k_{2}+2)}\oplus\omega^{(k_{1}+2,-k_{2})}\longrightarrow\omega^{(k_{1}+2,k_{2}+2)}\right]$

and its compactly-supported analogue $\operatorname{BGG}^{\bullet}(-D)$, which
are quasi-isomorphic to the logarithmic de Rham complexes
$V\otimes\Omega^{\bullet}\langle D\rangle$ and
$V\otimes\Omega^{\bullet}\langle-D\rangle$ respectively. (We shall give
explicit formulae for the differentials of the BGG complex, and the quasi-
isomorphism relating it to the de Rham complex, in Section 7.1 below, but we
shall not need this just yet.)

The BGG complex is equipped with a natural decreasing
$\mathbf{Z}^{2}$-filtration $\operatorname{Fil}^{\bullet\bullet}$, which gives
rise to a $\mathbf{Z}^{2}$-filtration on its cohomology. We can (and do)
normalise so that the nontrivial graded pieces are in bidegrees
$\\{(0,0),(k_{1}+1,0),(0,k_{2}+1),(k_{1}+1,k_{2}+1)\\}$, and are given by

$\displaystyle\operatorname{Gr}^{(k_{1}+1,k_{2}+1)}=H^{0}\left(X_{G},\omega^{(k_{1}+1,k_{2}+1)}(-D)\right)[\Pi_{\mathrm{f}}],$
$\displaystyle\operatorname{Gr}^{(k_{1}+1,0)}=H^{1}\left(X_{G},\omega^{(k_{1}+2,-k_{2})}(-D)\right)[\Pi_{\mathrm{f}}],\qquad\qquad\operatorname{Gr}^{(0,k_{2}+1)}=H^{1}\left(X_{G},\omega^{(-k_{1},k_{2}+2)}(-D)\right)[\Pi_{\mathrm{f}}],$
$\displaystyle\operatorname{Gr}^{(0,0)}=H^{2}\left(X_{G},\omega^{(-k_{1},-k_{2})}(-D)\right)[\Pi_{\mathrm{f}}],$

where $\omega^{(r,s)}$ is the sheaf of Hilbert modular forms of weight $(r,s)$
(so that $\omega^{(r,s)}(-D)$ is the subsheaf of cusp forms). The induced
single filtration, with graded pieces in degrees
$\\{0,k_{1}+1,k_{2}+1,k_{1}+k_{2}+2\\}$, is the Hodge filtration.

######  Definition 3.3.2.

We let $\nu$ be a basis of the 1-dimensional $L$-vector space
$H^{1}\left(X_{G},\omega^{(-k_{1},k_{2}+2)}\right)[\Pi_{\mathrm{f}}]$.

######  Remark 3.3.3 (Occult periods).

After base-extension to $\mathbf{C}$, these graded pieces have canonical bases
arising from the comparison between sheaf cohomology and Dolbeault cohomology.
The one which will interest us is $\operatorname{Gr}^{(0,k_{2}+1)}$, which is
spanned by the differential form associated to the real-analytic Hilbert
modular form $\mathcal{F}^{\mathrm{ah,1}}$ (anti-holomorphic at the place
$\sigma_{1}$ and holomorphic at $\sigma_{2}$) having the same
Fourier–Whittaker coefficients as the holomorphic newform $\mathcal{F}$
generating $\Pi$; see Lemma 5.2.1 of [LLZ18] for further details. Harris’
_occult period_ for $\Pi$ at $\sigma_{1}$ (cf. [Har90]) is the ratio between
$\nu$ and $\mathcal{F}^{\mathrm{ah,1}}$, well-defined as an element of
$\mathbf{C}^{\times}/L^{\times}$.

###  3.4. The Asai Galois representation

We now fix (for the remainder of this paper) a prime $p$, and a prime
$\mathfrak{P}$ of the coefficient field $L$ above $p$.

######  Definition 3.4.1.

  1. (i)

Let $V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ denote the four-dimensional
Asai Galois representation associated to $\Pi$ as in [LLZ18, Definition
4.4.2], defined as the $\Pi_{\mathrm{f}}$-eigenspace in the $p$-adic étale
cohomology of the Hilbert modular variety
$Y_{G,1}(\mathcal{N}_{\Pi})\otimes\overline{\mathbf{Q}}$ (with coefficients in
the étale local system of $L_{\mathfrak{P}}$-vector spaces determined by
$(k_{1},k_{2})$).

  2. (ii)

Let
$D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)=L_{\mathfrak{P}}\otimes_{L}D_{L}^{\operatorname{As}}(\Pi)$,
so that there is a canonical comparison isomorphism (compatible with the
filtrations)

$\mathbf{D}_{\mathrm{dR}}\left(\mathbf{Q}_{p},V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)\right)\cong
D^{\operatorname{As}}_{\mathfrak{P}}(\Pi).$

######  Remark 3.4.2.

More precisely, this is $M_{L_{\mathfrak{P}}}(\mathcal{F})$ in the notation of
_op.cit._ , where $\mathcal{F}$ is the normalised newform generating $\Pi$.

By results of Brylinski–Labesse and Nekovář recalled in _op.cit._ , the Galois
representation $V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ can be
characterised, up to isomorphism, as the unique semisimple Galois
representation whose $L$-series is
$L_{\operatorname{As}}\left(\Pi,s-\tfrac{k_{1}+k_{2}+2}{2}\right)$. (However,
since we want to consider Euler system classes for
$V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$, it is important to fix not only
an abstract isomorphism class of Galois representations but a specific
realisation of this isomorphism class in étale cohomology.) By Poincaré
duality for the Hilbert modular surface, there is a canonical nondegenerate
symmetric bilinear form

(3.1)
$\lambda:\operatorname{Sym}^{2}\left(V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)\right)\to
L_{\mathfrak{P}}(-2-k_{1}-k_{2}),$

equivariant for the action of
$\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. (The pairing is
symmetric since $V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ appears in
cohomology in even degree.)

######  Remark 3.4.3.

By [LLZ18, Proposition 9.4.3], our hypotheses on $\Pi$ imply that
$V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ is either irreducible, or the
direct sum of a character and a three-dimensional irreducible representation,
with the latter case occurring if and only if $\Pi$ is a twist of a base-
change from $\operatorname{GL}_{2}/\mathbf{Q}$.

###  3.5. The split-prime case

For the remainder of this paper, we suppose that $p$ is split in $F$, and
$\Pi$ is unramified at the primes above $p$. Since the embeddings $\sigma_{i}$
of $F$ take values in $L\subset\mathbf{C}$, and we have fixed a prime
$\mathfrak{P}\mid p$ in $L$, we can number the primes above $p$ as
$\mathfrak{p}_{1},\mathfrak{p}_{2}$ such that
$\sigma_{i}(\mathfrak{p}_{i})\subset\mathfrak{P}$.

We define

$a_{\mathfrak{p}_{i}}(\Pi)=p^{(k_{i}+1)/2}a_{\mathfrak{p}_{i}}^{\circ}(\Pi)\in
L,\qquad\text{so
}a_{p}(\Pi)=a_{\mathfrak{p}_{1}}(\Pi)a_{\mathfrak{p}_{2}}(\Pi).$

(Note this normalisation depends on the choice of prime $\mathfrak{P}\mid p$,
and the $a_{\mathfrak{p}_{i}}$ are integral at $\mathfrak{P}$, but possibly
not at other primes of $L$ above $p$.) We likewise define

$\alpha_{i}=p^{\tfrac{1+k_{i}}{2}}\alpha^{\circ}_{\mathfrak{p}_{i}}$

and similarly $\beta_{i}$. Extending $L$ if necessary, we may suppose that the
$\alpha_{i}$ and $\beta_{i}$ also lie in $L$.

######  Corollary 3.5.1.

The representation $V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ is crystalline
at $p$, so $D_{\mathfrak{P}}(\Pi)$ is naturally a filtered $\varphi$-module;
and the eigenvalues of $\varphi$ on this module are the pairwise products
$\\{\alpha_{1}\alpha_{2},\dots,\beta_{1}\beta_{2}\\}.$∎

######  Lemma 3.5.2.

Let $0\leqslant j\leqslant\min(k_{1},k_{2})$ be an integer. Then $p^{j}$ is
not an eigenvalue of $\varphi$ on $D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$.
Moreover, if $p^{(1+j)}$ is an eigenvalue of $\varphi$, then we must have
$k_{1}=k_{2}=j$.

###### Proof.

This follows from the fact that the Satake parameters $\alpha_{i}^{\circ}$,
$\beta_{i}^{\circ}$ have complex absolute value 1.∎

This implies that for the Galois representation
$V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)$, the Bloch–Kato subspaces
$H^{1}_{\mathrm{e}}(\mathbf{Q}_{p},-)$ and
$H^{1}_{\mathrm{f}}(\mathbf{Q}_{p},-)$ agree; and these are also equal to
$H^{1}_{\mathrm{g}}$, except possibly in the boundary case $k_{1}=k_{2}=j$ in
which case $H^{1}_{\mathrm{g}}$ can be strictly larger. Moreover, the inverse
of the Bloch–Kato exponential map for
$V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)$ is an isomorphism

(3.2)
$\log:H^{1}_{\mathrm{f}}\left(\mathbf{Q}_{p},V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)\right)\xrightarrow{\
\cong\
}\left(\operatorname{Fil}^{1+j}D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)\right)^{*}=\left(\operatorname{Fil}^{1}D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)\right)^{*},$

with both sides 3-dimensional over $L_{\mathfrak{P}}$.

###  3.6. Partial Frobenii

We may identify $D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$, as a Frobenius
module (forgetting the filtration), with the rigid cohomology of the special
fibre of $Y_{G}$ at $p$. This special fibre has two commuting endomorphisms,
the partial Frobenii $\varphi_{1}$ and $\varphi_{2}$ at the primes
$\mathfrak{p}_{i}$, whose composite is the Frobenius $\varphi$; more
precisely, $\varphi_{i}$ corresponds to sending a Hilbert–Blumenthal abelian
surface $A$ to the quotient $A/\left(\ker(\varphi_{A})\cap
A[\mathfrak{p}_{i}]\right)$. We refer to [TX16] or [Nek18] for detailed
accounts of this construction.

The operators $\varphi_{i}$ induce commuting linear operators on
$D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$, with
$\varphi=\varphi_{1}\varphi_{2}$; and it follows from the “partial
Eichler–Shimura” comparison result proved in [Nek18] that for each $i$ we have

$(\varphi_{i}-\alpha_{i})(\varphi_{i}-\beta_{i})=0\quad\text{on
}D_{\mathfrak{P}}^{\operatorname{As}}(\Pi).$

One checks easily that the partial Frobenii satisfy
$\lambda(\varphi_{i}x,\varphi_{i}y)=p^{k_{i}+1}\lambda(x,y)$, where $\lambda$
is the Poincaré duality form. This identifies the $\beta_{i}$-generalised
eigenspace with the dual of that for $\alpha_{i}$. Hence, if
$\alpha_{i}\neq\beta_{i}$, the $\varphi_{i}=\alpha_{i}$ and
$\varphi_{i}=\beta_{i}$ eigenspaces are both 2-dimensional, and each is
isotropic with respect to $\lambda$.

######  Remark 3.6.1.

We are using a slightly different normalisation of the partial Frobenii here
from [LZ22]: the $\varphi_{i}$ here is $p^{-t_{i}}\varphi_{i}$ in the notation
of _op.cit._ , where $(t_{1},t_{2})$ is an auxiliary choice of integers such
that $w=k_{i}+2t_{i}$ is independent of $i$. This reflects the fact that the
$\alpha_{i}$ of _op.cit._ is $p^{(w+1)/2}\alpha_{\mathfrak{p}_{i}}^{\circ}$,
while $\alpha_{i}$ here is $p^{(k_{i}+1)/2}\alpha_{\mathfrak{p}_{i}}^{\circ}$.
The present normalisation is more convenient for comparison with higher Hida
theory, since it matches the minimal integral normalisation of the
$U_{\mathfrak{p}_{i}}^{\prime}$ operators.

###  3.7. A lifting of $\nu$

We write $v_{p}$ for the valuation on $L_{\mathfrak{P}}$ normalized by
$v_{p}(p)=1$; and we fix an ordering of the Satake parameters at
$\mathfrak{p}_{1}$. Then we have $0\leqslant v_{p}(\alpha_{1})\leqslant
k_{1}+1$.

######  Proposition 3.7.1.

Suppose that $\alpha_{1}\neq\beta_{1}$, and $v_{p}(\alpha_{1})<k_{1}$. Then
the vector space

$D_{\mathfrak{P}}(\Pi)^{(\varphi_{1}=\alpha_{1})}\cap\operatorname{Fil}^{(0,k_{2}+1)}D_{\mathfrak{P}}(\Pi)$

is one-dimensional, and surjects onto
$\operatorname{Gr}^{(0,k_{2}+1)}D_{\mathfrak{P}}(\Pi)$.

Hence there exists a uniquely determined vector $\nu_{\mathrm{dR}}\in
D_{\mathfrak{P}}$ with the following properties:

  * •

$\varphi_{1}\nu_{\mathrm{dR}}=\alpha_{1}\cdot\nu_{\mathrm{dR}}$;

  * •

$\nu_{\mathrm{dR}}\in\operatorname{Fil}^{(0,k_{2}+1)}D_{\mathfrak{P}}$;

  * •

the image of $\nu_{\mathrm{dR}}$ in the graded piece

$\operatorname{Gr}^{(0,k_{2}+1)}D_{\mathfrak{P}}\cong
L_{\mathfrak{P}}\otimes_{L}H^{1}\left(X_{G},\omega^{(-k_{1},k_{2}+2)}\right)[\Pi_{\mathrm{f}}]$

coincides with the $\nu$ of Definition 3.3.2.

###### Proof.

This is a special case of the first main theorem of [LZ22], which states that
(over totally-real fields of any degree) the $\mathbf{Z}^{d}$-indexed
filtration of $D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ has a canonical
splitting given by intersecting with partial Frobenius eigenspaces. The
assumption $v_{p}(\alpha_{1})<k_{1}$ is the _strictly small slope_ assumption
of _op.cit._. ∎

######  Remark 3.7.2.

Note that
$\ker\left((\varphi-\alpha_{1}\alpha_{2})(\varphi-\alpha_{1}\beta_{2}):D_{p}\to
D_{p}\right)$ contains $D_{p}^{(\varphi_{1}=\alpha_{1})}$, but it may be
larger; this always occurs if $\Pi$ is a twist of a base-change from
$\mathbf{Q}$ (so that $\alpha_{1}/\beta_{1}=\alpha_{2}/\beta_{2}$).

##  4\. Definition of the $p$-adic regulator

###  4.1. Euler system classes

Let $0\leqslant j\leqslant\min(k_{1},k_{2})$. We refer to [LLZ18, Definition
4.4.6] for the definition of the _étale Asai–Flach class_

$\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}\in
H^{1}\left(\mathbf{Z}[1/\Sigma],V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)\right),$

where $\Sigma$ is the set of primes dividing $pN\operatorname{disc}(F)$. (More
precisely, this class is defined for any normalised eigenform $\mathcal{F}$,
not necessarily new, and we are defining
$\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}$ as
$\operatorname{AF}^{[\mathcal{F},j]}_{\text{{\'{e}t}}}$ where $\mathcal{F}$ is
the unique newform generating $\Pi$, consistently with our definition of the
Galois representation $V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$.)

######  Remark 4.1.1.

In _op.cit._ we showed that $\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}$
was the $m=1$ case of a family of classes defined over $\mathbf{Q}(\mu_{m})$
with norm-compatibility properties as $m$ varies, but we shall not use this
here.

######  Note 4.1.2.

We briefly recall the definition of
$\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}$. Letting $\mathcal{V}_{G}$ and
$\mathcal{V}_{H}$ denote the relative Chow motives (over $L$) associated to
the representations $V_{G}$ and $V_{H}$ defined in Section 2.2, we have a
pushforward map

$\iota^{[j]}_{*}:H^{1}_{\operatorname{mot}}\left(Y_{H,1}(N),\mathcal{V}_{H}(1+t)\right)\otimes_{\mathbf{Q}}L\longrightarrow
H^{3}_{\operatorname{mot}}\left(Y_{G,1}(\mathcal{N}_{\Pi}),\mathcal{V}_{G}(2+k_{1}+k_{2}-j)\right),$

where $t=k_{1}+k_{2}-2j$ as before. We have an analogous pushforward map in
étale cohomology with $L_{\mathfrak{P}}$-coefficients, and the two are
compatible via the étale regulator map $r_{\text{{\'{e}t}}}$. Since the
$\Pi_{\mathrm{f}}$-generalised eigenspace in the cohomology of
$Y_{G,1}(\mathcal{N}_{\Pi})$ vanishes outside degree 2, the Hochschild–Serre
spectral sequence gives a projection map from
$H^{3}_{\text{{\'{e}t}}}\left(Y_{G,1}(\mathcal{N}_{\Pi}),\mathcal{V}_{G}(2+k_{1}+k_{2}-j)\right)$
to
$H^{1}\left(\mathbf{Z}[1/\Sigma],V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)\right)$.
We can then define $\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}$ as the
image of the weight $t$ Eisenstein class under this chain of maps.

###  4.2. Localisation at $p$ and syntomic cohomology

######  Proposition 4.2.1.

The localisation of $\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}$ at $p$
lies in the Bloch–Kato subspace

$H^{1}_{\mathrm{f}}(\mathbf{Q}_{p},V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j))\subseteq
H^{1}(\mathbf{Q}_{p},V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)^{*}(-j)).$

###### Proof.

This follows from the comparison between étale and syntomic cohomology, since
the map from syntomic to étale cohomology factors through the Bloch–Kato
exponential map; cf. [KLZ20, Proposition 5.4.1] in the analogous case of
Beilinson–Flach elements. ∎

Now fix a choice of $\alpha_{1}$ satisfying the conditions of Proposition
3.7.1, and let
$\nu_{\mathrm{dR}}\in\operatorname{Fil}^{(0,k_{2}+1)}D_{\mathfrak{P}}(\Pi)$ be
the ensuing lifting of $\nu$. Our goal will be to compute the pairing

(4.1)
$\left\langle\nu_{\mathrm{dR}},\log\left(\operatorname{loc}_{p}\operatorname{AF}^{[\Pi,j]}_{\text{{\'{e}t}}}\right)\right\rangle_{D_{\mathfrak{P}}(\Pi)},$

where $\log$ is the Bloch–Kato logarithm (3.2), and
$\langle-,-\rangle_{D_{\mathfrak{P}}(\Pi)}$ denotes the canonical pairing
between $D_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$ and its dual.

Let $\mathbb{Y}_{G}$ denote the canonical $\mathbf{Z}_{p}$-model of
$Y_{G,1}(\mathcal{N}_{\Pi})$. This is a smooth $\mathbf{Z}_{p}$-scheme, and we
may choose an arithmetic toroidal compatification $\mathbb{X}_{G}$ such that
$(\mathbb{Y}_{G},\mathbb{X}_{G})$ is a _smooth pair_ over $\mathbf{Z}_{p}$ in
the sense of [KLZ20] (i.e. $\mathbb{X}_{G}$ is smooth, and the cuspidal
divisor $\mathbb{X}_{G}-\mathbb{Y}_{G}$ is a smooth normal-crossing divisor
relative to $\operatorname{Spec}\mathbf{Z}_{p}$). Thus Besser’s theory of
_rigid syntomic_ and _finite-polynomial_ cohomology applies to
$\mathbb{Y}_{G}$, and rigid-syntomic cohomology has a natural comparison to
étale cohomology.

######  Notation 4.2.2.

Let
$P(T)=\left(1-\frac{T}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{T}{\alpha_{1}\beta_{2}}\right)\in
L[T]$, and let $P_{1+j}(T)=P(p^{1+j}T)$.

######  Proposition 4.2.3.

There is a unique lift $\nu_{\operatorname{fp}}$ of $\nu_{\mathrm{dR}}$ to the
space

$H^{2}_{\operatorname{fp},c}\left(\mathbb{Y}_{G},\mathcal{V}_{G};1+j,P_{1+j}\right)[\Pi_{f}].$

######  Remark 4.2.4.

The above group is actually independent of $j$ in the range $0\leqslant
j\leqslant\min(k_{1},k_{2})$: Besser’s cohomology for twist $r$ and polynomial
$Q$ is defined using the mapping fibre of $Q(p^{-r}\varphi)$, and we have
$P_{1+j}(p^{-1-j}\varphi)=P(\varphi)$ for any $j$. However, different values
of $j$ will correspond to the étale cohomology of different twists of
$V_{\mathfrak{P}}^{\operatorname{As}}(\Pi)$.

###### Proof.

Since $\Pi$ is cuspidal, the $\Pi_{\mathrm{f}}$-generalised eigenspace in de
Rham (or, equivalently, rigid) cohomology vanishes outside degree 2. So the
natural map

$H^{2}_{\operatorname{fp},c}\big{(}\mathbb{Y}_{G},\mathcal{V}_{G};1+j,P\big{)}\to\operatorname{Fil}^{(1+j)}H^{2}_{\mathrm{dR},c}\big{(}\mathbb{Y}_{G},\mathcal{V}_{G}\big{)}^{P(\varphi)=0}$

is an isomorphism after localising at the $\Pi_{\mathrm{f}}$-eigenspace. Since
$P(\varphi)$ annihilates $\nu_{\mathrm{dR}}$ the result follows. ∎

The compatibility of étale and syntomic Abel–Jacobi maps for smooth pairs (cf.
Proposition 5.4.1 of [KLZ20]) then implies that

(4.2)
$\displaystyle\left\langle\nu_{\mathrm{dR}},\,\log\left(\operatorname{loc}_{p}\operatorname{AJ}^{[\Pi,j]}_{\text{{\'{e}t}}}\right)\right\rangle_{D_{\mathfrak{P}}(\Pi)}$
$\displaystyle=\left\langle\nu_{\mathrm{dR}},\,\log\circ\operatorname{pr}^{\operatorname{As}}_{\Pi}\circ\iota^{[j]}_{*}\left(\mathrm{Eis}^{t}_{\text{{\'{e}t}},N}\right)\right\rangle_{\mathrm{dR},Y_{G}}$
$\displaystyle=\left\langle\nu_{\operatorname{fp}},\,\iota^{[j]}_{*}\left(\mathrm{Eis}^{t}_{\operatorname{syn},N}\right)\right\rangle_{\operatorname{fp},\mathbb{Y}_{G}}$
$\displaystyle=\left\langle\iota^{[j],*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle_{\operatorname{fp},\mathbb{Y}_{H}},$

where the last equality follows from the adjunction between pushforward and
pullback. Note that

$\iota^{[j],*}(\nu_{\operatorname{fp}})\in
H^{2}_{\operatorname{fp},c}(\mathbb{Y}_{H},\mathcal{V}_{H};1+j,P_{1+j}),$

and the coefficient module $\mathcal{V}_{H}$ depends on $j$.

##  5\. Lifting $\nu$ to the $\mathfrak{p}_{1}$-ordinary locus

In this section, we shall lift the coherent class $\nu$ (and its cousins
$\nu_{\mathrm{dR}}$ and $\nu_{\operatorname{fp}}$) from the cohomology of the
full variety $Y_{G}$, to cohomology groups associated to certain open subsets
of the special fibre of $Y_{G}$.

###  5.1. Geometry of the Hilbert modular variety

Let $Y_{G,0}$ be the special fibre of $\mathbb{Y}_{G}$, which is a smooth
$\mathbb{F}_{p}$-variety; and similarly for the compactification $X_{G,0}$.

######  Notation 5.1.1.

  * •

For $i=1,2$, denote by $X^{i-\mathrm{ss}}_{G,0}\subset X_{G,0}$ the
$\mathfrak{p}_{i}$-supersingular locus (the vanishing locus of the partial
Hasse invariant, as constructed in [TX16, §3.2]).

  * •

Let $X^{i-\mathrm{ord}}_{G,0}$ be the complement of $X^{i-\mathrm{ss}}_{G,0}$,
and $X^{\mathrm{ord}}_{G,0}=X^{1-\mathrm{ord}}_{G}\cap
X^{2-\mathrm{ord}}_{G,0}$.

We write $Y^{i-\mathrm{ord}}_{G,0}$ etc for the intersection of these
subvarieties with $Y_{G,0}\subset X_{G,0}$.

The following results on the geometry of the supersingular loci are well-known
(see e.g. [TX16]):

######  Lemma 5.1.2.

For $i=1,2$, $X^{i-\mathrm{ss}}_{G,0}$ is a smooth codimension 1 closed
subscheme of $X_{G,0}$, disjoint from the toroidal boundary; and
$X^{1-\mathrm{ss}}_{G,0}\cap X^{2-\mathrm{ss}}_{G,0}$ is a smooth closed
subvariety of codimension $2$ (i.e. a finite disjoint union of points).

######  Remark 5.1.3.

The preimage of either $X^{1-\mathrm{ord}}_{G,0}$ or
$X^{2-\mathrm{ord}}_{G,0}$ under the finite map $\iota:X_{H,0}\to X_{G,0}$ is
the ordinary locus $X_{H,0}^{\mathrm{ord}}$.

######  Proposition 5.1.4.

The extension-by-0 map

$R\Gamma_{\mathrm{rig},c}(Y^{1-\mathrm{ord}}_{G,0},\mathcal{V}_{G})\longrightarrow
R\Gamma_{\mathrm{rig},c}(Y_{G,0},\mathcal{V}_{G})\cong
R\Gamma_{\mathrm{dR},c}(Y_{G},\mathcal{V}_{G})$

is a quasi-isomorphism on the $\Pi$ generalised eigenspace for the prime-
to-$Np$ Hecke operators.

###### Proof.

This is a special case of Proposition 4.3 of [LZ22]. ∎

######  Notation 5.1.5.

Write $\nu^{(1-\mathrm{ord})}_{\mathrm{dR}}\in
H^{2}_{\mathrm{rig},c}(Y^{1-\mathrm{ord}}_{G,0},\mathcal{V}_{G})$ for the
preimage of $\nu_{\mathrm{dR}}$ under the isomorphism in Proposition 5.1.4.

###  5.2. Coherent cohomology of the 1-ordinary locus

We write $\mathcal{X}_{G}$ for the dagger space associated to
$X_{G}/\mathbf{Q}_{p}$; and we denote the tubes in $\mathcal{X}_{G}$ of the
various subvarieties of $X_{G,0}$ considered above by the corresponding
superscripts on $\mathcal{X}_{G}$, so $\mathcal{X}_{G}^{i-\mathrm{ord}}$ is
the tube of $X_{G,0}^{i-\mathrm{ord}}$ in $\mathcal{X}_{G}$ etc.

######  Theorem 5.2.1.

Suppose $v_{p}(\alpha_{1})<k$, and $\alpha_{1}\neq\beta_{1}$. Then there
exists a unique class

$\nu^{1-\mathrm{ord}}\in
H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\omega^{(-k_{1},k_{2}+2)}(-D)\right)$

satisfying the following properties:

  1. (1)

it is a $\varphi_{1}$-eigenvector with eigenvalue $\alpha_{1}$;

  2. (2)

its image in $H^{1}(\mathcal{X}_{G},\omega^{(-k_{1},k_{2}+2)}(-D))$ is $\nu$.

###### Proof.

This is an instance of Proposition 5.2 of [LZ22]. ∎

######  Remark 5.2.2.

Observe that we have chosen a lifting of the de Rham class $\nu_{\mathrm{dR}}$
to $X_{G,0}^{(1-\mathrm{ord})}$ characterised by information about the action
of Hecke operators _away_ from $p$; and, separately, we have lifted the
coherent class $\nu$ to $\mathcal{X}_{G}^{(1-\mathrm{ord})}$ using information
about the action of the Frobenius $\varphi_{1}$ at $\mathfrak{p}_{1}$. So it
is not obvious how these liftings are related, and our next task is to find a
way to reconcile the two, which we will carry out in Section 5.4 – see
Proposition 5.4.1 below.

###  5.3. Comparison with a higher-level class

We now compare the class $\nu^{1-\mathrm{ord}}$ with algebraic coherent
classes at level $\mathfrak{p}_{1}$. If $X_{G}(\mathfrak{p}_{1})$ denotes the
Shimura variety of level
$U_{1}(\mathcal{N}_{\Pi})\cap\\{\left(\begin{smallmatrix}*&*\\\
0&*\end{smallmatrix}\right)\bmod\mathfrak{p}_{1}\\}$, then the special fibre
$X_{G}(\mathfrak{p}_{1})_{0}$ has a stratification with three strata,

$X_{G}(\mathfrak{p}_{1})_{0}=X_{G}(\mathfrak{p}_{1})^{m}_{0}\cup
X_{G}(\mathfrak{p}_{1})^{\text{{\'{e}t}}}_{0}\cup
X_{G}(\mathfrak{p}_{1})^{\alpha}_{0},$

on which the level structure is multiplicative, étale, or $\alpha_{p}$
respectively; and this gives a corresponding decomposition of the dagger space
$\mathcal{X}_{G}(\mathfrak{p}_{1})$, which restricts to an isomorphism of
dagger spaces

$\mathcal{X}_{G}(\mathfrak{p}_{1})^{m}\xrightarrow{\ \cong\
}\mathcal{X}_{G}^{1-\mathrm{ord}}.$

(The inverse map is given by the “canonical subgroup” construction.) From the
functoriality of pushforward maps we have a commutative square of cohomology
groups (all with coefficients in $\omega^{-k_{1},k_{2}+2}(-D)$)

${H^{1}_{c}(\mathcal{X}_{G}(\mathfrak{p}_{1})^{m})}$${H^{1}(\mathcal{X}_{G}(\mathfrak{p}_{1}))}$${H^{1}_{c}(\mathcal{X}_{G}^{1-\mathrm{ord}})}$${H^{1}(\mathcal{X}_{G})}$$\scriptstyle{\pi_{\mathfrak{p}_{1},*}}$

where the horizontal maps are extension-by-zero (i.e. pushforwards along open
embeddings) and the right-hand vertical map is pushforward along the natural
degeneracy map
$\pi_{\mathfrak{p}_{1}}:\mathcal{X}_{G}(\mathfrak{p}_{1})\to\mathcal{X}_{G}$.
By [LZ22, Lemma 5.7], our Frobenius lift $\varphi_{1}$ on
$H^{1}_{c}(\mathcal{X}_{G}^{1-\mathrm{ord}})$ corresponds to the Hecke
operator $U_{\mathfrak{p}_{1}}^{\prime}$ at level
$\mathcal{X}_{G}(\mathfrak{p}_{1})$. Hence we have the following
compatibility:

######  Proposition 5.3.1.

The image of $\nu^{1-\mathrm{ord}}$ in
$H^{1}(\mathcal{X}_{G}(\mathfrak{p}_{1}))$ is the unique class which lies in
the $\Pi_{\mathrm{f}}$-eigenspace away from $p$, is a
$(U_{\mathfrak{p}_{1}}^{\prime}=\alpha_{1})$-eigenvector, and maps to $\nu$
under the trace map.∎

We briefly compare this with the choice of basis vector used in [GLZ23, §6–7],
since this will be needed for our final formula. In §7.3 of _op.cit._ we
define a $U_{\mathfrak{p}_{1}}^{\prime}=\alpha_{1}$ eigenvector
$\breve{\nu}_{\Pi,\alpha}\in H^{1}(\mathcal{X}_{G}(\mathfrak{p}_{1}))$,
depending on a choice of basis $W_{\mathrm{f}}^{(p)}$ of the Whittaker model
away from $p\infty$. If we choose this basis to be the normalised new-vector,
then by construction we have

$\breve{\nu}_{\Pi,\alpha}=\left(1-\tfrac{\beta_{1}}{U^{\prime}_{\mathfrak{p}_{1}}}\right)\pi_{\mathfrak{p}_{1}}^{*}(\nu).$

Since $\breve{\nu}_{\Pi,\alpha}$ and the image of $\nu^{1-\mathrm{ord}}$ lie
in the same one-dimensional space, we may compare them by computing their
images in $H^{1}(\mathcal{X}_{G})$. An elementary computation shows that the
map

$\pi_{\mathfrak{p}_{1},*}\circ\left(1-\tfrac{\beta}{U^{\prime}_{\mathfrak{p}_{1}}}\right)\circ\pi_{\mathfrak{p}_{1}}^{*}$

acts on the $\Pi$-eigenspace as multiplication by
$p(1-\tfrac{\beta_{1}}{p\alpha_{1}})$ (which is not zero, since
$\alpha_{1}/\beta_{1}$ has complex absolute value 1); so we have

$\breve{\nu}_{\Pi,\alpha}=p(1-\tfrac{\beta_{1}}{p\alpha_{1}})\cdot\text{image}\left(\nu^{1-\mathrm{ord}}\right).$

###  5.4. The de Rham spectral sequence for $\mathcal{X}_{G}^{1-\mathrm{ord}}$

Since $(\mathbb{X}_{G},\mathbb{Y}_{G})$ is a smooth pair, and our coefficient
system $\mathcal{V}_{G}$ extends to a vector bundle on $\mathcal{X}_{G}$ whose
connection has log poles along the boundary divisor $D$, we can compute rigid
cohomology of $Y_{G,0}$ using the analytification of the BGG complex on
$\mathcal{X}_{G}$ (just as we did for de Rham cohomology above). By taking the
mapping fibre of the restriction map we obtain the same result for compactly-
supported cohomology of $Y_{G,0}^{1-\mathrm{ord}}$; that is, we have

$R\Gamma_{\mathrm{rig},c}(Y_{G,0}^{1-\mathrm{ord}},\mathcal{V})\cong
R\Gamma_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)\cong
R\Gamma_{c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\operatorname{BGG}^{\bullet}(-D)).$

This gives rise to a first-quadrant spectral sequence converging to
$H^{*}_{\mathrm{rig},c}(Y^{1-\mathrm{ord}}_{G,0},\mathcal{V}_{G})$, whose
$E_{1}^{mn}$ terms are
$H^{n}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\operatorname{BGG}^{m}(-D)\right)$.

We denote by
$\widetilde{R\Gamma}_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)$
the cohomology of the truncated complex

$\tau_{\geqslant
1}\operatorname{BGG}^{\bullet}(-D)=\left[0\longrightarrow\omega^{(-k_{1},k_{2}+2)}\oplus\omega^{(k_{1}+2,-k_{2})}\longrightarrow\omega^{(k_{1}+2,k_{2}+2)}\right](-D),$

which is quasi-isomorphic to the filtered de Rham complex
$\operatorname{Fil}^{1+j}\operatorname{BGG}^{\bullet}(-D)$, for any $j$ in our
range. Since $\mathcal{X}_{G}$ is connected and non-compact,
$H^{0}_{c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},-)$ is zero for all locally-free
sheaves, and so we obtain an isomorphism

$\alpha_{\mathrm{dR}}^{(1-\mathrm{ord})}:H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\operatorname{BGG}^{1}(-D)\right)^{\nabla=0}\cong\widetilde{H}^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle).$

Moreover, the inclusion of the subcomplex $\tau_{\geqslant
1}\operatorname{BGG}$ into the full BGG complex gives a commutative square of
maps

${\widetilde{H}^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)}$${\operatorname{Fil}^{1}H^{2}_{\mathrm{dR}}(X_{G},\mathcal{V}\langle-D\rangle)_{L_{\mathfrak{P}}}}$${H^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)}$${H^{2}_{\mathrm{dR}}(X_{G},\mathcal{V}\langle-D\rangle)_{L_{\mathfrak{P}}},}$

in which the top horizontal arrow is compatible, via
$\alpha_{\mathrm{dR}}^{(1-\mathrm{ord})}$, with the natural map

$H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\operatorname{BGG}^{1}(-D)\right)\to
H^{1}_{c}\left(\mathcal{X}_{G},\operatorname{BGG}^{1}(-D)\right)=\operatorname{Fil}^{1}H^{2}_{\mathrm{dR}}/\operatorname{Fil}^{k_{1}+k_{2}+2}.$

Since the partial Frobenius $\varphi_{1}$ lifts to
$\mathcal{X}_{G}^{(1-\mathrm{ord})}$, there is an action of $\varphi_{1}$ on
both of the spaces in the left-hand column, compatible with the action on
$H^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle,1+j)$
given by comparison with the rigid cohomology of $X_{G,0}$.

######  Proposition 5.4.1.

If $\nu^{1-\mathrm{ord}}$ is as in Theorem 5.2.1, then the class
$(\nu^{1-\mathrm{ord}},0)$ in

$H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\operatorname{BGG}^{1}(-D)\right)=H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\omega^{-k_{1},k_{2}+2}(-D)\right)\oplus
H^{1}_{c}\left(\mathcal{X}_{G}^{(1-\mathrm{ord})},\omega^{k_{1}+2,-k_{2}}(-D)\right)$

is in the kernel of $\nabla$, and hence defines a class in
$\widetilde{H}^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle)$.
The image of this class in
$H^{2}_{\mathrm{rig},c}(Y_{G,0}^{(1-\mathrm{ord})},\mathcal{V}_{G})$, under
the left vertical map of the above diagram, is
$\nu^{(1-\mathrm{ord})}_{\mathrm{dR}}$.

###### Proof.

We first show that $\nu^{1-\mathrm{ord}}$ is in the kernel of $\nabla$. This
follows from the fact that it has strictly small slope for $\varphi_{1}$: the
slopes of $\varphi_{1}$ on $\omega^{(k_{1}+2,k_{2}+2)}$ are all at least
$k_{1}+1$, and the operator $\nabla$ commutes with the Frobenius, so it must
be zero on all Frobenius eigenspaces of slope smaller than $k_{1}+1$.

This shows that $\nu^{1-\mathrm{ord}}$ has a well-defined image in
$H^{2}_{\mathrm{rig},c}(Y_{G,0}^{(1-\mathrm{ord})},\mathcal{V}_{G})$. Let us
temporarily write $\hat{\nu}^{(1-\mathrm{ord})}_{\mathrm{dR}}$ for this image;
our goal is to show that it coincides with
$\nu_{\mathrm{dR}}^{1-\mathrm{ord}}$. Since the latter is characterised as the
unique lifting of $\nu_{\mathrm{dR}}$ compatible with Hecke operators away
from $p\mathcal{N}$, it suffices to show that
$\hat{\nu}^{(1-\mathrm{ord})}_{\mathrm{dR}}$ lies in the correct Hecke
eigenspace, and that it maps to $\nu_{\mathrm{dR}}$ in
$H^{2}_{\mathrm{dR},c}(Y_{G},\mathcal{V}_{G})$.

It follows readily from the construction of $\nu^{1-\mathrm{ord}}$ that it
lies in the $\Pi$-eigenspace for the Hecke operators away from
$\mathfrak{p}_{1}$ (including $T_{\mathfrak{p}_{2}}$), since these operators
commute with $\varphi_{1}$. So $\hat{\nu}^{(1-\mathrm{ord})}_{\mathrm{dR}}$
has the correct Hecke action. Moreover, its image in
$H^{2}_{\mathrm{dR},c}(\mathcal{Y}_{G},\mathcal{V}_{G})$ is in the
$\varphi_{1}=\alpha_{1}$ eigenspace (because $\nu^{1-\mathrm{ord}}$ is); and
it lies in $\operatorname{Fil}^{(0,1+k_{2})}$, and maps to $\nu$ in
$\operatorname{Gr}^{(0,1+k_{2})}$, so it must be equal to $\nu_{\mathrm{dR}}$.
∎

###  5.5. FP-cohomology of the $\mathfrak{p}_{1}$-ordinary locus

We now consider a modified form of Besser’s finite-polynomial cohomology,
namely _Gros fp-cohomology_ , for the $\mathfrak{p}_{1}$-ordinary locus (with
compact supports). This cohomology, denoted by
$\widetilde{R\Gamma}_{\operatorname{fp},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})$
can be defined as the mapping fibre of the map

$\widetilde{R\Gamma}_{\mathrm{dR},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle)\xrightarrow{\
P_{1+j}(p^{-1-j}\varphi)\
}R\Gamma_{\mathrm{rig},c}(X_{G,0}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle).$

(As before, this is in fact independent of $j$ in the stated range, despite
the notations.) Note that although $\varphi_{1}$ lifts to
$\mathcal{X}_{G}^{1-\mathrm{ord}}$, the full Frobenius $\varphi$ does not; so
although
$R\Gamma_{\mathrm{rig},c}(X_{G,0}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle)$
is isomorphic to de Rham cohomology of $\mathcal{X}_{G}^{1-\mathrm{ord}}$, the
action of Frobenius (given by the functoriality of rigid cohomology) cannot be
‘seen’ via this description.

For proper schemes such as $\mathbb{X}_{G}$, there is no difference between
Gros fp-cohomology and the usual fp-cohomology, so there is an extension-by-
zero map

$\widetilde{R\Gamma}_{\operatorname{fp},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})\to
R\Gamma_{\operatorname{fp},c}(\mathbb{X}_{G},\mathcal{V}_{G}\langle-D\rangle;1+j,P_{1+j}).$

######  Proposition 5.5.1.

There exists a class

$\tilde{\nu}_{\operatorname{fp}}^{1-\mathrm{ord}}\in\widetilde{H}^{2}_{\operatorname{fp},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})$

with the following properties:

  * •

Its image in
$H^{2}_{\operatorname{fp},c}(\mathbb{X}_{G},\mathcal{V}_{G}\langle-D\rangle;1+j,P_{1+j})$
is the $\nu_{\operatorname{fp}}$ of Proposition 4.2.3.

  * •

Its image in
$\widetilde{H}^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle)$
is the class $(\nu^{1-\mathrm{ord}},0)$ of Proposition 5.4.1.

######  Remark 5.5.2.

The reader may be relieved to hear that the class
$\tilde{\nu}_{\operatorname{fp}}^{1-\mathrm{ord}}$ is “the ultimate among
liftings of $\nu$”: all other variants of $\nu$ will be images of this one.

###### Proof.

From the mapping-fibre definition of Gros fp-cohomology we have a long exact
sequence

$\dots\to
H^{1}_{\mathrm{rig},c}(X_{G,0}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)\to\widetilde{H}^{2}_{\operatorname{fp},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})\to\widetilde{H}^{2}_{\mathrm{dR},c}(\mathcal{X}_{G}^{(1-\mathrm{ord})},\mathcal{V}\langle-D\rangle)\to\dots,$

in which the boundary map is $P(\varphi)\circ\iota$. Moreover, this is
compatible under extension-by-zero with the corresponding sequence for fp-
cohomology of $\mathbb{X}_{G}$.

We have seen that the image of $(\nu^{1-\mathrm{ord}},0)$ under $\iota$ is the
class $\nu_{\mathrm{dR}}^{1-\mathrm{ord}}$, which is annihilated by
$P(\varphi)$. Hence it lifts to $\widetilde{H}^{2}_{\operatorname{fp},c}$.
This lift is unique up to the image of an element of
$H^{1}_{\mathrm{rig},c}(X_{G,0}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle)$;
but from Proposition 5.1.4 it follows that this group has trivial
$\Pi_{\mathrm{f}}$-eigenspace for the prime-to-$p$ Hecke operators, so there
is a unique Hecke-equivariant lifting of $(\nu^{1-\mathrm{ord}},0)$ to Gros-fp
cohomology. The image of this class under extension-by-0 is therefore a Hecke-
equivariant lifting of $\nu_{\mathrm{dR}}$ to fp-cohomology of
$\mathbb{X}_{G}$, so it must be $\nu_{\operatorname{fp}}$. ∎

It is easily checked that
$\iota^{-1}(\mathcal{X}_{G}^{\mathrm{ord}})=\mathcal{X}_{H}^{\mathrm{ord}}$,
so we have the following:

######  Corollary 5.5.3.

The pairing (4.1) is equal to
$\left\langle\iota^{[j],*}\left(\tilde{\nu}_{\operatorname{fp}}^{1-\mathrm{ord}}\right),\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn}}\right\rangle$,
where $\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn}}$ is the
image of $\mathrm{Eis}^{t}_{\operatorname{syn}}$ in the Gros fp-cohomology of
$\mathcal{X}_{H}^{\mathrm{ord}}$.∎

Here we define $\iota^{[j],*}$ for classes in Gros fp-cohomology using the
quasi-isomorphism from the BGG complex to the full de Rham complex. We shall
give explicit formulae in Section 7.1 below, but first we need to give an
explicit form for $\tilde{\nu}_{\operatorname{fp}}^{1-\mathrm{ord}}$, which
can only be done after restricting to
$\mathcal{X}_{G}^{\mathrm{ord}}\subset\mathcal{X}_{G}^{1-\mathrm{ord}}$.

##  6\. Restricting to the fully-ordinary locus

###  6.1. Cohomology with partial compact support

We now consider the cohomology of the fully ordinary locus
$X_{G,0}^{\mathrm{ord}}$. Since the complement
$X_{G,0}-X_{G,0}^{\mathrm{ord}}$ is the disjoint union of a closed subvariety
$X_{G,0}^{(1-\mathrm{ss})}$ and the open subvariety
$X_{G,0}^{(2-\mathrm{ss})}\cap X_{G,0}^{(1-\mathrm{ord})}$, we can apply the
formalism of [LZ20, §13] to define “cohomology of
$\mathcal{X}_{G}^{\mathrm{ord}}$ with compact support towards
$\mathcal{X}_{G}^{(1-\mathrm{ss})}$” (with coefficients in any abelian sheaf
on $\mathcal{X}_{G}$). We write this as $R\Gamma_{c1}(\mathcal{X}_{G},-)$. By
construction, this comes equipped with a restriction map

$R\Gamma_{c}\left(\mathcal{X}_{G}^{1-\mathrm{ord}},-\right)\to
R\Gamma_{c1}\left(\mathcal{X}_{G}^{\mathrm{ord}},-\right),$

which fits into a triangle whose third term is the compactly-supported
cohomology of
$\mathcal{X}_{G}^{1-\mathrm{ord}}\cap\mathcal{X}_{G}^{2-\mathrm{ss}}$. In
particular, by the same argument as Proposition 5.1.4, we have isomorphisms

$R\Gamma_{\mathrm{rig},c1}\left(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle\right)[\Pi_{\mathrm{f}}]\longleftarrow
R\Gamma_{\mathrm{rig},c}\left(\mathcal{X}_{G}^{1-\mathrm{ord}},\mathcal{V}\langle-D\rangle\right)[\Pi_{\mathrm{f}}]\longrightarrow
R\Gamma_{\mathrm{rig},c}\left(\mathcal{X}_{G},\mathcal{V}\langle-D\rangle\right)[\Pi_{\mathrm{f}}].$

The advantage of working with $\mathcal{X}_{G}^{\mathrm{ord}}$ is that both
$\varphi_{1}$ and $\varphi_{2}$ have liftings.

######  Notation 6.1.1.

Write $\nu^{\mathrm{ord}}\in
H^{1}_{c1}\Big{(}\mathcal{X}_{G}^{\mathrm{ord}},\omega^{(-k_{1},k_{2}+2)}(-D)\Big{)}$
for the image of $\nu^{1-\mathrm{ord}}$ under the above restriction map.

######  Note 6.1.2.

Over the ordinary locus, we have commuting liftings of both $\varphi_{1}$ and
$\varphi_{2}$, which both act on
$R\Gamma_{c1}\left(\mathcal{X}_{G}^{\mathrm{ord}},\omega^{(\dots)}\right)$;
and the operator $T_{\mathfrak{p}_{2}}$ decomposes as
$T_{\mathfrak{p}_{2}}=U_{\mathfrak{p}_{2}}+\varphi_{2},$ with
$U_{\mathfrak{p}_{2}}\circ\varphi_{2}=p^{k_{2}+1}\langle\mathfrak{p}_{2}\rangle$.

######  Corollary 6.1.3.

The class $P(\varphi)\cdot\nu^{\mathrm{ord}}$ lies in the kernel of the Hecke
operator $U_{\mathfrak{p}_{2}}$.

###### Proof.

We have $\varphi=\varphi_{1}\varphi_{2}$. The result follows easily from
Theorem 5.2.1 and Note 6.1.2, using the fact that $\nu^{(1-\mathrm{ord})}$ is
a $T_{\mathfrak{p}_{2}}$-eigenvector. ∎

###  6.2. The Poznan spectral sequence

We now recall a spectral sequence (introduced in [LZ20]) relating Gros fp-
cohomology to coherent cohomology. Here Gros fp-cohomology is defined in the
same way as for the $\mathfrak{p}_{1}$-ordinary locus above, but now with
$c1$-support.

######  Definition 6.2.1.

We define groups
$\mathscr{C}^{m,n}_{\operatorname{fp},c1}(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})$,
for $m,n\geqslant 0$, by

$\mathscr{C}^{m,n}_{\operatorname{fp},c1}(\dots)=H^{n}_{c1}\left(\mathcal{X}_{G}^{\mathrm{ord}},(\tau_{\geqslant
1}\operatorname{BGG})^{m}(-D)\right)\oplus
H^{n}_{c1}\left(\mathcal{X}_{G}^{\mathrm{ord}},\operatorname{BGG}^{m-1}(-D)\right);$

and we define differentials
$\mathscr{C}^{m,n}_{\operatorname{fp},c1}(\dots)\to\mathscr{C}^{m+1,n}_{\operatorname{fp},c1}(\dots)$
by

$(x,y)\mapsto\left(\nabla x,P(\varphi/p^{n})\iota(x)-\nabla y\right),$

where $\iota$ is the inclusion of
$\operatorname{Fil}^{1+j}\operatorname{BGG}^{\bullet}$ into
$\operatorname{BGG}^{\bullet}$, and $\nabla$ the differential of the BGG
complex.

Note that $\mathscr{C}^{m,n}_{\operatorname{fp},c1}$ is zero for $m\leqslant
0$ (this is obvious for $m\leqslant-1$, and holds for $m=0$ since
$\operatorname{Fil}^{(1+j)}\operatorname{BGG}^{0}=0$). It is also zero for
$n\leqslant 0$, since $H^{0}_{c1}$ vanishes for locally-free sheaves.

######  Proposition 6.2.2.

There is a first-quadrant spectral sequence, the _Poznań spectral sequence_ ,
with

${}^{\operatorname{Pz}}E_{1}^{mn}=\mathscr{C}^{m,n}_{\operatorname{fp},c1}(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle;1+j,P_{1+j}),$

and the differentials on the $E_{1}$ page given by the formula above. This
spectral sequence abuts to the Gros fp-cohomology
$\widetilde{H}^{m+n}_{\operatorname{fp},c1}(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})$.

######  Definition 6.2.3.

We define a _coherent fp-pair_ (of degree $(m,n)$, twist $1+j$ and
$c1$-support) to be an element of the kernel of the differential
$\mathscr{C}^{m,n}_{\operatorname{fp},c1}\to\mathscr{C}^{m+1,n}_{\operatorname{fp},c1}$;
we write the group of these as
$\mathscr{Z}^{m,n}_{\operatorname{fp},c1}(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})$.

Thus an fp-pair is a pair of elements

$x\in
H^{n}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},{\mathscr{F}il}^{1+j}\operatorname{BGG}^{m}(-D)),\qquad
y\in
H^{n}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},{\mathscr{F}il}^{1+j}\operatorname{BGG}^{m-1}(-D))$

which satisfy

(6.1) $\nabla(x)=0\qquad\text{and}\qquad\nabla(y)=P(p^{-1-j}\varphi)\iota(x).$

######  Note 6.2.4.

Given $x$, the equation (6.1) does not determine the element $y$ uniquely: it
is determined up to an element of
$H^{n}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\operatorname{BGG}^{m-1}(-D))^{\nabla=0}$.

######  Proposition 6.2.5.

For $0\leqslant j\leqslant\min\\{k_{1},k_{2}\\}$, the spectral sequence gives
rise to an isomorphism

$\alpha_{G,\operatorname{fp}}:\mathscr{Z}^{1,1}_{\operatorname{fp},c1}(\mathcal{X}_{G}^{\mathrm{ord}},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})\xrightarrow{\
\cong\
}\widetilde{H}^{2}_{\operatorname{fp},c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle;1+j,P_{1+j}).$

###### Proof.

Since the $E_{1}$ page of the spectral sequence is zero for $m\leqslant 0$,
the term ${}^{\operatorname{Pz}}E_{2}^{1,1}$ is the kernel of the differential
on ${}^{\operatorname{Pz}}E_{1}^{1,1}$, which is the group of fp-pairs. Since
this is the only nonzero term with $m+n=2$, and the incoming and outgoing
differentials at ${}^{\operatorname{Pz}}E_{r}^{1,1}$ are trivially zero for
all $r\geqslant 1$, we conclude that ${}^{\operatorname{Pz}}E_{2}^{1,1}$
coincides with $\widetilde{H}^{2}$ of the abutment, as required. ∎

######  Corollary 6.2.6.

Every cohomology class in
$\widetilde{H}^{2}_{\operatorname{fp},c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle;1+j,P_{1+j})$
can be uniquely represented by a coherent fp-pair of degree $(1,1)$.

###### Proof.

Since the $E_{1}$ terms of the spectral sequence are supported in the region
$m,n\geqslant 1$, there are no other terms of total degree 2 except
$(m,n)=(1,1)$; and clearly $E_{2}^{(1,1)}=E_{\infty}^{(1,1)}$ since the
differentials on the $E_{2}$ page and beyond land outside this region. ∎

An exactly analogous argument shows that for the truncated de Rham cohomology
groups $\widetilde{H}^{i}_{\mathrm{dR},c1}$ (the hypercohomology of
$\tau_{\geqslant 1}\operatorname{BGG}^{\bullet}(-D)$), we have an isomorphism

$H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\operatorname{BGG}^{1}(-D))\xrightarrow{\
\alpha_{G,\mathrm{rig}}\
}\widetilde{H}_{\mathrm{dR},c1}^{2}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle,1+j).$

######  Lemma 6.2.7.

Let $0\leqslant j\leqslant\min\\{k_{1},k_{2}\\}$. We have a commutative
diagram

${\mathscr{Z}^{1,1}_{\operatorname{fp},c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-D\rangle;1+j,P_{1+j})}$${\widetilde{H}^{2}_{\operatorname{fp},c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle;1+j,P_{1+j})}$${H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\operatorname{BGG}^{1}(-D))}$${\widetilde{H}_{\mathrm{dR},c1}^{2}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle,1+j)}$$\scriptstyle{\alpha_{G,\operatorname{fp}}}$$\scriptstyle{\alpha_{G,\mathrm{rig}}}$

where the vertical arrows are the natural projection maps.

###### Proof.

Clear from the constructions. ∎

###  6.3. Construction of a coherent fp-pair

######  Notation 6.3.1.

Define

$\tilde{\nu}^{(\mathrm{ord})}_{\operatorname{fp}}\in\widetilde{H}^{2}_{\operatorname{fp},c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}\langle-\mathcal{D}\rangle;1+j,P_{1+j})$

to be the restriction of the class
$\tilde{\nu}^{(1-\mathrm{ord})}_{\operatorname{fp}}$ constructed above.

######  Corollary 6.3.2.

There exists a uniquely determined class

$\xi\in
H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\omega^{(-k_{1},-k_{2})}(-\mathcal{D})),$

which is independent of $j$, such that $(\nu^{(\mathrm{ord})},\xi)$ forms an
fp-pair representing the class
$\tilde{\nu}^{(\mathrm{ord})}_{\operatorname{fp}}$, and such that $\xi$ lies
in the $\Pi$-eigenspace for the Hecke operators away from $pN$.

###### Proof.

The existence of $\xi$ is immediate from Corollary 6.2.6 and Lemma 6.2.7; the
independence from $j$ is clear by construction.

Now, if $\xi^{\prime}$ is another element such that
$(\nu^{(\mathrm{ord})},\xi^{\prime})$ also represents
$\tilde{\nu}^{(\mathrm{ord})}_{\operatorname{fp}}$, then

$\xi-\xi^{\prime}\in
H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\operatorname{BGG}^{0}(-D))^{\nabla=0}\cong
H^{1}_{\mathrm{rig},c1}(X_{G,0},\mathcal{V}\langle-\mathcal{D}\rangle).$

As we have seen, the $\Pi_{\mathrm{f}}$-eigenspace in this cohomology is zero,
so there is a unique choice of $\xi$ which is Hecke-equivariant. ∎

######  Lemma 6.3.3.

The element $\xi$ has the following properties:

(6.2) $\varphi_{1}.\xi=\alpha_{1}\,\xi\qquad\text{and}\qquad
U_{\mathfrak{p}_{2}}.\xi=0.$

###### Proof.

We first show the latter statement. We deduce from Corollary 6.1.3 that

$U_{\mathfrak{p}_{2}}\cdot\xi\in
H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\operatorname{BGG}^{0}(-\mathcal{D}))^{\nabla=0}[\Pi_{f}^{\prime}].$

But as observed above, this space is zero. The former statement follows
analogously, by considering the element $(\varphi_{1}-\alpha_{1})\cdot\xi$. ∎

We now lift these classes from the BGG complex to the full de Rham complex:

######  Definition 6.3.4.

  1. (1)

Write $\dot{\xi}$ for the image of $\xi$ in
$H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},\mathcal{V}(-D))$.

  2. (2)

Similarly, for $0\leqslant j\leqslant\min\\{k_{1},k_{2}\\}$ write
$\dot{\nu}^{(\mathrm{ord})}_{j}$ for the image of $\nu^{(\mathrm{ord})}$ in

$H^{1}_{c1}(\mathcal{X}^{\mathrm{ord}}_{G},{\mathscr{F}il}^{j}\mathcal{V}\otimes\Omega^{1}\langle-\mathcal{D}\rangle).$

######  Lemma 6.3.5.

The element $\dot{\xi}$ also satisfies

(6.3) $\varphi_{1}.\dot{\xi}=\alpha_{1}\,\dot{\xi}\qquad\text{and}\qquad
U_{\mathfrak{p}_{2}}.\dot{\xi}=0.$

The following result will be very important for the regulator evaluation:

######  Proposition 6.3.6.

We have

$U_{p}\circ\iota^{*}(\dot{\xi})=0.$

###### Proof.

As maps $H^{\bullet}(\mathcal{X}^{\mathrm{ord}}_{G})\rightarrow
H^{\bullet}(\mathcal{X}^{\mathrm{ord}}_{H})$, we have the following identity:

(6.4)
$U_{p}\circ\iota^{*}\circ\varphi_{\mathfrak{p}_{1}}=\iota^{*}\circ(\langle\mathfrak{p}_{1}\rangle
U_{\mathfrak{p}_{2}}).$

Since $\dot{\xi}$ is an eigenvector of $\varphi_{\mathfrak{p}_{1}}$ with non-
zero eigenvalue, and it is in the kernel of $\varphi_{\mathfrak{p}_{2}}$, the
Proposition follows. ∎

##  7\. Expression via coherent cohomology

###  7.1. Relating the BGG and de Rham complexes

We need to recall some formulae relating the de Rham and BGG complexes for
$\operatorname{GL}_{2}$. Let $k\in\mathbf{Z}_{\geqslant 0}$. The BGG complex
for weight $k\geqslant 0$ is given by
$\left[\omega^{-k}\xrightarrow{\Theta}\omega^{k+2}\right]$, where $\Theta$ is
a differential operator given in terms of $q$-expansions by

$\Theta=\tfrac{(-1)^{k}}{k!}\theta^{k+1},\qquad\theta=q\tfrac{\mathrm{d}}{\mathrm{d}q}.$

(Cf. [TX16, Remark 2.17] for example.) The map
$\operatorname{BGG}^{\bullet}\to\operatorname{DR}^{\bullet}$ is given as
follows.

  * •

In degree 1, it is given by the tensor product of the natural inclusion
$\omega^{k}=\operatorname{Fil}^{k}(\operatorname{Sym}^{k}\mathcal{W}_{H})\hookrightarrow\mathcal{W}_{H}$
and the Kodaira–Spencer isomorphism $\omega^{2}\cong\Omega^{1}(D)$.

  * •

In degree 0, a section $s$ of $\omega^{-k}$ is mapped to the unique section of
$\operatorname{Sym}^{k}\mathcal{W}_{H}$ whose image in
$\operatorname{Sym}^{k}\mathcal{W}_{H}/\operatorname{Fil}^{0}\cong\omega^{-k}$
is $s$, and whose image under the differential $\nabla$ lands in
$\omega^{k}\otimes\Omega^{1}(D)$.

We now recall (and somewhat reformulate) some results from §4 of [KLZ20]
giving a completely explicit description of these maps. We write
$\mathscr{X}^{\mathrm{ord}}_{H}$ for the ordinary locus _as a classical rigid
space_ (not a dagger space), i.e. neglecting overconvergence. Passing to the
Igusa tower $\mathscr{I\\!G}_{H}$ (the canonical
$\mathbf{Z}_{p}^{\times}$-covering of $\mathscr{X}^{\mathrm{ord}}_{H}$
parametrising ordinary elliptic curves with a trivialization of their formal
group), we obtain a canonical section $v$ of $\omega$, corresponding to the
invariant differential form $\tfrac{dT}{T}$ on the Tate curve
$\mathbf{G}_{m}/q^{\mathbb{Z}}$; note that $\varphi^{*}(v)=pv$. If we let $w$
be the unique section such that $\nabla(v)=u\otimes\xi$, where $\xi$ the local
basis of $\Omega^{1}_{\mathscr{X}^{\mathrm{ord}}_{H}}(D)$ corresponding to
$\tfrac{\mathrm{d}q}{q}$, then $v$ and $w$ are a basis of sections of
$\mathcal{W}_{H}$ over the Igusa tower, with $w$ spanning the unit-root
subspace (we have $\varphi^{*}(w)=w$) and $\nabla w=0$. Hence we obtain a
basis of sections $(v^{a}w^{k-a})_{0\leqslant a\leqslant k}$ of
$\operatorname{Sym}^{k}\mathcal{W}_{H}$ in which the actions of $\nabla$ and
$\varphi$ are completely explicit.

In these coordinates, the map from $\omega^{-k}$ to
$\operatorname{Sym}^{k}\mathcal{W}_{H}$ sends a section $\mathcal{G}$ of
$\omega^{-k}$ to the section

$\sum_{i=0}^{k}\tfrac{(-1)^{i}}{i!}\theta^{i}(\mathcal{G})\cdot v^{i}w^{k-i}.$

One verifies easily that the image of this under $\nabla$ is
$\theta^{k+1}(\mathcal{G})v^{k}\otimes\xi$, as expected. (Note that it is not
_a priori_ obvious that such a sum is overconvergent if $\mathcal{G}$ is,
since neither $\theta$ nor the local bases $v^{i}w^{k-i}$ have any meaning
outside the ordinary locus.)

###  7.2. The Eisenstein class as a coherent fp-pair

Recall that the $\operatorname{GL}_{2}$ Eisenstein class
$\mathrm{Eis}^{t}_{\operatorname{syn},N}$ lies in
$H^{1}_{\operatorname{syn}}(\mathbb{Y}_{H},\mathcal{V}_{H}(1+t))$, where
$\mathbb{Y}_{H}$ is the $\mathbf{Z}_{p}$-model of the modular curve
$Y_{1}(N)$, and $\mathcal{V}_{H}$ is the sheaf corresponding to the $t$-th
symmetric power of the standard representation.

If we restrict to the open subscheme $\mathbb{Y}_{H}^{\mathrm{ord}}$ given by
removing the supersingular points of the special fibre (so
$\mathbb{Y}_{H}^{\mathrm{ord}}$ has the same generic fibre as
$\mathbb{Y}_{H}$), and work with Gros syntomic cohomology, then we have a
convenient explicit description: a class
$x\in\widetilde{H}^{1}_{\operatorname{syn}}(\mathcal{X}_{1}(N)^{\mathrm{ord}},\mathcal{V}_{H},1+t)$
is given by a pair $(x_{0},x_{1})$, where

$x_{0}\in H^{0}(\mathcal{X}_{1}(N)^{\mathrm{ord}},\mathcal{V}),\qquad x_{1}\in
H^{0}(\mathcal{X}_{1}(N)^{\mathrm{ord}},\operatorname{Fil}^{t}\mathcal{V}\otimes\Omega^{1}\langle
D\rangle),\qquad\nabla x_{0}=(1-p^{-1-t}\varphi)x_{1}.$

The sheaf $\operatorname{Fil}^{t}\mathcal{V}\otimes\Omega^{1}\langle D\rangle$
is simply $\omega^{t+2}$, so $x_{1}$ is an overconvergent $p$-adic modular
form of weight $t+2$; via this description, $\varphi$ acts on overconvergent
forms as $p^{t+1}\langle p\rangle V_{p}$, where $\langle p\rangle$ is the
diamond operator for $p\bmod N$.

We let $\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn},N}$ be
the image of $\mathrm{Eis}^{t}_{\operatorname{syn},N}$ in this group; we shall
now write down an explicit representing pair. The following is a reformulation
of Theorem 4.5.7 of [KLZ20]; the formulations differ because we are using
symmetric powers here rather than symmetric tensors (the basis vector
$v^{[r,s]}$ of _op.cit._ corresponds to $\tfrac{v^{s}w^{r}}{r!s!}$ in our
present notation) and because we use a slightly different notation for
Eisenstein series following [Kat04].

######  Proposition 7.2.1.

The Eisenstein class
$\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn},N}$ is
represented by the pair of sections
$\left(\epsilon_{0}^{t},\epsilon_{1}^{t}\right)$ whose restrictions to
$\mathscr{X}_{H}^{\mathrm{ord}}$ are

$\displaystyle\epsilon_{0}$
$\displaystyle=-N^{k}\sum_{u=0}^{t}\frac{(-1)^{t-u}}{u!}\theta^{u}\left(E^{-t,\mathrm{ord}}_{0,1/N}\right)\cdot
v^{t-u}w^{u},$ $\displaystyle\epsilon_{1}$
$\displaystyle=-\tfrac{N^{t}}{t!}F^{t+2}_{0,1/N}\cdot v^{t}\otimes\xi.$

Here $F^{t+2}_{0,1/N}$ is the algebraic Eisenstein series with $q$-expansion

$\zeta(-1-t)+\sum_{n>0}q^{n}\sum_{d|n}(\tfrac{n}{d})^{t+1}(\zeta_{N}^{d}+(-1)^{t}\zeta_{N}^{-d}),$

and $E^{-t,\mathrm{ord}}_{0,1/N}$ is an ordinary $p$-adic Eisenstein series of
weight $-t$, satisfying $\theta^{t+1}(E^{-t,\mathrm{ord}}_{0,1/N})=(1-\langle
p\rangle V_{p})F^{t+2}_{0,1/N}$.∎

######  Remark 7.2.2.

Note that the individual terms $\theta^{u}(E^{-t,\mathrm{ord}}_{0,1/N})$ are
$p$-adic modular forms, but they are not overconvergent (unless $u=0$).
Nonetheless, $\epsilon_{0}$ is an overconvergent section of $\mathcal{V}_{H}$;
the non-overconvergence arises because the sections $v,w$ we are using to
trivialize $\mathcal{V}_{H}$ over the ordinary locus are not themselves
overconvergent.

###  7.3. Explicit formulae for the Clebsch–Gordan map

The map $\operatorname{CG}^{[j]}$ is given explicitly by the following formula
(c.f. [KLZ20, Prop. 5.1.2])444The factorials appear slightly different from
_op.cit._ since we are here working with symmetric powers $v^{m}$ rather than
symmetric tensors $v^{[m]}$. This is also the reason for the presence of $t!$
in the formula for the Eisenstein series.: for $0\leqslant s\leqslant t$, we
have

(7.1) $\iota^{[j]}(v^{s}w^{t-s})=\\\ \sum_{\begin{subarray}{c}0\leqslant
r\leqslant k_{1}-j\\\ 0\leqslant r^{\prime}\leqslant k_{2}-j\\\
r+r^{\prime}=s\end{subarray}}\sum_{i=0}^{j}(-1)^{i}\frac{s!(t-s)!}{r!(r^{\prime})!(k_{1}-r-j)!(k_{2}-r^{\prime}-j)!i!(j-i)!}v^{r+i}w^{k_{1}-r-i}\boxtimes
v^{r^{\prime}+j-i}w^{k_{2}-r^{\prime}-j+i}\otimes e_{-j}.$

######  Lemma 7.3.1.

For given values of $k_{1},k_{2},j$, the image of $\iota^{[j]}(v^{s}w^{t-s})$
in the line spanned by the basis vector $v^{k_{1}}\boxtimes w^{k_{2}}\otimes
e_{-j}$ is zero for all $s$ except $s=k_{1}-j$, in which case it is equal to
$\tfrac{(-1)^{j}}{j!}\cdot v^{k_{1}}\boxtimes w^{k_{2}}\otimes e_{-j}$.

###### Proof.

This is a superficially modified version of Proposition 5.1.2 of [KLZ20]. ∎

######  Remark 7.3.2.

In particular, this shows that the Clebsch–Gordan map is not in general
defined integrally (i.e. does not respect the lattice given by the
$\mathbf{Z}$-span of the basis vectors). However, the coefficient $j!$ is the
worst possible – one can check that $j!\iota^{[j]}$ is integral.

###  7.4. Reduction to a pairing in coherent cohomology

Recall that we want to evaluate the pairing

(7.2)
$\left\langle(\iota^{[j]})^{*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle.$

Now we have

(7.2)
$\displaystyle=\left\langle(\iota^{[j]})^{*}\left(\nu^{(1-\mathrm{ord})}_{\operatorname{fp}}\right),\,\mathrm{Eis}^{t,\mathrm{ord}}_{\operatorname{syn},N}\right\rangle.$
$\displaystyle=\left\langle(\iota^{[j]})^{*}\left(\tilde{\nu}^{(1-\mathrm{ord})}_{\operatorname{fp}}\right),\,\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn},N}\right\rangle$
$\displaystyle=\left\langle(\iota^{[j]})^{*}\left(\tilde{\nu}^{(\mathrm{ord})}_{\operatorname{fp}}\right),\,\widetilde{\mathrm{Eis}}^{t,\mathrm{ord}}_{\operatorname{syn},N}\right\rangle$
(7.3)
$\displaystyle=\left\langle(\iota^{[j]})^{*}\left(\dot{\nu}^{(\mathrm{ord})}_{j},\dot{\xi}\right),\,(\epsilon_{1},\epsilon_{0})\right\rangle.$

Here, (7.3) takes values in the group

$\widetilde{H}^{2}_{\operatorname{fp},c}(\mathcal{X}^{\mathrm{ord}}_{H},\mathbf{Q}_{p}\langle-\mathcal{D}_{H}\rangle,2;P_{j})\cong^{\operatorname{tr}}\mathbf{Q}_{p}.$

We evaluate (7.3) using the formalism of cup products in fp-cohomology (cf.
[Bes00]). We have

$P_{j}(st)=a(t,s)P_{j}(t)+b(t,s)(1-s)$

with $a(t,s)=s$ and

$b(t,s)=\frac{P_{j}(st)-sP_{j}(t)}{1-s}=1-b\cdot t^{2}s,\qquad
b=\frac{p^{2j+2}}{\alpha_{1}^{2}\alpha_{2}\beta_{2}}=\frac{\beta_{1}}{\alpha_{1}}\cdot\frac{1}{p^{(k_{1}+k_{2}-2j)}}.$

Hence

$\displaystyle
P_{j}(p^{-1})\times\left\langle(\iota^{[j]})^{*}\left(\dot{\nu}^{(\mathrm{ord})}_{j},\dot{\xi}\right),\,(\epsilon_{1}^{t},\epsilon_{0}^{t})\right\rangle$
$\displaystyle=(\iota^{[j]})^{*}(\dot{\xi})\cup\varphi_{H}^{*}(\epsilon_{1}^{t})+\left(1-\frac{\beta_{1}\cdot(\varphi_{H}^{*,2}\otimes\varphi_{H}^{*})}{p^{k_{1}+k_{2}-2j}\langle
p\rangle\alpha_{1}}\right)\left((\iota^{[j]})^{*}(\dot{\nu}^{(\mathrm{ord})}_{j})\cup\epsilon_{0}^{t}\right).$

######  Lemma 7.4.1.

We have $(\iota^{[j]})^{*}(\dot{\xi})\cup\varphi_{H}^{*}(\epsilon_{0}^{t})=0$.

###### Proof.

Observe that $U_{p}$ acts on the top-degree rigid cohomology as multiplication
by a power of $p$. But we also have

$U_{p}\left((\iota^{[j]})^{*}(\dot{\xi})\cup\varphi_{H}^{*}(\epsilon_{0}^{t})\right)=U_{p}\left((\iota^{[j]})^{*}(\dot{\xi})\right)\cup\epsilon_{0}^{t},$

which is equal to zero by Proposition 6.3.6. The Proposition follows. ∎

######  Proposition 7.4.2.

Equation (7.3) is equal to

$\frac{\left(1-\tfrac{\beta_{1}}{p\alpha_{1}}\right)}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)}\left\langle(\iota^{[j]})^{*}(\dot{\nu}^{(\mathrm{ord})}_{j}),\epsilon_{0}^{t}\right\rangle.$

###### Proof.

Since any $p$-depleted form will pair to 0 with a form that’s in a direct sum
of finite-slope $\varphi$-eigenspaces, we only care about the $\epsilon_{0}$
term modulo exact forms. We also only care about its projection to the
$\langle p\rangle=1$ eigenspace, because we are pairing it with a class in the
$\Pi$-eigenspace and $\Pi$ has trivial central character.

On the space
$H^{2}_{\mathrm{rig},c}(\mathcal{X}_{H}^{\mathrm{ord}}\langle-\mathcal{D}\rangle,\mathbf{Q}_{p}(2))$,
the Frobenius acts as $p^{-1}$, and $\langle p\rangle$ acts trivially; so we
can replace
$\frac{\beta_{1}\cdot(\varphi_{H}^{*,2}\otimes\varphi_{H}^{*})}{p^{k_{1}+k_{2}-2j}\langle
p\rangle\alpha_{1}}$ with
$\frac{\beta_{1}\cdot(1\otimes\varphi_{H}^{-1})}{p^{k_{1}+k_{2}-2j+2}\alpha_{1}}$.

The operator $\varphi^{-1}$ makes sense modulo $p$-depleted forms, and in this
quotient we have

$\varphi^{-1}\cdot\epsilon_{0}^{m}=p^{m+1}\langle
p\rangle^{-1}\epsilon_{0}^{m}.$

So we are done. ∎

######  Remark 7.4.3.

Note that $\dot{\nu}^{(\mathrm{ord})}_{j}$ makes sense over the
$\mathfrak{p}_{1}$-ordinary locus $\mathcal{X}_{G}^{1-\mathrm{ord}}$; the
antiderivative $\dot{\xi}$ is only defined over the fully ordinary locus
$\mathcal{X}_{G}^{\mathrm{ord}}$, but this term has disappeared from our
formula. So we can also interpret the pairing of Proposition 7.4.2 as a cup-
product in the cohomology of $\mathcal{X}_{G}^{1-\mathrm{ord}}$, which will
allow us to compare with the construction of $p$-adic $L$-functions from
[GLZ23].

###  7.5. A partial unit root splitting

We saw above that over $\mathscr{X}_{H}^{\mathrm{ord}}$ (the ordinary locus as
a classical rigid space) the Hodge filtration of $\mathcal{V}_{H}$ has a
canonical splitting given by the unit-root subspace. For $X_{G}$ we have the
more refined structure of a $\mathbf{Z}^{2}$-filtration, and we can ask for
splittings of either factor. We state below the case of interest.

######  Proposition 7.5.1.

Over $\mathscr{X}_{G}^{1-\mathrm{ord}}$, the natural inclusion map
$\omega^{(k_{1},-k_{2})}\hookrightarrow\mathcal{V}_{G}/\operatorname{Fil}^{k_{1}+1}$
admits a canonical splitting.

###### Proof.

We have a projection map from $\mathcal{V}_{G}$ onto
$\mathcal{V}_{G}/\operatorname{Fil}^{(0,1)}\mathcal{V}_{G}\cong\operatorname{Sym}^{k_{1}}\mathcal{W}_{G}\boxtimes\omega^{-k_{2}}$.
So it suffices to show that the filtration on
$\operatorname{Sym}^{k_{1}}\mathcal{W}_{G}$ is splittable over the 1-ordinary
locus; but this is clear, since a complement is provided by the unit-root
subspace for the operator $\varphi_{1}$. ∎

###  7.6. Comparison of pushforward maps

To begin with, we observe that there exist pushforward-maps in coherent
cohomology:

######  Proposition 7.6.1.

There exists a pushforward map on coherent cohomology

$\iota_{\operatorname{coh},*}:H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},\omega^{k_{1}-k_{2}})\longrightarrow
H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}}).$

###### Proof.

Clear since the conormal sheaf of the embedding $\iota$ is isomorphic to
$\omega^{2}$. ∎

We want to compare this map with the pushforward maps $\iota_{*}^{[j]}$ on the
cohomology of the de Rham sheaves.

######  Notation 7.6.2.

We use the notation of Section 2.2, and we write

  * •

$\mathcal{V}_{G}=\operatorname{Sym}^{[k_{1},k_{2}]}\mathcal{H}(\mathcal{A})_{\mathbf{Q}_{p}}$;

  * •

$\mathcal{V}_{H}=\operatorname{Sym}^{t}\mathcal{H}(\mathcal{E})_{\mathbf{Q}_{p}}$
for $0\leqslant j\leqslant\min\\{k_{1},k_{2}\\}$.

for for vector bundles attached to $V_{H}$ and $V_{G}$, respectively.

######  Proposition 7.6.3.

Pushforward along $\iota^{[j]}$ induces a commutative diagram

${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},V_{H}/\operatorname{Fil}^{k_{1}-j+1}V_{H})}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},V_{G}/\operatorname{Fil}^{k_{1}+1}V_{G}\otimes\Omega^{1}\langle
D\rangle)}$${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},\operatorname{Gr}^{k_{1}-j}V_{H})}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\operatorname{Gr}^{k_{1}}V_{G}\otimes\Omega^{1}\langle
D\rangle)}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}}).}$$\scriptstyle{\iota^{[j]}_{*}}$$\scriptstyle{\tfrac{1}{j!}\iota_{\operatorname{coh},*}}$

###### Proof.

For the commutativity of the top square, we note that the Clebsch–Gordan map
is compatible with filtrations and hence induces a commutative diagram of
$B_{H}$-representations

${V_{H}/\operatorname{Fil}^{k_{1}-j}V_{H}}$${V_{G}/\operatorname{Fil}^{k_{1}}V_{G}}$${\operatorname{Gr}^{k_{1}-j}V_{H}}$${\operatorname{Gr}^{k_{1}}V_{G}.}$

The identification of the diagonal map as
$\tfrac{1}{j!}\iota_{\operatorname{coh},*}$ arises similarly, using the
explicit formula (7.1). ∎

######  Proposition 7.6.4.

The diagram in Proposition 7.6.3 is compatible with the unit root splitting

$u_{H}:H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},V_{H}/\operatorname{Fil}^{k_{1}-j+1}V_{H})\longrightarrow
H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},\operatorname{Gr}^{k_{1}-j}V_{H})$

and the partial unit root splitting

$u_{G}:H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},V_{G}/\operatorname{Fil}^{k_{1}+1}V_{G}\otimes\,\Omega^{1})\longrightarrow
H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}}):$

we have a commutative diagram

${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},V_{H}/\operatorname{Fil}^{k_{1}-j+1}V_{H})}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},V_{G}/\operatorname{Fil}^{k_{1}+1}\otimes\,\Omega^{1})}$${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},\operatorname{Gr}^{k_{1}-j}V_{H})}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\operatorname{Gr}^{k_{1}}V_{G}\otimes\Omega^{1})}$${H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}})}$$\scriptstyle{\iota_{*}^{[j]}}$$\scriptstyle{u_{H}}$$\scriptstyle{u_{G}}$$\scriptstyle{\tfrac{1}{j!}\iota_{\operatorname{coh},*}}$

###### Proof.

Over $\mathscr{X}_{H}^{\mathrm{ord}}$, we have a canonical splitting of the
Hodge filtration of $\mathcal{W}_{H}$, as above. To prove the proposition, it
is sufficient to check that the unit root splittings induce a commutative
diagram

${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},V_{H}/\operatorname{Fil}^{k_{1}-j+1}V_{H})}$${H^{0}\left(\mathscr{X}^{\mathrm{ord}}_{H},\iota^{*}(V_{G}/\operatorname{Fil}^{k_{1}+1}\otimes\,\Omega^{1})\right)}$${H^{0}(\mathscr{X}_{H}^{\mathrm{ord}},\operatorname{Gr}^{k_{1}-j}V_{H})}$${H^{0}\left(\mathscr{X}^{\mathrm{ord}}_{H},\iota^{*}(\operatorname{Gr}^{k_{1}}V_{G}\otimes\Omega^{1})\right)}$${H^{0}\left(\mathscr{X}^{\mathrm{ord}}_{H},\iota^{*}(\omega^{(k_{1}+2,-k_{2})}\right)}$

which boils down to an explicit computation with the Clebsch–Gordan map using
equation (7.1). ∎

It follows from the classicity theorem of higher Hida theory proved in [GLZ23]
that the natural restriction map (forgetting overconvergence)

$H^{1}(\mathcal{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}})\to
H^{1}(\mathscr{X}^{1-\mathrm{ord}}_{G},\omega^{k_{1}+2,-k_{2}})$

is an isomorphism on the ordinary (slope 0) eigenspace for
$U_{\mathfrak{p}_{1}}$. If $\Pi$ is ordinary at $\mathfrak{p}_{1}$ (and
$\alpha_{1}$ is, necessarily, the unit root) then pairing with
$\nu^{\mathrm{ord}}$ factors through this eigenspace, so we obtain the
following:

######  Corollary 7.6.5.

Assume $\Pi$ is ordinary at $\mathfrak{p}_{1}$. Then the linear functional on
$H^{0}(\mathcal{X}_{H}^{\mathrm{ord}},V_{H}/\operatorname{Fil}^{k_{1}-j+1}V_{H})$
given by pairing with $(\iota^{[j],*})(\dot{\nu}^{\mathrm{ord}})$ factors
through the unit root splitting into
$:H^{0}(\mathcal{X}_{H}^{\mathrm{ord}},\operatorname{Gr}^{k_{1}-j}V_{H})$, and
it is given by

$\frac{k_{1}!\,k_{2}!}{j!}\cdot\,\left\langle\nu^{(1-\mathrm{ord})},\iota_{\operatorname{coh},*}(-)\right\rangle_{\mathcal{X}_{G}^{1-\mathrm{ord}}}.\qed$

######  Remark 7.6.6.

The factors $k_{i}!$ arise from the pairing of the basis vectors $v^{k_{i}}$
and $w^{k_{i}}$ of $\operatorname{Sym}^{k_{i}}W_{G}$.

###  7.7. Relation to $p$-adic $L$-functions

We use Corollary 7.6.5 in order to relate the formula in Proposition 7.4.2 to
values of $p$-adic $L$-functions. We assume henceforth that $\Pi$ is ordinary
at $\mathfrak{p}_{1}$.

######  Proposition 7.7.1.

The pairing (7.2) is given by

$\left\langle(\iota^{[j]})^{*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle=\\\
\frac{N^{(k_{1}+k_{2}-2j)}(-1)^{k_{2}-j+1}\left(1-\tfrac{\beta_{1}}{p\alpha_{1}}\right)}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)}\binom{k_{1}}{j}\,k_{2}!\,\cdot\left(\nu^{(\mathrm{ord})}\cup\iota_{\operatorname{coh},*}(\theta^{(k_{1}-j)}E_{0,1/N}^{-t,\mathrm{ord}})\right).$

Compare Corollary 6.5.7 in [KLZ20].

###### Proof.

We deduce from Proposition 7.2.1 that the image of
$\epsilon_{0}^{k_{1}+k+2-2j}$ under projection to
$H^{0}(\mathcal{X}^{\mathrm{ord}}_{H},\operatorname{Gr}^{k_{1}-j}\mathcal{V}_{H})$
is given by
$-N^{k}(-1)^{k_{2}-j}\frac{(t)!}{(k_{1}-j)!}\theta^{k_{1}-j}(E^{-t,\mathrm{ord}}_{0,1/N})$.
Combining this with Proposition 7.4.2 and Corollary 7.6.5 gives the result. ∎

######  Note 7.7.2.

By adjunction, we can write the formula in Proposition 7.7.1 as

$\left\langle(\iota^{[j]})^{*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle=\\\
\frac{N^{(k_{1}+k_{2}-2j)}(-1)^{k_{2}-j+1}\left(1-\tfrac{\beta_{1}}{p\alpha_{1}}\right)}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)}\binom{k_{1}}{j}\,k_{2}!\,\cdot\left(\iota_{\operatorname{coh}}^{*}(\nu^{(\mathrm{ord})})\cup\theta^{(k_{1}-j)}E_{0,1/N}^{-t,\mathrm{ord}}\right).$

In order to relate this formula to a (non-critical) value of a $p$-adic
$L$-function, we need to replace

$\theta^{(k_{1}-j)}E_{0,1/N}^{-t,\mathrm{ord}}=E^{j-k_{2},\mathrm{ord}}_{0,1/N}$

by its $p$-depletion

$E^{j-k_{2},[p]}_{0,1/N}=(1-\varphi\circ
U_{p})E^{j-k_{2},\mathrm{ord}}_{0,1/N}.$

We adapt the argument from [KLZ20, §6.5]. Let
$V_{\mathfrak{p}_{2}}=p^{-1-k_{2}}\langle\mathfrak{p}_{2}\rangle^{-1}\varphi_{2}$,
so $V_{\mathfrak{p}_{2}}$ is a right inverse of $U_{\mathfrak{p}_{2}}$. Let
$\lambda,\mu$ be constants such that

$U_{\mathfrak{p}_{2}}(\nu^{\mathrm{ord}})=\lambda\nu^{\mathrm{ord}}-\mu
V_{\mathfrak{p}_{2}}(\nu^{\mathrm{ord}})$

(explicitly, we have $\lambda=a_{\mathfrak{p}_{2}}(\mathcal{F})$ and
$\mu=p^{1+k_{2}}$), and let $\gamma=p^{k_{1}-j}$. Using an analogue of Lemma
6.5.8 from _op.cit._ , we deduce the following result:

######  Lemma 7.7.3.

We have

$\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\theta^{k_{1}-j}E^{-t,[p]}_{0,1/N}=\left(1-\lambda\gamma
p^{-k_{1}}\beta_{\mathfrak{p}_{1}}\cdot
V_{p}+\mu\gamma^{2}(p^{-k_{1}}\beta_{\mathfrak{p}_{1}})^{2}V_{p}^{2}\right)\left(\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\theta^{k_{1}-j}E^{-t,\mathrm{ord}}_{0,1/N}\right).$

###### Proof.

Arguing as in §6.5 in _op.cit._ , we see that

$\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\theta^{k_{1}-j}E^{-t,[p]}_{0,1/N}=\left(1-\lambda\gamma
p^{-k_{1}}\beta_{\mathfrak{p}_{1}}\cdot
V_{p}+\mu\gamma^{2}(p^{-k_{1}}\beta_{\mathfrak{p}_{1}})^{2}V_{p}^{2}\right)\left(\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\theta^{k_{1}-j}E^{-t,\mathrm{ord}}_{0,1/N}\right)\\\
+(1-V_{p}U_{p})\left(\mu\varphi^{2}\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\theta^{k_{1}-j}E^{-t,\mathrm{ord}}_{0,1/N}-\iota^{*}_{\operatorname{coh}}(\nu^{\mathrm{ord}})\cup\varphi\theta^{k_{1}-j}E^{-t,\mathrm{ord}}_{0,1/N}\right).$

But the operator $1-V_{p}U_{p}$ acts as the zero map, which proves the result.
∎

We deduce the following formula:

######  Proposition 7.7.4.

We have

$\left\langle(\iota^{[j]})^{*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle=N^{(k_{1}+k_{2}-2j)}(-1)^{k_{2}-j+1}\times\\\
\frac{\left(1-\tfrac{\beta_{1}}{p\alpha_{1}}\right)}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)\left(1-\tfrac{\beta_{1}\alpha_{2}}{p^{j+1}}\right)\left(1-\tfrac{\beta_{1}\beta_{2}}{p^{j+1}}\right)}\binom{k_{1}}{j}\,k_{2}!\,\cdot\left(\iota_{\operatorname{coh}}^{*}(\nu^{(\mathrm{ord})})\cup\theta^{k_{1}-j}E_{0,1/N}^{-t,[p]}\right).$

We can also express this in terms of the class $\breve{\nu}_{\Pi,\alpha}$ on
the higher-level variety $\mathcal{X}_{G}(\mathfrak{p}_{1})$ described above.
If $\pi_{\mathfrak{p}_{1}}$ denotes the degeneracy map
$\mathcal{X}_{G}(\mathfrak{p}_{1})\to\mathcal{X}_{G}$, then we have
$\operatorname{pr}_{\mathfrak{p}_{1},*}\left(\breve{\nu}_{\Pi,\alpha}\right)=p\left(1-\frac{\beta_{1}}{p\alpha_{1}}\right)\nu^{(\mathrm{ord})}$;
so the formula in Proposition 7.7.4 is equivalent to

$\left\langle(\iota^{[j]})^{*}(\nu_{\operatorname{fp}}),\,\mathrm{Eis}^{t}_{\operatorname{syn},N}\right\rangle=N^{(k_{1}+k_{2}-2j)}(-1)^{k_{2}-j+1}p^{-1}\times\\\
\frac{1}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)\left(1-\tfrac{\beta_{1}\alpha_{2}}{p^{j+1}}\right)\left(1-\tfrac{\beta_{1}\beta_{2}}{p^{j+1}}\right)}\binom{k_{1}}{j}\,k_{2}!\,\cdot\left(\iota_{\operatorname{coh}}^{*}\circ\operatorname{pr}_{\mathfrak{p}_{1},*}\left(\nu^{(\mathrm{ord})}_{\operatorname{Iw}(\mathfrak{p}_{1})}\right)\cup\theta^{k_{1}-j}E_{0,1/N}^{-t,[p]}\right).$

We now recall the definition of the (imprimitive) $p$-adic Asai $L$-function
in [GLZ23, Def. 7.3.1]. For a $p$-adic character
$\sigma:\mathbf{Z}_{p}^{\times}\to\mathbf{C}_{p}^{\times}$, we have

(7.4)
$L_{p,\mathrm{Asai}}^{\mathrm{imp}}(\Pi,\sigma)=\tfrac{1}{p}(\sqrt{D})^{-1-(k_{1}+k_{2})/2-\sigma}(-1)^{\sigma}\left\langle\breve{\nu}_{\Pi,\alpha},\iota_{\operatorname{coh},*}^{{(p)}}\left(\mathcal{E}_{k_{1}-k_{2}}(\sigma)\right)\right\rangle,$

where

  * •

$X_{H}(p)$ is the modular curve of level $\Gamma_{1}(N)\cap\Gamma_{0}(p)$;

  * •

$\iota^{(p)}:\,X_{H}(p)\rightarrow X_{G}(\mathfrak{p}_{1})$ is the natural
embedding induced by $\iota$;

  * •

$\mathcal{E}_{k_{1}-k_{2}}$ is a $p$-adic family of $p$-depleted Eisenstein
series, of constant weight $k_{1}-k_{2}$, defined in [GLZ23, §7.2], with the
Schwartz function $\Phi^{(p)}$ chosen as in §7.3.2 of _op.cit._ ; at
$\sigma=-\tfrac{t}{2}$, this specialises to
$\theta^{k_{1}-j}\left(E_{0,1/N}^{-t,[p]}\right)$.

######  Lemma 7.7.5.

The following diagram is cartesian:

${X_{H}(p)}$${X_{G}(\mathfrak{p}_{1})}$${X_{H,\mathrm{sph}}}$${X_{G,\mathrm{sph}}}$$\scriptstyle{\iota^{(p)}}$$\scriptstyle{\pi_{p}}$$\scriptstyle{\pi_{\mathfrak{p}_{1}}}$$\scriptstyle{\iota}$

so $\pi_{\mathfrak{p}_{1}}^{*}\circ\iota_{*}=\iota^{(p)}_{*}\circ\pi^{*}_{p}$.

Comparing the previous proposition with (7.4), we hence deduce the main
theorem of this paper:

######  Theorem 7.7.6.

Let $0\leqslant j\leqslant\min\\{k_{1},k_{2}\\}$, and let
$t=k_{1}+k_{2}-2j\geqslant 0$. Then we have

$\left\langle\nu_{\mathrm{dR}},\,\log\left(\operatorname{loc}_{p}\operatorname{AF}_{\text{{\'{e}t}}}^{[\Pi,j]}\right)\right\rangle=\frac{(\sqrt{D})^{j+1}(-1)^{(k_{1}-k_{2})/2+1}N^{t}\binom{k_{1}}{j}\,k_{2}!}{\left(1-\frac{p^{j}}{\alpha_{1}\alpha_{2}}\right)\left(1-\frac{p^{j}}{\alpha_{1}\beta_{2}}\right)\left(1-\tfrac{\beta_{1}\alpha_{2}}{p^{j+1}}\right)\left(1-\tfrac{\beta_{1}\beta_{2}}{p^{j+1}}\right)}\,\cdot
L_{p,\mathrm{Asai}}^{\mathrm{imp}}\left(\Pi,-\frac{t}{2}\right).$

######  Remark 7.7.7.

Note that in [GLZ23] we also defined a primitive $p$-adic $L$-function,
interpolating the critical values of the primitive Asai $L$-function (with the
optimal Euler factors at ramified primes). One can likewise modify the
construction of the Asai–Flach class by incorporating more general test data
away from $p$ (see [Gro20]), and prove a refinement of Theorem 7.7.6, relating
Asai–Flach classes with appropriate “optimal” test data to non-critical values
of the primitive $p$-adic Asai $L$-function. We leave the details to the
interested reader.

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