2.1 Groups and varieties

Let ${\mathrm {GL}_2}$ be the group scheme over ${\mathbb {Z}}$ of invertible 2-by-2 matrices. For $E/F$ a quadratic extension of totally real fields, we let

$$\begin{align*}H := \mathrm{Res }_{\mathcal{O}_F/{\mathbb{Z}}} \mathrm{GL}_2,\end{align*}$$

$$\begin{align*}G := \mathrm{Res }_{\mathcal{O}_E/{\mathbb{Z}}} \mathrm{GL}_2. \end{align*}$$

We denote by $Z_{H}, Z_G$ the centres of H and G respectively.

Note that we have the natural diagonal embedding $\iota : H \hookrightarrow G$ obtained from viewing G as the restriction of scalars of H with respect to $E/F$ . Now, let L be a field over which H and G are split. Then, $\iota : H_L \to G_L $ is induced by the diagonal embedding

$$\begin{align*}\prod_{\sigma \in \Sigma_F} \mathrm{GL}_{2,L} \hookrightarrow \prod_{\sigma \in \Phi} \Big( \mathrm{GL}_{2,L} \times \mathrm{GL}_{2,L} \Big), \; (g_\sigma)_\sigma \mapsto (g_\sigma,g_{\sigma'})_\sigma,\end{align*}$$

where $\Phi $ is the F-type defined by (2).

We denote by $Y_H$ (resp. $Y_G$ ) the Hilbert modular variety associated with H (resp. G). Recall that the associated Hermitian symmetric domains are the product of $[F:{\mathbb {Q}}]=d$ (resp. $[E:{\mathbb {Q}}]=2d$ ) copies of the Poincaré upper-half plane $\mathfrak {H}$ .

Definition 2.1. We define $K_{\infty }^+$ to be the connected component containing the identity of the maximal compact subgroup $K_\infty $ of $G({\mathbb {R}})$ . Let K be a compact-open subgroup of $G({\mathbb {A}}_{f})$ . We write

$$\begin{align*}G({\mathbb{A}}_f)= \bigcup_i G({\mathbb{Q}}) t_i K K_{\infty}^+. \end{align*}$$

We say that K is neat if

$$\begin{align*}\overline{\Gamma}_i(K):=\mathrm{Im}( t_i K K_{\infty}^+t_i^{-1} \cap G({\mathbb{Q}}) \rightarrow \mathrm{PGL}_2 (E)) \end{align*}$$

acts without fixed points on $\mathfrak {H}^{2d}$ .

An analogous definition can be made for a compact-open subgroup of $H({\mathbb {A}}_{f})$ . When K and $K'$ are neat compact-open subgroups respectively of $G({\mathbb {A}}_{f})$ and $H({\mathbb {A}}_{f})$ , $Y_G(K)$ and $Y_H(K')$ are smooth quasi-projective schemes over $\mathrm {Spec}\, {\mathbb {Q}}$ of dimension $2d$ and d respectively.

2.2 Representations

From now on, L denotes a field for which H and G are split. Let V be the standard representation $\textbf {GL}_{2,L}$ acting on the left on $V \simeq L^2$ with given basis $(e_1, e_2)$ , and denote $\mathrm {det} := \bigwedge ^2 V$ by minor abuse of notation. Every irreducible algebraic representation of $\textbf {GL}_{2,L}$ is of the form $V^{k,m}:= \mathrm {Sym}^k(V) \otimes \mathrm {det}^m$ , with $(k,m) \in {\mathbb {Z}}_{\geq 0} \times {\mathbb {Z}}$ . Here we denote by $\mathrm {Sym}^k(V)$ the quotient module of co-invariants of $V^{\otimes k}$ for the action of the symmetric group $\mathfrak {S}_k$ on k elements.

Definition 2.2. For $ (\underline {k},m) \in ({\mathbb {Z}}_{\geq 0})^{\Sigma _E} \times {\mathbb {Z}}$ s.t. $k_\sigma \equiv m$ (mod $2$ ), where we write ${\underline {k} = (k_\sigma )_{\sigma \in \Sigma _E}}$ , we define

$$\begin{align*}V^{\underline{k},m} := \bigotimes_{\sigma \in \Sigma_E} \mathrm{Sym}^{k_\sigma} (V_\sigma) \otimes \mathrm{det}_\sigma^{\frac{m-k_{\sigma}}{2}}.\end{align*}$$

We can realize $V^{\underline {k},m}$ explicitly as follows. The L-vector space $L[X_\sigma ,Y_\sigma ]^{=k_\sigma }$ of homogeneous polynomials of degree $k_\sigma $ is a model for the dual representation of $\mathrm {Sym}^{k_\sigma } (V_\sigma )$ . The element $g \in \textbf {GL}_{2,L}$ acts on $L[X_\sigma ,Y_\sigma ]^{=k_\sigma }$ via the left action given by

$$\begin{align*}g \cdot P(X,Y)=\mathrm{det}(g)^{-k_\sigma} P\left( \left(g^* \left(\begin{array}{c} X \\ Y \end{array} \right)\right)^t \right)=\mathrm{det}(g)^{-k_\sigma} P(dX-bY,-cX+aY), \end{align*}$$

where for $g=\left (\begin {array}{cc} \!\! a & b \\ c & d \!\!\end {array} \right )$ , $g^*=\mathrm {det}(g)g^{-1}=\left (\begin {array}{cc} \!\!d & -b \\ -c & a \!\!\end {array} \right )$ . Observe that, as $\mathrm {SL}_{2,L}$ -representations, $\mathrm {Sym}^{k_\sigma } (V_\sigma )$ and $L[X_\sigma ,Y_\sigma ]^{=k_\sigma }$ are isomorphic via sending

$$\begin{align*}\alpha e_1+\beta e_2 \mapsto -\beta X+\alpha Y. \end{align*}$$

This implies that $L[X_\sigma ,Y_\sigma ]^{=k_\sigma } \simeq \mathrm {Sym}^{k_\sigma } (V_\sigma ) \otimes \mathrm {det}_\sigma ^{-k_\sigma }$ as $\textbf {GL}_{2,L}$ -representations. Via this identification, $e_{1,\sigma }$ and hence $Y_\sigma $ are the highest weight vectors for $V_\sigma $ and its dual. This implies that the dual of $\bigotimes _{\sigma \in \Sigma _E} \mathrm {Sym}^{k_\sigma } (V_\sigma )$ as a $G_L$ -representation is isomorphic to $\bigotimes _{\sigma \in \Sigma _E} L[X_\sigma ,Y_\sigma ]^{=k_\sigma }$ .

We introduce some useful lattices in $\mathrm {Sym}^k(V)$ . Let $\mathcal {O} := \mathcal {O}_L$ be the ring of integers of L, and identify $\mathcal {O}^2$ with $\mathcal {O}e_1 \oplus \mathcal {O}e_2$ inside V. We have

$$\begin{align*}\mathrm{TSym}^k(\mathcal{O}^2) \subset \mathrm{Sym}^k(\mathcal{O}^2)\subset \mathrm{Sym}^k(V). \end{align*}$$

Recall that $\mathrm {Sym}^k(\mathcal {O}^2)$ is the module of $\mathfrak {S}_k$ -coinvariants, and $\mathrm {TSym}^k(\mathcal {O}^2)$ is the submodule of $\mathfrak {S}_k$ -invariants in ${(\mathcal {O}^2)}^{\otimes k}$ , that we then identify with its image in $\mathrm { Sym}^k(\mathcal {O}^2)$ . Recall that a lattice in $\mathrm {Sym}^k(V)$ is admissible if it is invariant under the Kostant ${\mathbb {Z}}$ -form in the universal enveloping algebra of the Lie algebra of $\textbf {GL}_{2,L}$ . Every admissible lattice is a direct sum of its weight components, thus an admissible lattice in $\mathrm {Sym}^k(V)$ is proportional to one whose intersection with the highest weight subspace is $\mathcal {O} \cdot e_1^{\otimes k}$ . Hence, without loss of generality, we can suppose this intersection to be precisely $\mathcal {O} \cdot e_1^{\otimes k}$ . Finally, recall that any admissible lattice is invariant under the action of $\textbf {GL}_{2,\mathcal {O}}$ (for instance, see [Reference Lin30] for more details). Note that $\mathrm { Sym}^k(\mathcal {O}^2)$ is the maximal admissible lattice in $\mathrm {Sym}^k(V)$ , while $\mathrm {TSym}^k(\mathcal {O}^2)$ is the minimal admissible one. Indeed, by duality, we have two lattices in $L[X,Y]^{=k}$ : $\mathcal {O}[X,Y]^{=k}$ , the set of homogeneous polynomials of degree k with integer coefficients, which is a maximal lattice and it is dual to $\mathrm {TSym}^k(\mathcal {O}^2)$ , and $\mathcal {O}\left [ \binom {k}{i} X^{k-i}Y^i \right ]$ , which is a minimal lattice and is dual to $\mathrm {Sym}^k(\mathcal {O}^2)$ . It coincides with the duality of [Reference Getz and Goresky17, §7.5] (under the isomorphism above).

Given a finite order Hecke character $\chi $ of E of conductor dividing an ideal $\mathfrak {c}$ of $\mathcal {O}_E$ , we can define a character of

$$\begin{align*}K_0(\mathfrak{c}) : = \{ g \in \mathrm{GL}_2(\widehat{\mathcal{O}}_E)\,:\, g = {\matrix{a}{b}{c}{d}}, \text{ with } c \equiv 0 \text{ mod } \mathfrak{c}\}\end{align*}$$

via ${\matrix {a}{b}{c}{d}} \mapsto \chi (d)$ and consequently a one-dimensional representation $L({\chi })$ of $K_0(\mathfrak {c})$ . Following [Reference Getz and Goresky17, §6.8], we denote $V^{\underline {k},{m}}(\chi ) :=V^{\underline {k},{m}} \otimes _L L(\chi )$ . Explicitly, for $v \in V^{\underline {k},{m}}(\chi )$ and $g \in K_0(\mathfrak {c})$ , we have $g.v= \chi (g) v$ .

We conclude by defining two more representations of $G(L)$ , that are not necessarily algebraic.

Definition 2.6. For $ (\underline {k},m) \in ({\mathbb {Z}}_{\geq 0})^{\Sigma _E} \times {\mathbb {Z}}$ we define the complex representation of $G(L)$

$$\begin{align*}V^{\underline{k},m}_{\mathbb{C}} := \bigotimes_{\sigma \in \Sigma_E} \mathrm{Sym}^{k_\sigma} (V_\sigma \otimes {\mathbb{C}}) \otimes \mathrm{det}_\sigma^{\frac{m-k_{\sigma}}{2}}.\end{align*}$$

If, instead, we consider a p-adic field $L_{\mathfrak {m}}$ that contains all the square roots of elements of $E \otimes _{{\mathbb {Z}}} {\mathbb {Q}}_p$ (note that this is a finite extension of ${\mathbb {Q}}_p$ ) and denote by $\mathcal {O}_{\mathfrak {m}}$ its ring of integers, we define, for $ (\underline {k},m) \in ({\mathbb {Z}}_{\geq 0})^{\Sigma _E} \times {\mathbb {Z}}$ , the p-adic representation of $G(\mathcal {O})$

$$\begin{align*}V^{\underline{k},m}_{\mathcal{O}_{\mathfrak{m}}} := \bigotimes_{\sigma \in \Sigma_E} \mathrm{Sym}^{k_\sigma} (\mathcal{O}^2 \otimes \mathcal{O}_{\mathfrak{m}}) \otimes \mathrm{det}_\sigma^{\frac{m-k_{\sigma}}{2}}.\end{align*}$$

2.3 Branching laws

Recall that $\Phi $ denotes the F-type for E given by (2) and that the embedding $\iota :H_L \to G_L $ is induced by the diagonal embedding:

$$\begin{align*}\prod_{\sigma \in \Sigma_F} \mathrm{GL}_{2,L} \hookrightarrow \prod_{\sigma \in \Phi} \Big( \mathrm{GL}_{2,L} \times \mathrm{GL}_{2,L} \Big), \; (g_\sigma)_\sigma \mapsto (g_\sigma,g_{\sigma'})_\sigma.\end{align*}$$

Given the $G_L$ -representation $V^{\underline {k},m}$ , we can view it as an $H_L$ -representation via the embedding $\iota $ . It decomposes into a sum of irreducible sub- $H_L$ -representations. In the following, we characterize the set of representations of $G_L$ which contain a trivial $H_L$ -representation.

Lemma 2.7. The representation $V^{\underline {k}, m}_{|_{H_L}}$ contains a unique copy of the trivial $H_L$ -representation (up to a twist) if and only if $k_\sigma = k_{\sigma '}$ for all $\sigma \in \Phi $ . In this case, we have an embedding

$$\begin{align*}\iota^{\underline{k}} : \bigotimes_{\sigma \in \Sigma_F} \mathrm{det}_\sigma^{m} \hookrightarrow V^{\underline{k},m},\end{align*}$$

explicitly realized by the vector

(3) $$ \begin{align} \Delta^{\underline{k}}:=\bigotimes_{\sigma \in \Phi}(e_{1,\sigma}e_{2,\sigma'}-e_{2,\sigma}e_{1,\sigma'})^{\otimes k_\sigma} \in V^{\underline{k},m}. \end{align} $$

Proof. Recall that we have the decomposition as $\mathrm {SL}_2(L)$ -representations

$$\begin{align*}(\operatorname{Sym}^{k}(V) \otimes \operatorname{Sym}^{j}(V))_{|_{\mathrm{SL}_2(L)}} = \oplus_{r = 0}^{\mathrm{min}(k,j)} \operatorname{Sym}^{k+j-2r}(V),\end{align*}$$

see [Reference Fulton and Harris16, Exercise 11.11]. This in turn implies that

$$\begin{align*}\bigotimes_{\sigma \in \Phi} (\operatorname{Sym}^{k_\sigma}(V_\sigma) \otimes \operatorname{Sym}^{k_{\sigma'}}(V_{\sigma'}))_{|_{\mathrm{SL}_2(L)}} = \bigotimes_{\sigma \in \Sigma_F} \bigoplus_{r_\sigma = 0}^{\mathrm{ min}(k_\sigma,k_{\sigma'})} \operatorname{Sym}^{k_\sigma + k_{\sigma'} - 2 r_\sigma }(V_\sigma). \end{align*}$$

Hence we have an embedding of the trivial $\mathrm {SL}_2(L)^d$ -representation

$$\begin{align*}\mathbf{1} \hookrightarrow \bigotimes_{\sigma \in \Phi} (\operatorname{Sym}^{k_\sigma}(V_\sigma) \otimes \operatorname{Sym}^{k_{\sigma'}}(V_{\sigma'})) \end{align*}$$

iff $k_\sigma = k_{\sigma '}$ for all pairs of conjugate embeddings $\sigma , \sigma '$ , as $k_{\sigma } + k_{\sigma '} - {2 \mathrm {min}} (k_\sigma ,k_{\sigma '}) = 0$ iff $k_{\sigma } = k_{\sigma '}$ for all such pairs. By matching the central characters, we get an embedding of $H_L$ -representations

$$\begin{align*}\iota^{\underline{k}} : \bigotimes_{\sigma \in \Sigma_F} \mathrm{det}_\sigma^{m} \hookrightarrow \bigotimes_{\sigma \in \Phi} (\operatorname{Sym}^{k_\sigma}(V_\sigma) \otimes \operatorname{Sym}^{k_{\sigma}}(V_{\sigma}) \otimes \mathrm{det}_\sigma^{m- k_\sigma}) = V^{\underline{k},m}_{|_{H_L}}. \end{align*}$$

Finally, note that the invariant vector (cf. $\Delta $ in [Reference Getz and Goresky17, Proposition 9.1]) which generates the image of $\iota ^{\underline {k}}$ is explicitly given by:

$$ \begin{align*} \bigotimes_{\sigma \in \Phi}(e_{1,\sigma}e_{2,\sigma'}-e_{2,\sigma}e_{1,\sigma'})^{\otimes k_\sigma}. \end{align*} $$

Indeed, if $g = {\matrix {a}{b}{c}{d}} \in \mathrm {GL}(V_\sigma )$ , we have that

$$ \begin{align*} (g,g)(e_{1,\sigma}e_{2,\sigma'}-e_{2,\sigma}e_{1,\sigma'}) = & (ae_{1,\sigma}+ce_{2,\sigma})(be_{1,\sigma'}+d e_{2,\sigma'})\\&- (be_{1,\sigma}+d e_{2,\sigma})(ae_{1,\sigma'}+ce_{2,\sigma'})\\ = \ & ab e_{1,\sigma}e_{1,\sigma'}+cde_{2,\sigma}e_{2,\sigma'}+ade_{1,\sigma}e_{2,\sigma'} - (-bc)e_{2,\sigma}e_{1,\sigma'} +\\ & - ab e_{1,\sigma}e_{1,\sigma'}-cde_{2,\sigma}e_{2,\sigma'}-bce_{1,\sigma}e_{2,\sigma'}-ade_{2,\sigma}e_{1,\sigma'}\\ = \ &\mathrm{det}(g)(e_{1,\sigma}e_{2,\sigma'}-e_{2,\sigma}e_{1,\sigma'}).\\[-33pt] \end{align*} $$

Remark 2.8. Lemma 2.7 can be rephrased by saying that

$$\begin{align*}\mathrm{dim}( V^{\underline{k},m} \otimes \bigotimes_{\sigma \in \Sigma_F} \mathrm{det}_\sigma^{-m} )^H \leq 1\end{align*}$$

and it is equal to 1 if and only if $k_\sigma = k_{\sigma '}$ for every $\sigma \in \Phi $ .

Proof. Expanding $\Delta ^{\underline {k}}$ , we get

$$\begin{align*}\Delta^{\underline{k}}= \bigotimes_{\sigma \in \Phi} \sum_{i=0}^{k_{\sigma}}(-1)^i \binom{k_{\sigma}}{i} e_{1,\sigma}^{\otimes^ {k_{\sigma}-i}}e_{2,\sigma}^{\otimes^i}e_{1,\sigma'}^{\otimes^i}e_{2,\sigma'}^{k_{\sigma}-i} \end{align*}$$

which is a vector in $\left ( \bigotimes _{\sigma \in \Phi } \mathrm {Sym}^{k_\sigma }(\mathcal {O}^2) \otimes \mathrm {det}_\sigma ^{\frac {m-k_{\sigma }}{2}} \right ) \otimes \left (\bigotimes _{\sigma ' \in \Phi '} \mathrm {Sym}^{k_{\sigma '}}(\mathcal {O}^2) \otimes \mathrm {det}_{\sigma '}^{\frac {m-k_{\sigma '}}{2}}\right )$ .

Remark 2.10. Note that in general $\Delta ^{\underline {k}}$ is not in the minimal admissible lattice

$$\begin{align*}\bigotimes_\sigma \left( \mathrm{TSym}^{k_\sigma}(\mathcal{O}^2) \otimes \mathrm{det}_{\sigma}^{\tfrac{m-k_{\sigma}}{2}}\right) \otimes \left(\bigotimes_{\sigma'} \mathrm{TSym}^{k_{\sigma'}}(\mathcal{O}^2) \otimes \mathrm{det}_{\sigma'}^{\tfrac{m-k_{\sigma'}}{2}}\right). \end{align*}$$

2.4 Cohomology

Associated with the irreducible algebraic representation $V^{\underline {k},{m}}$ and its maximal admissible lattice $V^{\underline {k},{m}}_{\mathcal {O}}$ , we have two local systems $\mathscr {V}^{\underline {k},{m}}$ and $\mathscr {V}^{\underline {k},{m}}_{\mathcal {O}}$ on $Y_G(K)$ , which can be defined as in [Reference Getz and Goresky17, §6.8], as long as $K \subset G({\mathbb {A}}_f)$ is a neat open compact subgroup. Similarly, by [Reference Getz and Goresky17, Proposition 6.3], we can define local systemsFootnote ¹ $\mathscr {V}^{\underline {k},{m}}(\chi )$ on $Y_G(K_0(\mathfrak {c}))$ attached to a finite order Hecke character $\chi $ of E of conductor dividing $\mathfrak {c}$ and the representation $V^{\underline {k},{m}}(\chi )$ of $K_0(\mathfrak {c})$ . Similarly, we can define local systems over $Y_H(K')$ .

If $\underline {k} \in {\mathbb {Z}}^{\Sigma _E}$ is such that $k_\sigma = k_{\sigma '}$ for all $\sigma \in \Phi $ , then the branching map $\iota ^{\underline {k}}_{\mathcal {O}} : \mathcal {O} \hookrightarrow V^{\underline {k},0}_{\mathcal {O}}$ induces a map of local systems

$$\begin{align*}\iota^{\underline{k}}_{\mathcal{O}} : \mathcal{O} \to \iota^*\mathscr{V}^{\underline{k},0}_{\mathcal{O}}.\end{align*}$$

Twisting the vector $\Delta ^{\underline {k}}$ by the finite order Hecke character $\chi $ as in [Reference Getz and Goresky17, (9.2.5)], we can also define a map of local systems

$$\begin{align*}\tilde{\iota}^{\underline{k}}_{\mathcal{O}_\chi} : \mathcal{O}_\chi \to \iota^*\mathscr{V}^{\underline{k},0}_{\mathcal{O}_\chi}(\chi),\end{align*}$$

where $\mathcal {O}_\chi $ is the extension of $\mathcal {O}$ containing the values of $\chi $ .

3.1 Hilbert modular forms and their differential forms

In this section we quickly recall some notation for Hilbert modular forms, following mainly [Reference Getz and Goresky17, §5].

3.1.1 Hilbert cusp forms and their differential forms

Let F be a totally real field of degree d over $\mathbb {Q}$ and let $E=F(\sqrt {\Delta })$ , for $\Delta $ a totally positive element of F. Consider a weight $\kappa = (\underline {k},m) \in ({\mathbb {Z}}_{\geq 0})^{\Sigma _E} \times {\mathbb {Z}}$ . For an ideal $\mathfrak {c}$ of $\mathcal {O}_E$ , we consider the compact-open subgroup

$$\begin{align*}K_0(\mathfrak{c}) = \{ g \in \mathrm{GL}_2(\widehat{\mathcal{O}}_E)\,:\, g = {\matrix{a}{b}{c}{d}}, \text{ with } c \equiv 0 \text{ mod } \mathfrak{c}\}.\end{align*}$$

Let $\chi $ be a Hecke character for E of conductor dividing $\mathfrak {c}$ , such that its infinity type is given by $\chi _{\infty }(x) = \prod _{v \in \Sigma _E} x_v^{-m}$ . The space of cusp forms of weight $\kappa $ and Nebentypus $\chi $ is defined in [Reference Getz and Goresky17, (5.4.3)] and as in loc. cit. we denote it byFootnote ² $S_{\kappa }(K_0(\mathfrak {c}),\chi )$ . We shall denote by f an element of $S_{\kappa }(K_0(\mathfrak {c}),\chi )$ . It corresponds to a collection of classical Hilbert modular forms defined as functions on the product of upper-half planes. Note that, following loc. cit. our normalization on the weight is shifted by $2$ compared to the most common normalization for Hilbert modular forms as functions on the upper half-plane. For example, we associate elliptic modular forms of weight $0$ to elliptic curves over $\mathbb {Q}$ .

There is also a cohomological normalization, introduced in §5.5 of loc. cit., given by

$$\begin{align*}f\mapsto f^{-\iota}: x \mapsto f(x^{-\iota}) \end{align*}$$

for $x^{\iota }=\mathrm {det}(x)x^{-1}$ and $x^{-\iota }=\mathrm {det}(x)^{-1}x$ . This changes m to $-m$ (and $\chi $ to $\chi ^{-1}$ ). This different normalization is more natural when we associate differential forms and cohomology classes to elements of $S_{\kappa }(K_0(\mathfrak {c}),\chi )$ . Precisely, denote by $\chi _0$ the character of $K_0(\mathfrak {c})$ , which at each finite place v is defined by

$$\begin{align*}\chi_v\left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)= \begin{cases} \chi_v(d) & \text{ if } \mathfrak{p}_v \mid \mathfrak{c} \\ 1 & \text{ otherwise,} \end{cases} \end{align*}$$

with $\mathfrak {p}_v$ denoting the prime ideal associated with v. Then, for any subset $J \subset \Sigma _E$ , we have a map (Definition 6.8 of loc. cit.)

$$\begin{align*}S_{\kappa}(K_0(\mathfrak{c}),\chi) \to \Omega^{2d}(Y_G(K_0(\mathfrak{c})),\mathscr{V}^{\underline{k},{-m}}( \chi^{-1}_0)_{\mathbb{C}}),\, f \mapsto \omega_J(f^{-\iota}), \end{align*}$$

which defines a class $[\omega _J(f^{-\iota })]$ in $H^{2d}( Y_G(K_0(\mathfrak {c})),\mathscr {V}^{\underline {k},{-m}}( \chi ^{-1}_0)_{\mathbb {C}})$ . The map is Hecke equivariant in the sense of Proposition 7.13 of loc. cit. (see Lemma 6.1 below). The differential form is anti-holomorphic at places in J and holomorphic at places outside J and it is square integrable (Proposition 6.8 of loc. cit.), so that $[\omega _J(f^{-\iota })]$ defines a class in $L^2$ -cohomology of $Y_G(K_0(\mathfrak {c}))$ . Note that as f is a cusp form, the differential form $\omega _J(f^{-\iota })$ is rapidly decreasing, thus

(4) $$ \begin{align} [\omega_J(f^{-\iota})] \in H^{2d}_{\mathrm{rd}}(Y_G(K_0(\mathfrak{c})), \mathscr{V}^{\underline{k},{-m}}( \chi^{-1}_0)_{\mathbb{C}}). \end{align} $$

Recall that such a class can be represented by a compactly supported differential form:

Proposition 3.1 [Reference Borel6, Corollary 5.5].

Let $f \in S_{\kappa }(K_0(\mathfrak {c}),\chi )$ . There exists a $\mathscr {V}^{\underline {k},{-m}}( \chi ^{-1}_0)_{\mathbb {C}}$ -valued rapidly decreasing form $\Psi $ on $Y_G(K_0(\mathfrak {c}))$ such that $\omega _c := \omega _J(f^{-\iota }) + d \Psi $ is compactly supported. In other words,

$$\begin{align*}[\omega_c]=[\omega_J(f^{-\iota})] \in H^{2d}_{\mathrm{rd}}(Y_G(K_0(\mathfrak{c})), \mathscr{V}^{\underline{k},{-m}}( \chi^{-1}_0)_{\mathbb{C}}). \end{align*}$$

Similar normalizations and definitions can be made for Hilbert modular forms for F.

3.1.2 The Fourier expansion

We recall briefly the Fourier expansion of a form f over E, following [Reference Getz and Goresky17, §5.9]. We define an additive character $e_E(x)$ , for $x \in \mathbb {A}_E/E$ . For a finite place v of E, lying above $q \in \mathbb {Z}$ , we define

$$\begin{align*}\begin{array}{cccc} e_v: & E_v & \longrightarrow & \mathbb{C}^{\times} \\ & x &\mapsto & \exp(-2 \pi i \mathrm{tr}_{E_v}^{\mathbb{Q}_q}(x)) \end{array} \end{align*}$$

where we identify $\mathbb {Q}/\mathbb {Z}$ with $\mathbb {Q}_q/\mathbb {Z}_q$ . If v is an archimedean place, we set $e_v(x):=\exp (2 \pi i x) $ . We can finally define

$$\begin{align*}e_E(x):=\prod_{v}e_v(x_v), x = (x_v). \end{align*}$$

If f is of weight $\kappa = (\underline {k},m) \in ({\mathbb {Z}}_{\geq 0})^{\Sigma _E} \times {\mathbb {Z}}$ , for $\sigma \in \Sigma _E$ define

$$\begin{align*}\begin{array}{cccc} W_{\sigma}: & \mathbb{R}^{\times} & \rightarrow & \mathbb{C} \\ & y &\mapsto & |y|^{\frac{k_\sigma-m}{2}}e^{-2\pi|y|}, \end{array} \end{align*}$$

and finally, for $x \in \mathbb {A}_E$ and $y \in \mathbb {A}_E^{\times }$

$$\begin{align*}q_{\kappa}(x,y):=e_E(x)\prod_{\sigma \in \Sigma_E}W_{\sigma}(y_{\sigma}). \end{align*}$$

(Recall that our weight normalization is different from [Reference Getz and Goresky17].) We let $\mathrm {diff}_{E/\mathbb {Q}}$ be the different of E over $\mathbb {Q}$ . A modular form $f \in S_{\kappa }(K_0(\mathfrak {c}),\chi )$ has a Fourier expansion of the form

(5) $$ \begin{align} f\left( \begin{smallmatrix} y & x \\ 0 & 1 \end{smallmatrix} \right)= |y|_{\mathbb{A}_E}\sum_{\xi \in E^{\times}, \xi \gg 0} a (\xi y \mathrm{diff}_{E/\mathbb{Q}},f)q_{\kappa}(\xi x, \xi y). \end{align} $$

When $\mathfrak {n}=( \xi y_f \mathcal {O}_E) \mathrm {diff}_{E/\mathbb {Q}}$ is an integral ideal, we write

$$\begin{align*}a(\mathfrak{n},f)=a (\xi y \mathrm{diff}_{E/\mathbb{Q}},f). \end{align*}$$

Similar results hold for forms over F.

3.1.3 The Hecke algebra

We now define the Hecke algebra for E, following [Reference Hida21, §1]. Denote

$$\begin{align*}K_{11}(\mathfrak{c}) := \{ g \in K_0(\mathfrak{c})\,:\, g \equiv {\matrix{1}{\star}{0}{1}} \text{ mod } \mathfrak{c}\}.\end{align*}$$

Definition 3.2. Fix a level $K \supset K_{11}(\mathfrak {c})$ . For a prime ideal $\mathfrak {q}$ coprime with $\mathfrak {c}$ and a positive integer l we define the Hecke operators $T_{\mathfrak {q}^l}, S_{\mathfrak {q}^l}$ as the double coset operators

$$\begin{align*}T_{\mathfrak{q}^l}:=\left[K \left(\begin{array}{cc} \!\!\! \varpi_{\mathfrak{q}^l} & 0 \\ 0 & 1 \!\!\! \end{array} \right)K \right], \quad S_{\mathfrak{q}^l}:=\left[K \left(\begin{array}{cc} \!\!\! \varpi_{\mathfrak{q}^l} & 0 \\ 0 & \varpi_{\mathfrak{q}^l} \!\!\! \end{array} \right)K\right], \end{align*}$$

for $\varpi _{\mathfrak {q}}$ a uniformizer at $\mathfrak {q}$ . Note that as $\mathfrak {c}$ is coprime to $\mathfrak {q}$ , our definition does not depend on the choice of $\varpi _{\mathfrak {q}}$ .

Let $\mathcal {R}$ be an extension of E containing the Galois closure of E and such that all the ideals of E are principal in $\mathcal {R}$ ; let $\mathcal {V}$ be its ring of integers. For any $\mathcal {V}$ -algebra R we define the spherical Hecke algebra $\mathcal {H}^{\mathrm {sph}}_{K,R}$ of level K and coefficients in R as the R-algebra generated by the operators $T_{\mathfrak {q}^l}$ and $S_{\mathfrak {q}^l}$ as l and $\mathfrak {q} \nmid \mathfrak {c}$ vary. In $\mathcal {H}^{\mathrm {sph}}_{K,R}$ , we have the relation

$$\begin{align*}T_{\mathfrak{q}}^2-T_{\mathfrak{q}^2}=\mathrm{N}_{E/{\mathbb{Q}}}(\mathfrak{q})S_{\mathfrak{q}}. \end{align*}$$

We define similarly $U_{\mathfrak {q}^l}$ for ${\mathfrak {q}}$ dividing $\mathfrak {c}$ ; but note that it could depend on the choice of uniformizer $\varpi _{\mathfrak {q}}$ , see the explicit formula in [Reference Hida23, (2.2b)]. If p is coprime with $\mathfrak {q}$ , then there is no need of the normalization factor $\left \{ \phantom {e}\right \}^{-v}$ of [Reference Hida21, p. 142]. We shall define later the proper normalization when $\mathfrak {q}$ divides p. We define by $\mathcal {H}_{K,R}$ the abstract Hecke algebra over R generated by $U_{\mathfrak {q}^l}$ , for all $\mathfrak {q}\mid \mathfrak {c}$ and l, and $T_{\mathfrak {q}^l}$ , $S_{\mathfrak {q}^l}$ for all $\mathfrak {q} \nmid \mathfrak {c}$ and any l.

This Hecke algebra acts naturally on the space of Hilbert modular forms [Reference Hida23, §2] or on cohomology of Hilbert modular varieties [Reference Getz and Goresky17, §7.6] and these actions are compatible (in the sense of [Reference Getz and Goresky17, Proposition 7.13]).

Definition 3.3. Let $\chi '$ be another character of conductor $\mathfrak {c}$ . We see it as a character of $K_0(\mathfrak {c})$ via

$$\begin{align*}\chi^{\prime}_v\left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)= \begin{cases} \chi^{\prime}_v(a^{-1}d) & \text{ if } \mathfrak{p}_v \mid \mathfrak{c} \\ 1 & \text{ otherwise} \end{cases}. \end{align*}$$

We say that $f \in S_{\kappa }(K_0(\mathfrak {c}),\chi ,\chi ')$ if f is a cusp form of level $K_{11}(\mathfrak {c})$ such that

$$\begin{align*}f \vert \left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) = \chi(d)\chi'(a^{-1}d) f \end{align*}$$

for all $\left ( \begin {smallmatrix} a & b \\ c & d \end {smallmatrix} \right ) \in K_0(\mathfrak {c})$ , where we use the convention of [Reference Hida23] for the slash operator $\vert $ . We shall call $\chi $ the Nebentypus of f and $\chi '$ its adjoint character (because it factors via the adjoint group).

Definition 3.4. If $f \in S_{\kappa }(K_0(\mathfrak {c}),\chi ,\chi ')$ is an eigenform for all but finitely many Hecke operators, we denote by $\pi (f)$ the corresponding irreducible cuspidal automorphic representation of $\mathrm {GL}_2(\mathbb {A}_{E})$ of central character $\chi $ , as in [Reference Getz and Goresky17, (5.4.3)]. Note that $\pi (f)$ is not unitary in general, while its twist $\pi (f) \otimes | - |_{\mathbb {A}_{E}}^{m/2}$ is.

Similar definitions apply to Hilbert modular forms for F.

3.1.4 Atkin–Lehner operators

We recall here the Atkin–Lehner involution, following [Reference Getz and Goresky17, §7.4].

Choose a finite idèle $\tilde {c}$ such that it generates the ideal $\mathfrak {c}$ . The Atkin–Lehner operator $W^ *_{\mathfrak {c}}$ on $S_{\kappa }(K_1(\mathfrak {c}))$ is induced by right multiplication by the matrix

$$\begin{align*}W_{\mathfrak{c}}=\matrix{0}{-1}{\tilde{c}}{0}. \end{align*}$$

More precisely, it maps $ f \in S_{(\underline {k},m)}(K_0(\mathfrak {c}),\chi )$ to $ W^*_{\mathfrak {c}} (f) \in S_{(\underline {k},m)}(K_0(\mathfrak {c}),\psi )$ , where

$$\begin{align*}W^*_{\mathfrak{c}} (f) (\alpha)= \frac{|\chi(\mathrm{det}(\alpha))|}{\chi(\mathrm{det}(\alpha))}f(\alpha W_{\mathfrak{c}}), \end{align*}$$

and $\psi =\chi ^{-1}|\phantom {e}|^{-2m}$ (so at infinity it coincides with $\chi $ , but the finite part is the inverse). Similarly, right multiplication by $W_{\mathfrak {c}}$ on $Y_G(K_0(\mathfrak {c}))$ induces an isomorphism of local systems $\mathscr {V}^{\underline {k},{-m}}( \chi ^{-1}_0) \simeq \mathscr {V}^{\underline {k},{-m}}( \chi _0)$ which can be twisted to get a map on differential forms (cf. [Reference Getz and Goresky17, Proposition 7.6])

$$\begin{align*}W^*_{\mathfrak{c}} : \Omega^{2d}(Y_G(K_0(\mathfrak{c})),\mathscr{V}^{\underline{k},{-m}}( \chi^{-1}_0)_{\mathbb{C}}) \to \Omega^{2d}(Y_G(K_0(\mathfrak{c})),\mathscr{V}^{\underline{k},{-m}}( \chi_0)_{\mathbb{C}}), \end{align*}$$

so that we have a commutative diagram

(6)

Note that $W^*_{\mathfrak {c}}$ induces an isomorphism of the Hecke algebra of level $K_0(\mathfrak {c})$ by mapping every Hecke operator T to its adjoint $T^*$ (see the proof of [Reference Hida23, Lemma 9.3]), namely

$$\begin{align*}T^*= W_{\mathfrak{c},*} \circ T \circ {W^*_{\mathfrak{c}}}. \end{align*}$$

Remark 3.5. When $T=T_{\mathfrak {q}}$ for $\mathfrak {q}$ coprime with $\mathfrak {c}$ , then $T_{\mathfrak {q}}=T_{\mathfrak {q}}^*$ .

3.2 L-functions and base change

3.2.1 The adjoint L-function

Let $\mathfrak {c}_F$ be an ideal of F and f be a Hilbert cuspidal eigenform of level $K_{11}(\mathfrak {c}_F)$ , weight $(\underline {k},m)$ and Nebentypus $\chi $ . We assume that the form is minimal, i.e. that it has minimal conductor among all quadratic twists. For any Hecke character $\epsilon : F^\times \backslash \mathbb {A}^\times _F \to {\mathbb {C}}^\times $ of conductor $\mathfrak {f}(\epsilon )$ , we define the (imprimitive) adjoint L-function of f twisted by $\epsilon $ as

$$\begin{align*}L(\mathrm{Ad}(f) \otimes \epsilon, s):=\prod_{ \mathfrak{q} \not\mid \mathfrak{f}(\epsilon)} {L_{\mathfrak{q}}(\mathrm{Ad}(f) \otimes \epsilon, s)}. \end{align*}$$

If $\mathfrak {q}$ does not divide $\mathfrak {c}_F$ then ${L_{\mathfrak {q}}(\mathrm {Ad}(f) \otimes \epsilon , s)}^{-1}$ is defined as

$$\begin{align*}\left(1-\epsilon(\mathfrak{q})\frac{\alpha_{1,\mathfrak{q}}}{\alpha_{2,\mathfrak{q}}}\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right) \left(1-\epsilon(\mathfrak{q})\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right) \left(1-\epsilon(\mathfrak{q})\frac{\alpha_{2,\mathfrak{q}}}{\alpha_{1,\mathfrak{q}}}\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right) \end{align*}$$

for ${\alpha _{1,\mathfrak {q}}},{\alpha _{2,\mathfrak {q}}}$ the two Satake parameters of $\pi (f)_{\mathfrak {q}}$ defined viaFootnote ³

$$\begin{align*}1-a(\mathfrak{q},f)\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s-1/2}+\chi(\mathfrak{q})\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-2s}=(1-{\alpha_{1,\mathfrak{q}}}\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}) (1-{\alpha_{2,\mathfrak{q}}}\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}).\end{align*}$$

Note that the adjoint L-function is independent of the variable m.

If $\mathfrak {q}$ divides $\mathfrak {c}_F$ but not $\mathfrak {f}(\epsilon )$ , we consider the following cases. If the local automorphic representation $\pi (f)_{\mathfrak {q}}$ associated with f is a ramified principal series, then ${L_{\mathfrak {q}}(\mathrm {Ad}(f) \otimes \epsilon , s)}^{-1}$ is defined as

$$\begin{align*}(1-\epsilon(\mathfrak{q})\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}). \end{align*}$$

If $\pi (f)_{\mathfrak {q}}$ is special, (i.e. Steinberg), then ${L_{\mathfrak {q}}(\mathrm {Ad}(f) \otimes \epsilon , s)}^{-1}$ is defined as

$$\begin{align*}(1-\epsilon(\mathfrak{q})\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-1-s}). \end{align*}$$

If $\pi (f)_{\mathfrak {q}}$ is supercuspidal, we set the local L-factor to be equal to $1$ , even if the completed L-function could have an Euler factor different from $1$ at $\mathfrak {q}$ .

For $\sigma \in \Sigma _F$ we define

$$\begin{align*}{L_{\sigma}(\mathrm{Ad}(f) \otimes \epsilon, s)}:= {(2\pi)}^{-(s+k_\sigma+1)}\Gamma(s+k_\sigma+1)\pi^{-(s+1)/2}\Gamma((s+1)/2). \end{align*}$$

For any other ideal $\mathfrak {c}^{\prime }_F$ in $\mathcal {O}_F$ we define

$$\begin{align*}L^{\ast,\mathfrak{c}^{\prime}_F }(\mathrm{Ad}(f) \otimes \epsilon, s):=\prod_{\sigma \in \Sigma_F}L_{\sigma}(\mathrm{Ad}(f) \otimes \epsilon, s) \prod_{\mathfrak{q} \not\mid \mathfrak{c}^{\prime}_F \mathfrak{f}(\epsilon)} {L_{\mathfrak{q}}(\mathrm{Ad}(f) \otimes \epsilon, s)}. \end{align*}$$

Finally, note that one can also define the (partial) symmetric square L-function for f twisted by $\epsilon $ :

$$\begin{align*}L^{\mathfrak{c}\mathfrak{f}(\epsilon)} (\mathrm{Sym}^2(f) \otimes \epsilon, s):=\prod_{ \mathfrak{q} \not\mid \mathfrak{c}\mathfrak{f}(\epsilon)} {L_{\mathfrak{q}}(\mathrm{Sym}^2(f) \otimes \epsilon, s)}, \end{align*}$$

where ${L_{\mathfrak {q}}(\mathrm {Sym}^2(f) \otimes \epsilon , s)}^{-1}$ is given by

$$\begin{align*}\left(1-\epsilon(\mathfrak{q})\alpha_{1,\mathfrak{q}}^2\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right) \left(1-\epsilon(\mathfrak{q})\chi(\mathfrak{q})\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right) \left(1-\epsilon(\mathfrak{q})\alpha_{2,\mathfrak{q}}^2\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{q})^{-s}\right). \end{align*}$$

Note that at any prime ideal $\mathfrak {q}$ , unramified for f and $\chi $ , we have

$$\begin{align*}{L_{\mathfrak{q}}(\mathrm{Sym}^2(f) \otimes \chi^{-1}, s)} ={L_{\mathfrak{q}}(\mathrm{Ad}(f), s)}.\end{align*}$$

3.2.2 The Asai L-function

Here, we follow [Reference Getz and Goresky17, §5.12.4]. Let $\mathfrak {c}$ be an ideal of E, let $\chi _F$ be a Hecke character of ${\mathbb {A}}_F^\times $ , and $\chi := \chi _F \circ \mathrm {N}_{E/F}$ be the corresponding Hecke character of ${\mathbb {A}}_E^\times $ . We also let $\eta _{E/F}$ be the only non-trivial (quadratic) character of the Galois group of E over F, and, by abuse of notation, we use $\eta _{E/F}$ to denote the corresponding Hecke character of ${\mathbb {A}}^\times _F$ , trivial at the archimedean places. Recall that there is a distinguished F-type for E that we denoted by $\Phi $ in §2. Then for a given weight $\kappa = (\underline {k},m)$ for F, we have the corresponding weight for E: $\hat {\kappa } = (\hat {\underline {k}},m)$ , with $\hat {k}_\sigma =\hat {k}_{\sigma '}$ for every $\sigma \in \Phi $ .

To any Hilbert cuspidal eigenform g of level $K_{11}(\mathfrak {c})$ , weight $\hat {\kappa }$ and Nebentypus $\chi $ and to any quasi-character $\theta : E^\times \backslash {\mathbb {A}}_E^\times \to {\mathbb {C}}^\times $ with conductor $\mathfrak {f}(\theta )$ , we can associate the Asai L-function of g twisted by $\theta $ as followsFootnote ⁴

(7) $$ \begin{align} L(\mathrm{As}(g \otimes \theta) , s) := L^{\mathfrak{c}\mathfrak{f}(\theta) \cap \mathcal{O}_F }(\theta^2 \chi)|_F,2s) \sum_{\mathfrak{n} +\mathfrak{f}(\theta) = \mathcal{O}_F} \theta(\mathfrak{n}\mathcal{O}_E) a(\mathfrak{n}\mathcal{O}_E,g) \mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{n})^{-1-s}. \end{align} $$

The corresponding partial Asai L-function of g admits an Euler product

$$\begin{align*}L^{\mathfrak{c}\mathfrak{f}(\theta) \cap \mathcal{O}_F }(\mathrm{As}(g \otimes \theta) , s) :=\prod_{ \mathfrak{p} \not\mid \mathfrak{c} \cap \mathcal{O}_F } {L_{\mathfrak{p}}(\mathrm{As}(g \otimes \theta) , s)}, \end{align*}$$

where, if $\mathfrak {p} \nmid \mathfrak {d}_{E/F}(\mathfrak {c}\mathfrak {f}(\theta ) \cap \mathcal {O}_F) $ , we define

$$\begin{align*}&L_{\mathfrak{p}}(\mathrm{As}(g \otimes \theta) , s)^{-1} \\&\quad= \begin{cases} \prod_{i=1}^2\prod_{j=1}^2 (1 - \theta(\mathfrak{p}) \alpha_{i,\mathfrak{P}}\alpha_{j,\overline{\mathfrak{P}}} \mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{p})^{-s}) & \text{if } \mathfrak{p} = \mathfrak{P}\overline{\mathfrak{P}} \text{ is split,} \\ (1 - \theta(\mathfrak{p})^2 \chi(\mathfrak{p}) \mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{p})^{-2s})\prod_{i=1}^2 (1 - \theta(\mathfrak{p}) \alpha_{i,\mathfrak{p}}\mathrm{N}_{F/{\mathbb{Q}}}(\mathfrak{p})^{-s}) & \text{if } \mathfrak{p} \text{ is inert,} \end{cases} \end{align*}$$

with $\alpha _{1, \mathfrak {q}}, \alpha _{2, \mathfrak {q}}$ denoting the Satake parameters of $\pi (g)_{\mathfrak {q}}$ (cf. [Reference Getz and Goresky17, §5.12.4]).

3.2.3 Base change forms and their L-function

The main reference for this section is [Reference Getz and Goresky17, Appendix E]. If we let $G_1 = \mathrm {GL}_2 /F$ and $G_2= \mathrm {Res}_{E/F} \mathrm {GL}_{2}$ , we have

$$ \begin{align*} {}^LG_1 &= \mathrm{GL}_2({\mathbb{C}}) \times \mathrm{Gal}(E/F) \\ {}^LG_2 &= \mathrm{GL}_2({\mathbb{C}})^{\mathrm{Gal}(E/F)} \rtimes \mathrm{Gal}(E/F), \end{align*} $$

where in the second $\mathrm {Gal}(E/F)$ acts on $\mathrm {GL}_2({\mathbb {C}})^{\mathrm {Gal}(E/F)}$ by permuting the factors. The base change lifting from $G_1$ to $G_2$ is induced by the map of L-groups

$$\begin{align*}b:{}^LG_1 \to {}^LG_2, \end{align*}$$

given by the diagonal embedding on the $\mathrm {GL}_2({\mathbb {C}})$ factor and by the identity on the $\mathrm {Gal}(E/F)$ factor. For any place v of F, the map $ b:{}^LG_{1, F_v} \to {}^LG_{2, F_v}$ of L-groups over $F_v$ is defined similarly. Suppose that v is a finite place of F which is unramified for $E/F$ . Then b induces a morphism of local spherical Hecke algebras

$$\begin{align*}b: \mathcal{H}(\mathrm{GL}_2(E_v) // \mathrm{GL}_2( \mathcal{O}_{E_v})) \longrightarrow \mathcal{H}(\mathrm{GL}_2(F_v) // \mathrm{GL}_2( \mathcal{O}_{F_v})), \end{align*}$$

where we have denoted $E_v = E \otimes _F F_v$ . If $\pi _v$ is an irreducible unramified admissible representation of $\mathrm {GL}_2(F_v)$ , where $\mathcal {H}(\mathrm {GL}_2(F_v) // \mathrm {GL}_2( \mathcal {O}_{F_v}))$ acts via the character $\lambda _{\pi _v}$ , then an irreducible admissible representation $\Pi _v$ of $\mathrm {GL}_2(E_v) $ is the base change of $\pi _v$ if it is unramified and

$$\begin{align*}\mathcal{H}(\mathrm{GL}_2(E_v) // \mathrm{GL}_2( \mathcal{O}_{E_v})) \end{align*}$$

acts through the character

$$\begin{align*}\lambda_{\Pi_v} = \lambda_{\pi_v} \circ b. \end{align*}$$

Concretely, if $\pi _v$ is the principal series attached to unramified characters $\chi _1,\chi _2$ of $F_v^\times $ , $\Pi _v$ is the base change of $\pi _v$ if it is the principal series of characters $\chi _1 \circ \mathrm {N}_{E_v/F_v},\chi _2 \circ \mathrm {N}_{E_v/F_v}$ . Globally, we say that an automorphic representation $\Pi $ of $\mathrm {GL}_2({\mathbb {A}}_E)$ is a weak base change of an automorphic representation $\pi $ of $\mathrm {GL}_2({\mathbb {A}}_F)$ if $\Pi _v$ is the base change of $\pi _v$ almost everywhere. Note that, by strong multiplicity for $\mathrm {GL}_2$ , if there exists a weak base change $\Pi $ of $\pi $ , then it is unique.

Theorem 3.6 [Reference Getz and Goresky17, Theorem E.11].

Every automorphic representation $\pi $ of $\mathrm {GL}_2({\mathbb {A}}_F)$ has a unique base change lift to an automorphic representation $\widehat {\pi }$ of $\mathrm {GL}_2({\mathbb {A}}_E)$ . Moreover, any cuspidal automorphic representation $\Pi $ of $\mathrm {GL}_2({\mathbb {A}}_E)$ that is isomorphic to its conjugates under the action of $\mathrm {Gal}(E/F)$ is the base change lift of a cuspidal automorphic representation of $\mathrm {GL}_2({\mathbb {A}}_F)$ . The assignment $\pi \mapsto \widehat {\pi }$ satisfies the following: if $\pi $ and $\widehat {\pi }$ are both cuspidal, then

1. (1) If $\widehat {\pi }$ is also the base change lift of another automorphic representation $\pi '$ , then $\pi ' \simeq \pi \otimes \eta _{E/F}$ ;

2. (2) For every place v of F, $L(\widehat {\pi }_v,s) = L(\pi _v,s)L(\pi _v \otimes \eta _{E_v/F_v},s)$ .

Fix a weight $\kappa = (\underline {k},m)$ for F and consider the corresponding weight $\hat {\kappa } = (\hat {\underline {k}},m)$ for E. The theorem above can be used to prove the following proposition, which we will use repeatedly in the course of the manuscript.

Proposition 3.7. Suppose that a newform $g \in S_{\hat {\kappa }}(K_0(\mathfrak {c}), \chi )$ is an eigenform for all but finitely many Hecke operators for which the eigenvalues of $T_{\mathfrak {q}}$ and $T_{\mathfrak {q}^{\sigma }}$ , with $\sigma $ generator of $\mathrm {Gal}(E/F)$ , agree for almost all prime ideals $\mathfrak {q} \subset \mathcal {O}_E$ . Then there exists an ideal $\mathfrak {c}_1 \subseteq \mathcal {O}_F$ and a newform $f \in S_{\kappa }(K_0(\mathfrak {c}_1), \chi _F)$ such that $\widehat {\pi (f)} = \pi (g)$ . Conversely, if $f \in S_{\kappa }(K_0(\mathfrak {c}_F), \chi _F)$ is a newform, there exists a unique newform $f_E \in S_{\hat {\kappa }}(K_0(\mathfrak {c}), \chi )$ such that $\pi (f_E) = \widehat {\pi (f)}$ . Moreover, if $\theta :=\theta ^u|\phantom {ee}|_{\mathbb {A}_E}^{m/2}$ is a Hecke character of $\mathbb {A}^\times _E$ , with $\theta ^u$ unitary, such that its restriction to $\mathbb {A}^\times _F$ is $\chi _F \eta _{E/F}$ , then

$$\begin{align*}L^{\mathfrak{b}'}(\mathrm{As}(f_E \otimes \theta^{-1}) , s) = \zeta_F^{\mathfrak{d}_{E/F}(\mathfrak{c} \mathfrak{f}(\theta) \cap \mathcal{O}_F )}(s) L^{\mathfrak{d}_{E/F} ( \mathfrak{f}(\theta) \cap \mathcal{O}_F )}(\mathrm{Ad}(f) \otimes \eta_{E/F}, s), \end{align*}$$

with $\mathfrak {b}' := \prod _{\substack { \mathfrak {p} | \mathfrak {d}_{E/F} \\ \mathfrak {p} \nmid \mathfrak {f}(\theta ) \cap \mathcal {O}_F}} \mathfrak {p}$ . Hence,

$$\begin{align*}\mathrm{Res}_{s=1}L^{\mathfrak{b}'}(\mathrm{As}(f_E \otimes \theta^{-1}) , s) = L^{\mathfrak{d}_{E/F} ( \mathfrak{f}(\theta) \cap \mathcal{O}_F )}(\mathrm{Ad}(f) \otimes \eta_{E/F}, 1) \mathrm{Res}_{s=1}\zeta_F^{\mathfrak{d}_{E/F}(\mathfrak{c} \mathfrak{f}(\theta) \cap \mathcal{O}_F )}(s). \end{align*}$$

Proof. This follows from Corollary E.12 and Proposition 5.17 of [Reference Getz and Goresky17].

3.3 A cohomological formula for adjoint L-values

Let $\mathfrak {c}_F$ be an ideal of F and f be a Hilbert cuspidal eigenform of level $K_{11}(\mathfrak {c}_F)$ , weight $\kappa = (\underline {k},m)$ and Nebentypus $\chi _F$ . Let $\mathfrak {c}$ be an ideal of $\mathcal {O}_E$ , whose intersection with $\mathcal {O}_F$ is $\mathfrak {c}_F$ . We consider a Hecke character $\theta :=\theta ^u|\phantom {ee}|_{\mathbb {A}_E}^{m/2}$ of $\mathbb {A}^\times _E$ such that its restriction to $\mathbb {A}^\times _F$ is $\chi _F \eta _{E/F} $ . Here $\theta ^{u}$ denotes the unitary part of $\theta $ and m is as in the weight $(\underline {k},m)$ . Let $\mathfrak {b}$ be the conductor of $\theta ^{u}$ and $\mathfrak {c}'=\mathfrak {c}\mathfrak {b}^2$ . We write $\mathfrak {b}_F, \mathfrak {c}^{\prime }_F$ for the intersection of $\mathfrak {b}$ and $\mathfrak {c}'$ with $\mathcal {O}_F$ . We let $\chi := \chi _F \circ \mathrm {N}_{E/F}$ and we let $f_E \in S_{\hat {\kappa }}(K_0(\mathfrak {c}),\chi )$ be the unique newform on E which generates the automorphic representation of G given by the base change of $\pi (f)$ . Recall that $\hat {\kappa } = (\hat {\underline {k}},m)$ , with $\hat {k}_\sigma =\hat {k}_{\sigma '}$ for every $\sigma \in \Phi $ . We also have denoted by $\chi _0$ the character of $K_0(\mathfrak {c})$ , which at each finite place v is defined by

$$\begin{align*}\chi_v\left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)= \begin{cases} \chi_v(d) & \text{ if } \mathfrak{p}_v \mid \mathfrak{c} \\ 1 & \text{ otherwise,} \end{cases} \end{align*}$$

with $\mathfrak {p}_v$ denoting the prime ideal associated with v. Poincaré duality gives a perfect pairing

$$\begin{align*}\langle \, , \, \rangle : H^{2d}_c(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},-m}(\chi^{-1}_0)) \times H^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},m}(\chi_0)) \to {\mathbb{C}}, \end{align*}$$

where $H^{2d}_c(\bullet ) $ stands for compactly supported cohomology. Similarly, we have an intersection pairing for the middle intersection cohomology $\langle \, , \, \rangle _{IH}$ (with respect to the Baily–Borel compactification of $Y_G(K_0(\mathfrak {c}))$ )

$$\begin{align*}\langle \, , \, \rangle_{IH} : {IH}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},-m}(\chi^{-1}_0)) \times {IH}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},m}(\chi_0)) \to {\mathbb{C}}. \end{align*}$$

If we let $\alpha : H^{2d}_c(Y_G(K_0(\mathfrak {c})), \mathscr {V}_{\mathbb {C}}^{\hat {\underline {k}},-m}(\chi ^{-1}_0)) \to {IH}^{2d}(Y_G(K_0(\mathfrak {c})), \mathscr {V}_{\mathbb {C}}^{\hat {\underline {k}},-m}(\chi ^{-1}_0))$ be the composition of the natural morphism from $H^{2d}_c(\bullet )$ to $L^2$ -cohomology $H^{2d}_{(2)}(\bullet )$ with the isomorphism $\mathcal {Z}$ of Zucker’s conjecture ([Reference Getz and Goresky17, Theorem 4.2])

$$\begin{align*}\mathcal{Z}:{H}^{2d}_{(2)}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},-m}(\chi^{-1}_0)) \simeq {IH}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},-m}(\chi^{-1}_0)),\end{align*}$$

and $\beta $ the natural map ${IH}^{2d}(Y_G(K_0(\mathfrak {c})), \mathscr {V}_{\mathbb {C}}^{\hat {\underline {k}},m}(\chi _0)) \to {H}^{2d}(Y_G(K_0(\mathfrak {c})), \mathscr {V}_{\mathbb {C}}^{\hat {\underline {k}},m}(\chi _0))$ , then we have (cf. (3.4.7) of loc. cit.)

$$\begin{align*}\langle \omega_1 , \beta(\omega_2) \rangle =\langle \alpha(\omega_1) , \omega_2 \rangle_{IH}. \end{align*}$$

For any type J for $E/F$ , the differential form $\omega _J(f^{-\iota }_E)$ defines an intersection cohomology class via $\mathcal {Z}$ . Let $f_E \otimes \theta ^{-1}$ be the twist of $f_E$ by $\theta ^{-1}$ , see [Reference Getz and Goresky17, Lemma 5.14]. By [Reference Getz and Goresky17, Theorem 10.1], there exists an intersection cohomology class

$$\begin{align*}[Z(\mathfrak{c})_{\theta}] \in {IH}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\underline{k}},m}(\chi_0)) \end{align*}$$

(made up of a cycle $Z(\mathfrak {c})_{\theta }$ , given by $Y_H(K_0(\mathfrak {c}_F))$ twisted by $\theta $ and the H-invariant section $\Delta ^{\hat {\underline {k}}}$ of the restricted local system $\mathscr {V}^{\hat {\underline {k}},{m}}_{\mathbb {C}}$ ) such that

(8) $$ \begin{align} \langle \mathcal{Z}[ \omega_J(f^{-\iota}_E)], [Z(\mathfrak{c})_\theta] \rangle_{IH} &= C^{\prime}_2 \int_{Y_H(K_0(\mathfrak{c}_F'))} \tilde{\iota}^{\hat{\underline{k}},\ast}( \omega_J(f^{-\iota}_E \otimes \theta^{-1})), \end{align} $$

withFootnote ⁵ $C^{\prime }_2= [K_0(\mathfrak {c}^{\prime }_F):\mathcal {O}_F^{\ast , +} K_{1}(\mathfrak {c}^{\prime }_F)] [K_0(\mathfrak {c}'):\mathcal {O}_E^{\ast , +}K_{11}(\mathfrak {c}')]G(\theta ^{u}) |N_{E/{\mathbb {Q}}}(D_{E/{\mathbb {Q}}})|^{-m/2}$ and where $\tilde {\iota }^{\hat {\underline {k}},\ast }$ is induced by the map in cohomology corresponding to the pullback to $Y_H$ and at the level of vector bundles to the $\chi ^{-1}_0\theta _0^{-2}$ -twisted branching map $\tilde {\iota }^{\hat {\underline {k}}} : {\mathbb {C}} \hookrightarrow \iota ^{\ast } \mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi ^{-1}_0\theta _0^{-2})$ as in §2.4. Crucially, this integral recovers the algebraic part of the adjoint L-function of f twisted by $\eta _{E/F}$ .

Note that the above integral formula can be obtained without using middle intersection cohomology by means of the Hirzebruch–Zagier cycles in the cohomology of the open Hilbert modular variety, as we now explain. Since we ultimately need the following construction for weights $\hat {\kappa }$ of the form $(\hat {\underline {k}},0)$ , we assume that $m =0$ .

Let $\mathbf {1}_{K_0(\mathfrak {c}_F)}$ be the fundamental class

$$\begin{align*}[Y_H(K_0(\mathfrak{c}_F)) ] \in H^0(Y_H(K_0(\mathfrak{c}_F)), {\mathbb{Z}}),\end{align*}$$

corresponding to the Hilbert modular variety of level $K_0(\mathfrak {c}_F)$ and let $\iota ^{\hat {\underline {k}}}:{\mathbb {C}} \hookrightarrow \mathscr {V}^{\hat {\underline {k}},0}_{{\mathbb {C}}}$ be the branching map introduced in §2.4.

Definition 3.9. Let

$$\begin{align*}\mathscr{Z}^{\hat{\kappa}}(\mathfrak{c}) := \iota_\ast \circ \iota^{\hat{\underline{k}}} (\mathbf{1}_{K_0(\mathfrak{c}_F)}) \in {H}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}). \end{align*}$$

Similarly to the definition above, we can define a cycle in the cohomology with coefficients $\mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi _0)$ . To do so, let

$$\begin{align*}K_{1}(\mathfrak{c}) := \{ g \in K_0(\mathfrak{c})\,:\, g \equiv {\matrix{\star}{\star}{0}{1}} \text{ mod } \mathfrak{c}\},\end{align*}$$

and denote by

$$\begin{align*}P_{10,\ast}^{\chi_0}:{H}^{2d}(Y_G(K_1(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0)) \to {H}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0)) \end{align*}$$

the $\chi _0$ -twisted trace associated with the natural degeneracy map $P_{10}^{\mathfrak {c}}: Y_G(K_1(\mathfrak {c})) \to Y_G(K_0(\mathfrak {c}))$ (cf. [Reference Getz and Goresky17, (9.2.5)]). Note that $\mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi _0) = \mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}$ as local systems over $Y_G(K_1(\mathfrak {c}))$ .

Definition 3.10. Let

$$\begin{align*}\mathscr{Z}^{\hat{\kappa},\chi}(\mathfrak{c}) := P_{10,\ast}^{\chi_0} \circ \iota_\ast \circ \iota^{\hat{\underline{k}}} (\mathbf{1}_{K_1(\mathfrak{c}_F)}) \in {H}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0)). \end{align*}$$

This cycle can be further twisted by $\theta $ by using the cohomological correspondence of [Reference Getz and Goresky17, (9.4.1)]. Namely, choose a set $S = \{ t \}_{t \in \mathfrak {b}^{-1} \widehat {\mathcal {O}}_E}$ of representatives for the quotient $\mathfrak {b}^{-1} \widehat {\mathcal {O}}_E / \widehat {\mathcal {O}}_E$ . For each $t \in S$ , let $u(t) \in G({{\mathbb {A}}_{f}})$ be the matrix defined by

$$\begin{align*}u(t)_v = \begin{cases} {\matrix{1}{0}{0}{1}} & \text{ if } v \nmid \mathfrak{b} \\ {\matrix{1}{t_v}{0}{1}} & \text{ if } v \mid \mathfrak{b}. \end{cases} \end{align*}$$

Consider the correspondence

where $\pi _1,\pi _2$ are the natural degeneracy maps and $\mathfrak {c}'=\mathfrak {c}\mathfrak {b}^2$ , and define $\mathcal {P}_{\mathfrak {b}} := \sum _{t \in S} \pi _{2,\ast } u(t)_\ast \pi _1^\ast $ . Thanks to [Reference Getz and Goresky17, Lemma 9.3], $u(t)$ induces a map of local systems $\pi _1^* \mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi _0\theta _0^2) \to \pi _2^* \mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi _0)$ and so a cohomological correspondence

$$\begin{align*}\mathcal{P}_{\mathfrak{b}} : {H}^{2d}(Y_G(K_0(\mathfrak{c}')), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0\theta_0^2)) \to {H}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0)). \end{align*}$$

Thus we can give the following (cf. [Reference Getz and Goresky17, Proposition 9.5]).

Definition 3.11. Let

$$\begin{align*}\mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta : = \mathcal{P}_{\mathfrak{b}}(\mathscr{Z}^{\hat{\kappa},\chi\theta^2}(\mathfrak{c'})) \in {H}^{2d}(Y_G(K_0(\mathfrak{c})), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi_0)). \end{align*}$$

Let now $g \in S_{\hat {\kappa }}(K_0(\mathfrak {c}),\chi )$ be an eigenform for E and let J be an F-type for E. As $\omega _J(g^{-\iota })$ is rapidly decreasing and rapidly decreasing cohomology is isomorphic to the compactly supported one, see [Reference Borel6, Theorem 5.2], we can consider the value

$$\begin{align*}\langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta \rangle = \langle [ \omega_c ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta \rangle, \end{align*}$$

where $\omega _c$ is the compactly supported differential form given by Proposition 3.1.

Lemma 3.12. Let $g \in S_{\hat {\kappa }}(K_0(\mathfrak {c}),\chi )$ be an eigenform for E of weight $\hat {\kappa } = (\hat {\underline {k}},0)$ and let J be an F-type for E. Then

$$\begin{align*}\langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c}) \rangle = \int_{Y_H(K_0(\mathfrak{c}_F))} \iota^{\hat{\underline{k}},\ast}( \omega_J(g^{-\iota})). \end{align*}$$

Similarly, if $\theta $ is the Hecke character of ${\mathbb {A}}_E^\times $ introduced above,

$$\begin{align*}\langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta \rangle = C_2 \int_{Y_H(K_0(\mathfrak{c}_F'))} \tilde{\iota}^{\hat{\underline{k}},\ast}( \omega_J(g^{-\iota} \otimes \theta^{-1})), \end{align*}$$

where $C_2:= [K_0(\mathfrak {c}^{\prime }_F):\mathcal {O}_F^{\ast , +} K_{1}(\mathfrak {c}^{\prime }_F)] [K_0(\mathfrak {c}'):\mathcal {O}_E^{\ast , +}K_{11}(\mathfrak {c}')]G(\theta ^{u})$ and

$$\begin{align*}\tilde{\iota}^{\hat{\underline{k}},\ast} : H^{2d}(Y_G(K_0(\mathfrak{c}')), \mathscr{V}_{\mathbb{C}}^{\hat{\kappa}}(\chi^{-1}_0\theta_0^{-2})) \to H^{2d}(Y_H(K_0(\mathfrak{c}_F')), {\mathbb{C}})\end{align*}$$

is the map induced by pullback along $\iota $ at the level of varieties and by the $\chi ^{-1}_0\theta _0^{-2}$ -twisted branching map $\tilde {\iota }^{\hat {\underline {k}}} : {\mathbb {C}} \hookrightarrow \iota ^{\ast } \mathscr {V}_{\mathbb {C}}^{\hat {\kappa }}(\chi ^{-1}_0\theta _0^{-2})$ as in §2.4.

Proof. For the first part of the statement, we argue exactly as in [Reference Cauchi, Lemma and Rodrigues Jacinto9, Proposition 4.8]. Using Proposition 3.1, we get that

$$ \begin{align*} \langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c}) \rangle &= \langle [ \omega_c ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c}) \rangle \\ &= \langle [ \omega_c ] , \iota_\ast \circ \iota^{\hat{\underline{k}}} (\mathbf{1}_{K_0(\mathfrak{c}_F)}) \rangle \\ &= \langle \iota^{\hat{\underline{k}},\ast}[ \omega_c] , \mathbf{1}_{K_0(\mathfrak{c}_F)} \rangle \\ &= \int_{Y_H(K_0(\mathfrak{c}_F))}\!\!\!\! \iota^{\hat{\underline{k}},\ast}(\omega_c) \\ &= \int_{Y_H(K_0(\mathfrak{c}_F))}\!\!\!\! \iota^{\hat{\underline{k}},\ast}(\omega_J(g^{-\iota}) ) + \int_{Y_H(K_0(\mathfrak{c}_F))}\!\!\!\! \iota^{\hat{\underline{k}},\ast}(d \Psi), \end{align*} $$

with $\Psi $ a degree $2d-1$ rapidly decreasing form on $Y_G(K_0(\mathfrak {c}))$ . To conclude the proof, note that, as explained in [Reference Borel6, §5.6],

$$\begin{align*}\int_{Y_H(K_0(\mathfrak{c}_F))}\!\!\!\! \iota^{\hat{\underline{k}},\ast}(d \Psi) = 0.\end{align*}$$

Regarding the second formula, notice

$$ \begin{align*} \langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta \rangle &= \langle [ \omega_J(g^{-\iota}) ] , \mathcal{P}_{\mathfrak{b}}(\mathscr{Z}^{\hat{\kappa},\chi\theta^2}(\mathfrak{c'})) \rangle \\ &= \langle \mathcal{P}_{\mathfrak{b}}'[ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa},\chi\theta^2}(\mathfrak{c'}) \rangle \\ &= C_2 \langle [ \omega_J(g^{-\iota} \otimes \theta^{-1}) ] , \mathscr{Z}^{\hat{\kappa},\chi\theta^2}(\mathfrak{c'}) \rangle, \end{align*} $$

where $\mathcal {P}_{\mathfrak {b}}'$ denotes the adjoint to $\mathcal {P}_{\mathfrak {b}}$ and we have used that $\mathcal {P}_{\mathfrak {b}}'[ \omega _J(g^{-\iota }) ] = C_2 [ \omega _J(g^{-\iota } \otimes \theta ^{-1}) ]$ (cf. [Reference Getz and Goresky17, Proposition 9.4]). The commutativity of the diagram

implies that

$$ \begin{align*} P_{10, \ast}^{\mathfrak{c}'} \circ \iota_\ast(\mathbf{1}_{K_1(\mathfrak{c}_F')}) & = \iota_\ast \circ P_{10, \ast}^{\mathfrak{c}_F'}(\mathbf{1}_{K_1(\mathfrak{c}_F')}) \\ &= [K_0(\mathfrak{c}^{\prime}_F):\mathcal{O}_F^{\ast, +} K_{1}(\mathfrak{c}^{\prime}_F)] \iota_\ast(\mathbf{1}_{K_0(\mathfrak{c}_F')}), \end{align*} $$

with $[K_0(\mathfrak {c}^{\prime }_F):\mathcal {O}_F^{\ast , +} K_{1}(\mathfrak {c}^{\prime }_F)]$ being the degree of the natural degeneracy map $P_{10}^{\mathfrak {c}_F'}$ . This and the definition of $\tilde {\iota }^{\hat {\underline {k}},\ast } $ give that

$$ \begin{align*} \langle [ \omega_J(g^{-\iota} \otimes \theta^{-1}) ] , \mathscr{Z}^{\hat{\kappa},\chi\theta^2}(\mathfrak{c'}) \rangle &= \langle [ \omega_J(g^{-\iota} \otimes \theta^{-1}) ], P_{10,\ast}^{\chi_0\theta^2_0} \circ \iota_* \circ \iota^{\hat{\underline{k}}}(\mathbf{1}_{K_1(\mathfrak{c}_F')}) \rangle \\ &= [K_0(\mathfrak{c}^{\prime}_F):\mathcal{O}_F^{\ast, +} K_{1}(\mathfrak{c}^{\prime}_F)] \langle \tilde{\iota}^{\hat{\underline{k}},\ast} [ \omega_J(g^{-\iota} \otimes \theta^{-1}) ], \mathbf{1}_{K_0(\mathfrak{c}_F')} \rangle. \end{align*} $$

The result then follows as in the first part of the statement.

Remark 3.13. Lemma 3.12 implies in particular that

$$\begin{align*}\langle [ \omega_J(g^{-\iota}) ] , \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_\theta \rangle = \langle \mathcal{Z}[ \omega_J(g^{-\iota})], [Z(\mathfrak{c})_\theta] \rangle_{IH}. \end{align*}$$

Write

$$\begin{align*}\mathfrak{b}' := \prod \{ \mathfrak{p} \,:\, \mathfrak{p} \mid \mathfrak{d}_{E/F} \text{ and } \mathfrak{p} \nmid \mathfrak{f}(\theta) \}, \end{align*}$$

with $\mathfrak {f}(\theta )$ the conductor of $\theta $ . Finally, [Reference Getz and Goresky17, Theorem 10.1] reads as follows:

Theorem 3.14. For f as above, let $\chi _F$ be a Hecke character for F corresponding to the Nebentypus of f, and let $\theta $ be a Hecke character for E such that $\theta _{\vert _{{\mathbb {A}}_F^\times }}=\eta _{E/F} \chi _F $ . Then we have

$$\begin{align*}\langle \omega_J(f^{-\iota}_E), \mathscr{Z}^{\hat{\kappa}}(\mathfrak{c})_{\theta} \rangle=\frac{c_1}{2^{d}}L^{\ast,\mathfrak{d}_{E/F}\mathfrak{b}_F }(\mathrm{Ad}(f) \otimes \eta_{E/F}, 1) \prod_{\mathfrak{q} \mid \mathfrak{b}'} L_{\mathfrak{q}}(\mathrm{As}(f_E \otimes \theta^{-1}), 1), \end{align*}$$

where $L_{\mathfrak {q}}(\mathrm {As}(f_E \otimes \theta ^{-1}), 1)$ denotes the Euler factor at ${\mathfrak {q}}$ of the Asai L-function of $f_E$ twisted by $\theta ^{-1}$ and

$$ \begin{align*} c_1:=\frac{C_2 \mathrm{N}_{F/{\mathbb{Q}}}({\mathfrak{c}^{\prime}_F}^2 D_{E/F})D_{F}^3 \zeta_F^{\mathfrak{d}_{E/F}\mathfrak{c}^{\prime}_F}(1)^* }{2^{(\sum_{\sigma \in \Sigma_F} k_{\sigma})-2} R_F |(\mathcal{O}_F/\mathfrak{c}^{\prime}_F)^{\times}|},\end{align*} $$

for $R_F$ the regulator of F and $\zeta _F^{\mathfrak {d}_{E/F}\mathfrak {c}^{\prime }_F}(1)^*$ the residue at $s=1$ of the partial Dedekind zeta function of F with the Euler factors at $\mathfrak {d}_{E/F}\mathfrak {c}^{\prime }_F$ removed.

The proof of the theorem follows from comparing the left-hand side to the residue at $s=0$ of a Rankin–Selberg integral $I(f_E,s)$ which calculates the Asai L-function of $f_E \otimes \theta ^{-1}$ (cf. [Reference Getz and Goresky17, Chapter 10] and §7.1 below). Note that the adjoint L-value appears because of the decomposition of Proposition 3.7.

4.1 A spherical variety

Let $B_G = T_G \cdot N_G$ denote the upper-triangular Borel subgroup of G with Levi subgroup the maximal diagonal torus $T_G$ and unipotent radical $N_G$ . We also let $\overline {B}_G = T_G \cdot \overline {N}_G$ be the opposite parabolic. Similarly, denote by $B_H = T_H \cdot N_H = B_G \cap H$ the upper-triangular Borel subgroup of H. Consider the flag variety $\mathcal {F} := G / \overline {B}_G$ over $\mathrm {Spec}\, {\mathbb {Z}}_p$ . It has a natural left action by G and, thus, a left action by $H=G_F$ via the embedding $\iota $ . Recall that $H({\mathbb {Z}}_p) = \prod _{\mathfrak {p}|p} \mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}})$ over primes $\mathfrak {p}$ in F with ${\mathcal {O}}_{\mathfrak {p}} := {\mathcal {O}}_{F,\mathfrak {p}}$ , and $G({\mathbb {Z}}_p) = \prod _{\mathfrak {P}|p} \mathrm {GL}_2({\mathcal {O}}_{\mathfrak {P}})$ over primes $\mathfrak {P}$ in E with ${\mathcal {O}}_{\mathfrak {P}} := {\mathcal {O}}_{E,\mathfrak {P}}$ . Via these identifications $\iota : H({\mathbb {Z}}_p) \to G({\mathbb {Z}}_p)$ can be written as $(\iota _{\mathfrak {p}})_{\mathfrak {p}}$ where each embedding

$$\begin{align*}\iota_{\mathfrak{p}}: \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{p}}) \hookrightarrow \prod_{\mathfrak{P}|\mathfrak{p}} \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{P}}) \end{align*}$$

is the natural map induced by the inclusion of ${\mathcal {O}}_{\mathfrak {p}}$ in ${\mathcal {O}}_{\mathfrak {P}}$ . If $\mathfrak {p}$ splits in E, then $\mathfrak {p}=\mathfrak {P} \overline {\mathfrak {P}}$ (thus ${\mathcal {O}}_{\mathfrak {p}} = {\mathcal {O}}_{\mathfrak {P}}={\mathcal {O}}_{\overline {\mathfrak {P}}}$ ) and $\iota _{\mathfrak {p}}(g) = (g,g)$ . If $\mathfrak {p}$ is inert in E, there is only one prime $\mathfrak {P}$ above it and $\iota _{\mathfrak {p}}$ identifies $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}})$ with the ${\mathcal {O}}_{\mathfrak {P}}$ -valued matrices fixed by $\mathrm { Gal}(E_{\mathfrak {P}}/F_{\mathfrak {p}})$ .

Let $\Upsilon _{F,p}$ denote the set of primes of F above p. This can be divided into two subsets $\Upsilon _{F,p} = \Upsilon ^{\mathrm {s}}_{F,p} \cup \Upsilon ^{\mathrm {i}}_{F,p} $ , where $\Upsilon ^{\mathrm {s}}_{F,p}$ (resp. $\Upsilon ^{\mathrm { i}}_{F,p}$ ) is the set of primes above p which are split (resp. inert) in E. Then we may write

$$\begin{align*}G({\mathbb{Z}}_p)= \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}} \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{p}})^2 \times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{P}}). \end{align*}$$

Definition 4.1. Let $u \in G({\mathbb {Z}}_p)$ denote the element $(u_{\mathfrak {p}})_{\mathfrak {p} \in \Upsilon _{F,p}}$ given by

$$\begin{align*}u_{\mathfrak{p}} = \begin{cases} \left ( {\matrix {1} {1} { } {1}} , {\matrix {1} {} { } {1}} \right ) & \text{ if } \mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}, \\ {\matrix {1} {\delta_{\mathfrak{p}}} {} {1}} & \text{ if } \mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}, \end{cases} \end{align*}$$

with $\delta _{\mathfrak {p}} \in {\mathcal {O}}_{\mathfrak {P}} \setminus {\mathcal {O}}_{\mathfrak {p}} $ such that ${\mathcal {O}}_{\mathfrak {P}} = {\mathcal {O}}_{\mathfrak {p}} \oplus {\mathcal {O}}_{\mathfrak {p}} \delta _{\mathfrak {p}}$ , and $\delta _{\mathfrak {p}}^2 \in {\mathcal {O}}_{\mathfrak {p}}$ .

Finally, we let $L_H$ denote the torus of $H({\mathbb {Z}}_p)$ given by

$$\begin{align*}L_H = \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}}T_{(1,0)} \times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} T_{\delta_{\mathfrak{p}}}, \end{align*}$$

where $T_{(1,0)}$ denotes the conjugateFootnote ⁶ of the maximal split torus of $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}})$ given by

$$\begin{align*}\{ \left(\begin{smallmatrix}a & \\a-d & d\end{smallmatrix}\right)\in \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{p}})\}, \end{align*}$$

while $T_{\delta _{\mathfrak {p}}}$ denotes the maximal non-split torus of $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}})$ given by

$$\begin{align*}\{ \left(\begin{smallmatrix}a & c\delta_{\mathfrak{p}}^2 \\c & a\end{smallmatrix}\right)\,:\, (a,c) \in {\mathcal{O}}_{\mathfrak{p}}^2 \text{ such that } a+c\delta_{\mathfrak{p}} \in {\mathcal{O}}_{\mathfrak{P}}^\times \}. \end{align*}$$

Proof. We start with calculating the stabilizer of $u \overline {B}_G$ , which is

$$ \begin{align*}\mathrm{Stab}_{H} (u \overline{B}_G) = \iota( H) \cap u \overline{B}_G u^{-1}.\end{align*} $$

Let $\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}$ , then $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}}) \ni g={\matrix {a} { b} {c} {d}}$ fixes $[u_{\mathfrak {p}}]$ if g and $ {\matrix {1 } { -1} {} {1 }} \cdot g \cdot {\matrix {1 } {1} { } { 1}}$ are lower-triangular. As

$$\begin{align*}u_{\mathfrak{p}}^{-1} \cdot g \cdot u_{\mathfrak{p}} = \left({\matrix {a-c} {a+b-c-d} {c} {d+c}}, {\matrix {a} {b} {c} {d}} \right) ,\end{align*}$$

we deduce that $g ={\matrix {a} {} { a-d} {d}} $ . Therefore any element in $ u_{\mathfrak {p}}^{-1} \mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}}) u_{\mathfrak {p}} \cap \overline {B}_{\mathrm {GL}_2}({\mathcal {O}}_{\mathfrak {p}})^2$ is of the form $\left ( {\matrix {d} { } { a-d} {a}}, {\matrix {a} { } {a-d} {d}}\right )$ . In other words, $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}}) \cap u_{\mathfrak {p}} \overline {B}_{\mathrm {GL}_2}({\mathcal {O}}_{\mathfrak {p}})^2 u_{\mathfrak {p}}^{-1} = \iota _{\mathfrak {p}}(T_{(1,0)})$ .

Let $\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}$ . Keeping the notation as above,

$$\begin{align*}u_{\mathfrak{p}}^{-1} \cdot {\matrix {a} {b } {c} {d}} \cdot u_{\mathfrak{p}} = {\matrix { a - \delta_{\mathfrak{p}} c} {\delta_{\mathfrak{p}}(a-d) + b - \delta_{\mathfrak{p}}^2 c} {c} {d + \delta_{\mathfrak{p}} c}} \in \overline{B}_{\mathrm{GL}_2}({\mathcal{O}}_{\mathfrak{P}}) \end{align*}$$

implies that $b= \delta _{\mathfrak {p}}^2 c $ and that $d= a$ . Thus, in this case $g \in T_{\delta _{\mathfrak {p}}}$ . Hence $ \mathrm {GL}_2({\mathcal {O}}_{\mathfrak {p}}) \cap u_{\mathfrak {p}} \overline {B}_{\mathrm {GL}_2}({\mathcal {O}}_{\mathfrak {P}}) u_{\mathfrak {p}}^{-1}= \iota _{\mathfrak {p}}( T_{\delta _{\mathfrak {p}}} ) $ .

Summing up, we deduce that $\mathrm {Stab}_{H} (u \overline {B}_G)({\mathbb {Z}}_p) = \iota (L_H )$ . Finally, notice that the H-orbit of u is open for the Zariski topology as its dimension equals the dimension of $\mathcal {F}$ .

Proof. This follows directly from the proof of the lemma above. For instance, at an inert prime $\mathfrak {p}$ , if $g=\left (\begin {smallmatrix} a & c \delta _{\mathfrak {p}}^2 \\ c & a \end {smallmatrix}\right ) \in \iota _{\mathfrak {p}}( T_{\delta _{\mathfrak {p}}} )$ , then

$$\begin{align*}u_{\mathfrak{p}}^{-1} g u_{\mathfrak{p}} = {\matrix { a - \delta_{\mathfrak{p}} c} {} {c} {a + \delta_{\mathfrak{p}} c}}, \end{align*}$$

which is lower-triangular, with diagonal elements conjugate to each other as desired.

In the course of the proof of Theorem 5.4, we will need an explicit description of the action of $u^{-1}$ on the invariant vector $\Delta ^{\underline {k}} \in V^{\underline {k},0}_{\mathcal {O}_{\mathfrak {m}}}$ introduced in Formula (3). We record it here. The element u or its inverse acts on the $G(\mathcal {O}_{\mathfrak {m}})$ -lattice $V^{\underline {k},0}_{\mathcal {O}_{\mathfrak {m}}}$ via the embedding

$$\begin{align*}j: G({\mathbb{Z}}_p) = \mathrm{GL}_2( \mathcal{O}_E \otimes_{\mathbb{Z}} {\mathbb{Z}}_p) \hookrightarrow G(\mathcal{O}_{\mathfrak{m}}) = \mathrm{GL}_2(\mathcal{O}_E \otimes_{\mathbb{Z}} \mathcal{O}_{\mathfrak{m}}) \simeq \mathrm{GL}_2(\mathcal{O}_{\mathfrak{m}})^{2d}, \end{align*}$$

which we now recall. Write ${\mathbb {Z}}_p \otimes _{\mathbb {Z}} \mathcal {O}_E = \prod _{ \mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}} \mathcal {O}^2_{\mathfrak {p}} \times \prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}} \mathcal {O}_{\mathfrak {P}}$ as above, so that

$$\begin{align*}G({\mathbb{Z}}_p)= \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}} \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{p}})^2 \times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} \mathrm{GL}_2({\mathcal{O}}_{\mathfrak{P}}). \end{align*}$$

At each $\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}$ , then j is induced by the natural embedding of ${\mathcal {O}}_{\mathfrak {p}} \hookrightarrow {\mathcal {O}}_{\mathfrak {p}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}} \simeq {\mathcal {O}}_{\mathfrak {m}}^{d_{\mathfrak {p}}}$ . At each $\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}$ , j is also induced by the embedding $\mathrm {GL}_2({\mathcal {O}}_{\mathfrak {P}}) \hookrightarrow \mathrm {GL}_2({\mathcal {O}}_{\mathfrak {P}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}})$ together with the isomorphism ${\mathcal {O}}_{\mathfrak {P}} \otimes _{{\mathcal {O}}_{\mathfrak {p}}} ({\mathcal {O}}_{\mathfrak {p}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}}) \simeq ({\mathcal {O}}_{\mathfrak {p}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}})^2$ given by sending $\alpha \otimes \beta \mapsto (\beta i_{\mathfrak {P}}(\alpha ), \beta i_{\mathfrak {P}}(\bar {\alpha }))$ , with $i_{\mathfrak {P}} :{\mathcal {O}}_{\mathfrak {P}} \hookrightarrow {\mathcal {O}}_{\mathfrak {p}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}}$ , and the isomorphism ${\mathcal {O}}_{\mathfrak {p}} \otimes _{{\mathbb {Z}}_p} {\mathcal {O}}_{\mathfrak {m}} \simeq {\mathcal {O}}_{\mathfrak {m}}^{d_{\mathfrak {p}}}$ . Note that the sum of the $d_{\mathfrak {p}}$ ’s is d, and when p is totally split in F then $d_{\mathfrak {p}} =1$ for every $\mathfrak {p} \in \Upsilon _{F,p}$ .

Note that, for each $\mathfrak {p} \in \Upsilon _{F,p}$ , $\Sigma _{\mathfrak {p}}$ and $\Phi _{\mathfrak {p}}$ consist of $d_{\mathfrak {p}}$ elements.

At each $\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}$ , $u^{-1}_{\mathfrak {p}}$ acts on the components in $\Phi _{\mathfrak {p}}$ of $ \Delta ^{\underline {k}}$ via the action of $\left ( {\matrix {1} {-1} { } {1}}, {\matrix {1} {} { } {1}} \right )$ , which sends $\Delta _\sigma =(e_{1,\sigma }e_{2,\sigma '}-e_{2,\sigma }e_{1,\sigma '})$ to $(e_{1,\sigma }e_{1,\sigma '})+ \Delta _\sigma $ and so, taking the $k_\sigma $ tensor product, it sends $\Delta _\sigma ^{k_\sigma }=(e_{1,\sigma }e_{2,\sigma '}-e_{2,\sigma }e_{1,\sigma '})^{\otimes {k_\sigma }}$ to $(e_{1,\sigma }e_{1,\sigma '})^{\otimes {k_\sigma }} + R_\sigma ^{k_\sigma }$ , where $R_\sigma ^{k_\sigma }$ has zero projection to the highest weight eigenspace.

At each $\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}$ , recall we have fixed $\delta _{\mathfrak {p}} \in {\mathcal {O}}_{\mathfrak {P}}$ such that ${\mathcal {O}}_{\mathfrak {P}} = {\mathcal {O}}_{\mathfrak {p}} \oplus {\mathcal {O}}_{\mathfrak {p}} \delta _{\mathfrak {p}}$ and $\delta _{\mathfrak {p}}^2 \in {\mathcal {O}}_{\mathfrak {p}}$ . Then $u^{-1}_{\mathfrak {p}}$ acts on the components in $\Phi _{\mathfrak {p}}$ of $ \Delta ^{\underline {k}}$ via the action of $\left ( {\matrix {1} {-i_{\mathfrak {P}}(\delta _{\mathfrak {p}})} { } {1}}, {\matrix {1} {i_{\mathfrak {P}}(\delta _{\mathfrak {p}})} { } {1}} \right )$ , which sends $\Delta _\sigma $ to $(2 i_{\mathfrak {P}}(\delta _{\mathfrak {p}}))(e_{1,\sigma }e_{1,\sigma '}) + \Delta _\sigma $ . Summing up, we have shown the following:

Lemma 4.6. We have that

$$\begin{align*}u^{-1} \cdot \Delta^{\underline{k}} = a_{\underline{k}} \cdot v_{\underline{k}}^{\mathrm{hw}} + R^{\underline{k}}, \end{align*}$$

where $v_{\underline {k}}^{\mathrm {hw}}$ is the highest weight vector in $V^{\underline {k},0}_{\mathcal {O}_{\mathfrak {m}}}$ , $a_{\underline {k}} = \prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}} (2 i_{\mathfrak {P}}(\delta _{\mathfrak {p}}))^{\sum _{\sigma \in \Phi _{\mathfrak {p}}} k_\sigma } \in {\mathcal {O}}_{\mathfrak {m}}^\times $ , and $R^{\underline {k}} \in V^{\underline {k},0}_{\mathcal {O}_{\mathfrak {m}}}$ has no $v_{\underline {k}}^{\mathrm {hw}}$ -component.

4.2 Level subgroups and Hecke operators

Let $\eta $ denote the diagonal matrix ${\matrix {p} { } {} {1}} \in G({\mathbb {Q}}_p)$ and let

$$\begin{align*}\Gamma_0(p^r) := \{g \in G({\mathbb{Z}}_p)\,:\, g\,[\text{mod }p^r] \in B_G({\mathbb{Z}}/p^r{\mathbb{Z}})\} \end{align*}$$

be the Iwahori subgroup of $G({\mathbb {Z}}_p)$ . Similarly, $\Gamma ^0(p^r)$ denotes the opposite Iwahori subgroup of level $p^r$ of $G({\mathbb {Z}}_p)$ , while $\Gamma ^0_0(p^r,p^s)$ denotes the intersection of $\Gamma _0(p^r)$ with $\Gamma ^0(p^s)$ . Following [Reference Loeffler31], we define the following level subgroups.

Definition 4.7. Fix $K^{(p)} \subseteq G( { \hat {{\mathbb {Z}}} ^{(p)} } )$ a neat open compact subgroup and for $r\geq 0$ let

$$ \begin{align*} K_r &: = K^{(p)}\cdot \{ g \in \Gamma_0(p)\,:\, \eta^{-r} g \eta^r \in G ({\mathbb{Z}}_p),\; g\,[\text{mod }p^r] \in \overline{Q}_G({\mathbb{Z}}/p^r{\mathbb{Z}}) \}, \\ K_r' &: = K^{(p)}\cdot \{ g \in \Gamma_0(p)\,:\, \eta^{-r-1} g \eta^{r+1} \in G ({\mathbb{Z}}_p),\; g\,[\text{mod }p^r] \in \overline{Q}_G({\mathbb{Z}}/p^r{\mathbb{Z}}) \}, \\ V_r &:= \eta^{-r} K_r \eta^r, \\ V_r' &:= \eta^{-r-1} K_r' \eta^{r+1}. \end{align*} $$

Note that $ K_{r+1} \subseteq K_r' = K_r \cap \eta K_r \eta ^{-1} \subseteq K_r$ . This implies that right multiplication by $\eta $ gives a morphism

where $\mathrm {pr}$ denote natural projections.

We note that we can refine Definition 4.7 by defining towers of level subgroups for each subset of $ \Upsilon _{F,p}$ , as follows.

For each $\mathfrak {p} \in \Upsilon _{F,p}$ , let $\eta _{\mathfrak {p}} := \matrix {\pi _{\mathfrak {p}}}{}{}{1} \in \mathrm {GL}_2(F_{\mathfrak {p}})$ and denote in the same way the corresponding element of $H({\mathbb {Q}}_p)$ with trivial component at every other place in $\Upsilon _{F,p}$ . Moreover, for any $r \geq 0$ set $K_{\mathfrak {p},r}$ to be either

$$\begin{align*}\left\{(g_1,g_2) \in {\mathrm{GL}}_2(\mathcal{O}_{\mathfrak{p}})^2\,:\, (g_1,g_2) \equiv \left(\left(\begin{smallmatrix} a & \\ x_1 & d \end{smallmatrix} \right),\left(\begin{smallmatrix} d & \\ x_2 & a \end{smallmatrix} \right)\right)\,[\text{mod }\mathfrak{p}^r] \right\} \cap \Gamma_0(\pi_{\mathfrak{p}})^2\end{align*}$$

if $\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}$ , or

$$\begin{align*}\left\{g \in \mathrm{GL}_2(\mathcal{O}_{\mathfrak{P}})\,:\, g \equiv \left(\begin{smallmatrix} \alpha & \\ x & \tau(\alpha) \end{smallmatrix} \right)\,[\text{mod }\mathfrak{P}^r] \right\} \cap \Gamma_0(\pi_{\mathfrak{p}})\end{align*}$$

if $\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}$ , with $\mathfrak {P}$ denoting the prime of E above $\mathfrak {p}$ . We then let $K_{\mathfrak {p},r}' := K_{\mathfrak {p},r} \cap \eta _{\mathfrak {p}} K_{\mathfrak {p},r} \eta _{\mathfrak {p}}^{-1}$ , $V_{\mathfrak {p},r}:= \eta _{\mathfrak {p}}^{-r} K_{\mathfrak {p},r} \eta _{\mathfrak {p}}^r$ , and $V_{\mathfrak {p},r}':= \eta _{\mathfrak {p}}^{-r-1} K_{\mathfrak {p},r} \eta _{\mathfrak {p}}^{r+1}$ .

Definition 4.9. For any multi-index $\pmb {r} = (r_{\mathfrak {p}})_{\mathfrak {p}} \in {\mathbb {Z}}_{\geq 0}^{|\Upsilon _{F,p}|}$ , we define

$$ \begin{align*} K_{\pmb{r}}&:= K^{(p)} \cdot \prod_{\mathfrak{p} \in \Upsilon_{F,p}} K_{\mathfrak{p},r_{\mathfrak{p}}}, \\ V_{\pmb{r}}&:= K^{(p)} \cdot \prod_{\mathfrak{p} \in \Upsilon_{F,p}} V_{\mathfrak{p},r_{\mathfrak{p}}}, \end{align*} $$

and similarly for $K_{\pmb {r}}'$ and $V_{\pmb {r}}'$ .

When $\pmb {r} = (r)_{\mathfrak {p}}$ , then these level groups agree with those of Definition 4.7.

Later on, if $\mathfrak {S}$ is any subset of $ \Upsilon _{F,p}$ , we also consider

(9) $$ \begin{align} K_0(\mathfrak{S}) := K^{(p)} \cdot ( \prod_{\mathfrak{p} \in \mathfrak{S}} \Gamma^0_0(\pi_{\mathfrak{p}},\pi_{\mathfrak{p}}) \cdot \prod_{\mathfrak{p} \in \Upsilon_{F,p} \smallsetminus \mathfrak{S}} \Gamma_0( \pi_{\mathfrak{p}})). \end{align} $$

We have

$$\begin{align*}K_{\pmb{0}}' = K_0' \subseteq K_0(\mathfrak{S}) \subseteq K_{\pmb{0}} = K_0,\end{align*}$$

where the symbol $\pmb {0}$ denotes the multi-index $(0,\dots ,0)$ .

We now define certain Hecke operators that play a crucial role in the manuscript.

Lemma 4.10. There is a well-defined action of $p^{\sum _\sigma {(k_{\sigma }+m)/2}} \cdot \eta ^{-1}$ on $V_{\mathcal {O}_{\mathfrak {m}}}^{\underline {k},m}$ which induces a morphism of local systems

$$\begin{align*}\eta_\flat : \mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} \to \eta^*\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m}. \end{align*}$$

In particular we have a map

$$\begin{align*}\mathrm{Tr}_{\eta} : H^i(Y_G(K_{\pmb{r}}'),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ) \longrightarrow H^i(Y_G(K_{\pmb{r}}),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ) \end{align*}$$

defined by composing the map in cohomology induced by $\eta _\flat $ with the trace of $\mathrm {pr} \circ \eta : Y_G(K_{\pmb {r}}') \to Y_G(K_{\pmb {r}})$ in Betti cohomology.

Definition 4.11. Let the symbol $\pmb {1}$ denote the multi-index with all entries equal to $1$ . We define

$$\begin{align*}\mathrm{ Tr}_{[\eta]_{\pmb{r}+\pmb{1},\pmb{r}}}: H^i(Y_G(K_{\pmb{r}+\pmb{1}}),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ) \longrightarrow H^i(Y_G(K_{\pmb{r}}),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ) \end{align*}$$

as the composition of the trace of the natural projection $Y_G(K_{\pmb {r}+\pmb {1}}) \to Y_G(K_{\pmb {r}}')$ with $\mathrm {Tr}_\eta $ .

Note that $T_\eta $ is the correspondence associated with the Hecke operator $[K_{\pmb {r}} \eta ^{-1} K_{\pmb {r}}]$ in the Hecke algebra $\mathcal {H}(K_{\pmb {r}} \backslash G({\mathbb {A}}_f) / K_{\pmb {r}})_{\mathcal {O}_{\mathfrak {m}}}$ of $K_{\pmb {r}}$ -bi-invariant smooth compactly supported v-valued functions on $G({\mathbb {A}}_f)$ .

Remark 4.14. In the same analogous way, attached to each $\eta _{\mathfrak {p}}$ , we have the Hecke operator $T_{\eta _{\mathfrak {p}}}$ , which we normalize so that

$$\begin{align*}T_{\eta} = \prod_{\mathfrak{p} \in \Upsilon_{F,p}} T_{\eta_{\mathfrak{p}}}.\end{align*}$$

Concretely, we normalize $T_{\eta _{\mathfrak {p}}}$ as follows. Recall that, in Definition 4.5, we have fixed a partition of $\Sigma _F = \bigsqcup _{\mathfrak {p} \in \Upsilon _{F,p}} \Sigma _{\mathfrak {p}}$ . Then, we multiply the trace of $\eta _{\mathfrak {p}}$ by $p^{\sum _{\sigma \in \Sigma _{\mathfrak {p}}} (k_{\sigma } + k_{\sigma }^{\prime } + 2 m )/2}$ , where we identify $\Sigma _{\mathfrak {p}}$ with the subset $\Phi _{\mathfrak {p}}$ of the F-type $\Phi $ . Note that the (normalized) trace $\mathrm {Tr}_{[\eta ]_{\pmb {r}+\pmb {1},\pmb {r}}}$ is equal to the composition of the (normalized) traces

$$\begin{align*}\mathrm{Tr}_{[\eta_{\mathfrak{p}}]_{\pmb{r}+\pmb{1},{\pmb{r}}^{\prime}}}: H^i(Y_G(K_{\pmb{r}+\pmb{1}}),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ) \longrightarrow H^i(Y_G(K_{{\pmb{r}}^{\prime}}),\mathscr{V}_{\mathcal{O}_{\mathfrak{m}}}^{\underline{k},m} ),\end{align*}$$

with ${\pmb {r}}^{\prime }$ equal to $\pmb {r}+\pmb {1}$ away from $\mathfrak {p}$ and equal to $\pmb {r}$ at $\mathfrak {p}$ , as $\mathfrak {p}$ runs over $\Upsilon _{F,p}$ .

4.3 A few Cartesian diagrams

Let $u \in G({\mathbb {Z}}_p)$ denote the element of Definition 4.1. For any multi-index $\pmb {r} = (r_{\mathfrak {p}})_{\mathfrak {p}} \in {\mathbb {Z}}_{\geq 0}^{|\Upsilon _{F,p}|}$ , we have the morphism

Proof. Notice that it is enough to check it at the level of ${\mathbb {C}}$ -points. As right multiplication by u is clearly a bijection, the statement follows from showing that $\iota _{\pmb {r}}$ is injective. To ease the notation, let V denote $u K_{\pmb {r}}u^{-1}$ . Recall that, if we denote by $Y_H$ and $Y_G$ the varieties at infinite level, we have using [Reference Deligne11, (1.8.2)] and that the centre acts trivially

$$ \begin{align*}Y_H({\mathbb{C}}) = \tfrac{H({\mathbb{Q}})}{Z_H({\mathbb{Q}})} \backslash \left( X_H \times H({\mathbb{A}}_f) / \overline{Z_H({\mathbb{Q}})} \right),\end{align*} $$

where $\overline {Z_H({\mathbb {Q}})}$ denotes the closure of $Z_H({\mathbb {Q}})$ in $Z_H({\mathbb {A}}_f)$ and where $X_H$ is isomorphic to d copies of the union $\mathfrak {H} \cup \mathfrak {H}^{-}$ of upper and lower half planes. Similarly,

$$ \begin{align*}Y_G({\mathbb{C}}) = \tfrac{G({\mathbb{Q}})}{Z_G({\mathbb{Q}})} \backslash \left( X_G \times G({\mathbb{A}}_f) / \overline{Z_G({\mathbb{Q}})} \right),\end{align*} $$

with $X_G$ isomorphic to $2d$ copies of $\mathfrak {H} \cup \mathfrak {H}^{-}$ . Note that $\iota : H \to G$ induces an injection $Y_H({\mathbb {C}}) \hookrightarrow Y_G({\mathbb {C}})$ since, in this case,

$$ \begin{align*}\overline{Z_G({\mathbb{Q}})} \cap H({\mathbb{A}}_f) = \overline{Z_H({\mathbb{Q}})}.\end{align*} $$

The result thus follows from showing that $Y_H({\mathbb {C}}) \cap Y_H({\mathbb {C}})v = \emptyset $ for any $v \in V \smallsetminus V \cap H({\mathbb {A}}_f)$ . We show this fact as follows. Firstly, notice that the non-trivial element $\tau \in \mathrm {Gal}(E/F)$ induces a map $ \tau : Y_G({\mathbb {C}}) \to Y_G({\mathbb {C}})$ , which fixes $Y_H({\mathbb {C}})$ point-wise. Suppose now that $y,y v \in Y_H({\mathbb {C}})$ for $v \in V$ . Then $v^{-1} v^\tau $ fixes y. Since $v^{-1} v^\tau $ belongs to $u U u^{-1}$ and, by the neatness assumption, we deduce $v^{-1} v^\tau =1$ , so $v = v^\tau $ , which implies that $v \in H({\mathbb {A}}_f)$ , as desired.

Observe that, for instance, the hypothesis of Lemma 4.15 is satisfied whenever $K^{(p)}$ is stable under the non-trivial element $\tau \in \mathrm {Gal}(E/F)$ . From now on, we then assume that this is the case.

Proposition 4.17. Assume that $K^{(p)}$ is stable under $\tau \in \mathrm {Gal}(E/F)$ and let $\pmb {r} = (r_{\mathfrak {p}})_{\mathfrak {p}} \in {\mathbb {Z}}_{\geq 1}^{|\Upsilon _{F,p}|}$ . The commutative diagram

has Cartesian bottom square. Here $\pi _{\pmb {r}}^H, \pi _{\pmb {r}}^G$ denote the natural degeneracy maps.

Proof. To show that the diagram is Cartesian, it is enough to check it for the complex points of the varieties. Then the result follows from showing that

1. (1) $u \circ \iota _{\pmb {r}}$ and $\mathrm {pr} \circ u \circ \iota _{{\pmb {r}}+\pmb {1}}$ are closed immersions,

2. (2) The maps $\pi _{\pmb {r}}^H$ and $\pi _{\pmb {r}}^G$ are covering maps of the same degree.

By Lemma 4.15, $u \circ \iota _{\pmb {s}}$ is a closed immersion for any ${\pmb {s}}$ , therefore (1) follows from showing that $\mathrm {pr} \circ u \circ \iota _{{\pmb {r}}+\pmb {1}}$ is a closed immersion, i.e.

$$\begin{align*}u K^{\prime}_{\pmb{r}} u^{-1} \cap H = u K_{{\pmb{r}}+\pmb{1}} u^{-1} \cap H. \end{align*}$$

This follows as in the proof of [Reference Loeffler31, Lemma 4.4.1(i)] from the calculation of Lemma 4.2(2) applied modulo $p^{{\pmb {r}}+{\pmb {1}}}$ . Here, for any multi-index $\pmb {r}$ , we denote $p^{{\pmb {r}}}:= \prod _{\mathfrak {p}}\mathfrak {p}^{r_{\mathfrak {p}}}$ .

Notice that $K_{\pmb {r}}^{\prime } \cap Z_G({\mathbb {Q}}) = K_{\pmb {r}} \cap Z_G({\mathbb {Q}})$ . Indeed, if $\gamma \in K_{\pmb {r}}^{\prime } \cap Z_G({\mathbb {Q}})$ , then at p, $\eta ^{-\pmb {r}-\pmb {1}}\gamma \eta ^{\pmb {r}+\pmb {1}} = \gamma \in G({\mathbb {Z}}_p)$ , with $\eta ^{\pmb {s}}:=\prod _{\mathfrak {p}} \eta _{\mathfrak {p}}^{s_{\mathfrak {p}}}$ , and $ \gamma \,[\text {mod }p^{\pmb {r}}] \in \overline {Q}_G$ . These are exactly the conditions at p characterizing the intersection $K_{\pmb {r}} \cap Z_G({\mathbb {Q}})$ . Hence, by Lemma 4.18, the degrees of the maps $\pi _{\pmb {r}}^H$ and $\pi _{\pmb {r}}^G$ are equal to $[u K_{\pmb {r}} u^{-1} \cap H : u K_{\pmb {r}}^{\prime } u^{-1} \cap H ]$ and $[K_{\pmb {r}}: K_{\pmb {r}}^{\prime }]$ respectively. We are thus left to show that

$$\begin{align*}[u K_{\pmb{r}} u^{-1} \cap H : u K_{\pmb{r}}^{\prime} u^{-1} \cap H ] = [K_{\pmb{r}}: K_{\pmb{r}}^{\prime}]. \end{align*}$$

This follows as in the proof of [Reference Loeffler31, Lemma 4.4.1(ii)] from the fact that the H-orbit of $u \overline {B}_G$ is open as a ${\mathbb {Z}}_p$ -subscheme of $\mathcal {F}$ (cf. Lemma 4.2(1)).

Proposition 4.17 does not follow directly from [Reference Loeffler31] as our Shimura data do not satisfy the Axiom SV5. However, to circumvent this obstruction it suffices to use the following:

Remark 4.19. The Cartesian diagram of Proposition 4.17 is the composition of Cartesian diagrams involving each $T_{\eta _{\mathfrak {p}}}$ separately as $\mathfrak {p}$ varies in $\Upsilon _{F,p}$ . For each $\mathfrak {p}$ , the corresponding Cartesian diagram is

where ${\pmb {r}}^{\prime }$ is given by $r_{\mathfrak {q}}^{\prime } = r_{\mathfrak {q}}$ for $\mathfrak {q} \ne \mathfrak {p}$ and $r_{\mathfrak {p}}^{\prime } = r_{\mathfrak {p}} +1$ , $\mathrm {pr}^{\prime }$ denotes the natural projection $Y_G(K_{{\pmb {r}}^{\prime }}) \to Y_G(K_{\pmb {r}}^{(\mathfrak {p})} \cdot K_{\mathfrak {p},r_{\mathfrak {p}}}^{\prime })$ , and vertical arrows are the natural degeneracy maps.

Assume now that $\pmb {r}=\pmb {0}$ . As the level subgroups $K_{\pmb {0}}, K_{\pmb {0}}^{\prime }, $ and $K_{\pmb {1}}$ agree with $K_{0}, K_{0}^{\prime }, $ and $K_{1}$ , we adopt the latter notation. Note that in this case the bottom square of the diagram of Proposition 4.17 is not Cartesian as the degree of $\pi _{0}^H:Y_H(uK_{1}u^{-1} \cap H) \to Y_H(K_{0} \cap H)$ is strictly smaller than $\pi _{0}^G: Y_G(K_{0}^{\prime }) \to Y_G(K_{0})$ , as the next lemma shows.

Proof. We first note that, as in the proof of Proposition 4.17, we have

$$\begin{align*}u K^{\prime}_0 u^{-1} \cap H = u K_{1} u^{-1} \cap H. \end{align*}$$

Thus, by Lemma 4.18, the degree of $\pi _0^H$ equals to $[ K_0 \cap H : u K^{\prime }_0 u^{-1} \cap H]$ . To calculate the latter, since u is trivial outside p, it is enough to do the calculation prime by prime above p.

If $\mathfrak {p}$ is a prime of F above p which is split in E, the component at $\mathfrak {p}$ of $K_0 \cap H$ is given by $\Gamma _{\mathrm {GL}_2,0}(\pi _{\mathfrak {p}}) = \{ g={\matrix {a} { b} {c} {d}} \in \mathrm {GL}_2(\mathcal {O}_{\mathfrak {p}})\,:\,\pi _{\mathfrak {p}} \mid c \}$ , while the one of $ u K^{\prime }_0 u^{-1} \cap H$ is given by matrices $g={\matrix {a} { b} {c} {d}} \in \mathrm {GL}_2(\mathcal {O}_{\mathfrak {p}})$ with $\pi _{\mathfrak {p}} \mid c$ , $\pi _{\mathfrak {p}} \mid b$ , and $\pi _{\mathfrak {p}} \mid a-d$ (cf. proof of Lemma 4.2). The latter subgroup has index $(q_{\mathfrak {p}}-1)q_{\mathfrak {p}}$ in $\Gamma _{\mathrm {GL}_2,0}(\pi _{\mathfrak {p}})$ .

Similarly, when $\mathfrak {p}$ is inert in E the image of $u K_0^{\prime } u^{-1} \cap H$ in $\mathrm {GL}_2(\mathbb {F}_{q_{\mathfrak {p}}})$ is the centre, implying that the $\mathfrak {p}$ -component of $u K_0^{\prime } u^{-1} \cap H$ has index in $\Gamma _{\mathrm {GL}_2,0}(\pi _{\mathfrak {p}})$ equal to $q_{\mathfrak {p}}^2-q_{\mathfrak {p}}$ . Thus we have that $\pi _0^H$ has degree $\prod _{\mathfrak {p} \in \Upsilon _{F,p}} (q_{\mathfrak {p}}^2-q_{\mathfrak {p}})$ .

The degree of $\pi _0^G$ , by Lemma 4.18, equals to $[K_0 : K_0^{\prime }]$ . Since the p-component of $K_0^{\prime }$ is $\Gamma _0^0(p,p)$ , its $\mathfrak {p}$ -component has index in the Iwahori subgroup of $\mathrm {GL}_2(\mathcal {O}_{\mathfrak {p}})^2$ for $\mathfrak {p}$ split, resp. $\mathrm {GL}_2(\mathcal {O}_{\mathfrak {P}})$ for $\mathfrak {p}$ inert, equal to $q_{\mathfrak {p}}^2$ . Thus, we have that $[K_0 : K_0^{\prime }] = \prod _{\mathfrak {p} \in \Upsilon _{F,p}} q_{\mathfrak {p}}^2 $ .

Note the difference of the two degrees at each prime $\mathfrak {p}$ above p is $q_{\mathfrak {p}}$ , which is exactly the index of the Hecke correspondence $U_{\pi _{\mathfrak {p}}}$ at $\mathfrak {p}$ associated with $[\Gamma ^0_H(\pi _{\mathfrak {p}}){\matrix {\pi _{\mathfrak {p}}}{}{}{1}}\Gamma ^0_H(\pi _{\mathfrak {p}}) ]$ , with $\pi _{\mathfrak {p}}$ the uniformizer in $\mathcal {O}_{\mathfrak {p}}$ . A partial explanation of this phenomenon is given by the following proposition. Before that, we need to introduce some notation.

Let $\mathfrak {S}$ be a subset of $\Upsilon _{F,p}$ . Let $u_{\mathfrak {S}} \in G({\mathbb {Z}}_p)$ denote the element $(\gamma _{\mathfrak {p}})_{\mathfrak {p} \in \Upsilon _{F,p}}$ given by

$$\begin{align*}\gamma_{\mathfrak{p}} = \begin{cases} u_{\mathfrak{p}} & \text{ if } \mathfrak{p} \in \mathfrak{S}, \\ 1 & \text{ if } \mathfrak{p} \not\in \mathfrak{S}, \end{cases} \end{align*}$$

with $u_{\mathfrak {p}}$ given in Definition 4.1. Moreover, define

$$\begin{align*}K^0_H(\mathfrak{S}):= (K^{(p)} \cap H(\hat{{\mathbb{Z}}}^{(p)}))\cdot( u_{\mathfrak{S}}\Gamma^0_0(p,p)u_{\mathfrak{S}}^{-1} \cap H({\mathbb{Z}}_p)),\end{align*}$$

where we recall that $\Gamma ^0_0(p,p)$ is the p-component of $K_0^{\prime }$ . Finally, denote by $\tilde {\iota }_{\mathfrak {S}}: Y_H(K^0_H(\mathfrak {S})) \to Y_G(K_0^{\prime })$ the closed immersion induced by $u_{\mathfrak {S}} \circ \iota $ . A few remarks are in order.

The following lemma is needed in the proof of Proposition 4.23 and makes explicit the open and closed H-orbits on the flag $\mathcal {F}$ given in Lemma 4.2.

Proof. We start with the split case. Recall that $\mathbb {P}^1=\mathrm {GL}_2/\overline {B}_{\mathrm {GL}_2}$ . The action of $\mathrm {GL}_2$ is given by left multiplication:

$$\begin{align*}{\matrix{a}{b}{c}{d}} [x:y]^T=[ax+by:cx+dy]^T. \end{align*}$$

So we have that the open orbit is

$$\begin{align*}\left({\matrix{a}{a+b}{c}{c+d}},{\matrix{a}{b}{c}{d}} \right)([0:1]^T,[0:1]^T)=([a+b:c+d]^T,[b:d]^T) \end{align*}$$

and the closed one is

$$\begin{align*}\left( {\matrix{a}{b}{c}{d}},{\matrix{a}{b}{c}{d}} \right)([0:1]^T,[0:1]^T)=([b:d]^T,[b:d]^T) \end{align*}$$

In the inert case we have the open orbit

$$\begin{align*}{\matrix{a}{a\delta_{\mathfrak{p}} +b}{c}{c\delta_{\mathfrak{p}} + d}}[0:1]^T=[a\delta_{\mathfrak{p}} +b: c\delta_{\mathfrak{p}} + d]^T. \end{align*}$$

One sees that the $\delta _{\mathfrak {p}}$ -coefficient of the ratio $a\delta _{\mathfrak {p}} +b/ (c\delta _{\mathfrak {p}} + d)$ is $0$ if and only if $ad-bc=0$ , which is never the case. The closed orbit is instead

$$\begin{align*}{\matrix{a}{b}{c}{d}}[0:1]^T=[b: d]^T. \\[-33pt] \end{align*}$$

We can finally state the equivalent of Proposition 4.17 for when $r=0$ .

Proposition 4.23. Assume that $K^{(p)}$ is stable under $\tau \in \mathrm {Gal}(E/F)$ . The commutative diagram

is Cartesian. Moreover, it can be decomposed as a series of Cartesian diagrams indexed by the primes in $\Upsilon _{F,p}$ : if $\mathfrak {S}$ is a (possibly empty) subset of $ \Upsilon _{F,p}$ and $\mathfrak {p} \not \in \mathfrak {S}$ , then, denoting $\mathfrak {S}^{\prime } : = \mathfrak {S} \cup \{ \mathfrak {p}\}$ ,

is Cartesian, where $K_0(\mathfrak {S})$ is the level subgroup introduced in (9).

Proof. We mimic the proof of Proposition 4.17. By Remark 4.21(3), the degree of the map $(\mathrm { pr}_{\mathfrak {S}}^H)_{\mathfrak {S}}$ equals to

$$\begin{align*}\sum_{\mathfrak{S}} [ \prod_{\mathfrak{p} \not \in \mathfrak{S}} q_{\mathfrak{p}} \cdot \prod_{\mathfrak{p} \in \mathfrak{S}} (q_{\mathfrak{p}}^2-q_{\mathfrak{p}})] = \prod_{\mathfrak{p} \in \Upsilon_{F,p}} q_{\mathfrak{p}}^2, \end{align*}$$

where the last equality follows from the binomial theorem. This implies that the degree of $(\mathrm {pr}_{\mathfrak {S}}^H)_{\mathfrak {S}}$ equals to the degree of $\pi _0^G$ . To prove the result we are left to show that

is a closed immersion, where we recall that $\tilde {\iota }_{\mathfrak {S}} = u_{\mathfrak {S}} \circ \iota $ . This follows from the fact that each $\tilde {\iota }_{\mathfrak {S}}$ is a closed immersion and that the images of any two sub-varieties do not intersect in $Y_G(K_0^{\prime })$ . The first fact is clear (cf. Lemma 4.15), so we are left to show that the images do not intersect. This follows roughly from the fact that the number of orbits of H in the flag $\mathcal {F}=G/\overline {B}_G$ is in bijection with the subsets $\mathfrak {S}$ of $\Upsilon _{F,p}$ , with the open orbit corresponding to the choice of $\mathfrak {S}=\Upsilon _{F,p}$ . Precisely, note that the fibre of the natural degeneracy map $\pi _{\mathfrak {p}}^G$ on the right can be identified with

$$\begin{align*}\Gamma_0(p)/\Gamma_0^0(p,p)& \cong B_G(\mathbb{F}_p) / T_G(\mathbb{F}_p) \cong N_G(\mathbb{F}_p) \\ &\cong \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}} \left(N_{\mathrm{GL}_2}(\mathbb{F}_{\mathfrak{p}})\times N_{\mathrm{GL}_2}(\mathbb{F}_{\mathfrak{p}}) \right)\times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} N_{\mathrm{GL}_2}(\mathbb{F}_{\mathfrak{P}}), \end{align*}$$

for $\mathfrak {P}$ the only prime in E above $\mathfrak {p}$ . Moreover, as $N_G$ can be realized as an open affine of $\mathcal {F}$ , we can view the right-hand side explicitly as the subspace of $\prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}} \left (\mathbb {P}^1(\mathbb {F}_{\mathfrak {p}})\times \mathbb {P}^1(\mathbb {F}_{\mathfrak {p}}) \right )\times \prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}} \mathbb {P}^1(\mathbb {F}_{\mathfrak {P}})$ given by

$$\begin{align*}\prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}} \{ \left ([a:1],[a^{\prime}:1] \right ),\, a,a^{\prime} \in \mathbb{F}_{\mathfrak{p}} \} \times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} \{ [b:1],\, b \in \mathbb{F}_{\mathfrak{P}} \}\, (=: \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{s}}_{F,p}} N_{\mathfrak{p}}^{\mathrm{s}} \times \prod_{\mathfrak{p} \in \Upsilon^{\mathrm{i}}_{F,p}} N_{\mathfrak{p}}^{\mathrm{i}}). \end{align*}$$

We are left to show that the $\mathfrak {p}$ -component of any element in the image of $Y_H(K_0^H(\mathfrak {S}))$ is characterized by landing into the closed, resp. open, orbit for $\mathrm {GL}_2(\mathbb {F}_{\mathfrak {p}})$ when $\mathfrak {p} \not \in \mathfrak {S}$ , resp. $\mathfrak {p} \in \mathfrak {S}$ . This follows from our definition of $u_{\mathfrak {S}}$ and the explicit calculation of Lemma 4.22: indeed, when $\mathfrak {p} \in \mathfrak {S}$ , the image of the $\mathfrak {p}$ -component of $\Gamma _{H,0}(p)$ into the $\mathfrak {p}$ -component of $\Gamma _0(p)/\Gamma _0^0(p,p)$ equals to $t_{\mathfrak {p}}(B_{\mathrm {GL}_2}(\mathbb {F}_{\mathfrak {p}}))$ , which realizes the intersection of the open orbit with $N_{\mathfrak {p}}^{\mathrm {s}}$ , resp. $N_{\mathfrak {p}}^{\mathrm {i}}$ , if $\mathfrak {p}$ is split, resp. inert. When $\mathfrak {p} \not \in \mathfrak {S}$ , it equals to $\iota _{\mathfrak {p}}(B_{\mathrm {GL}_2}(\mathbb {F}_{\mathfrak {p}}))$ , which gives the intersection of the closed orbit with $N_{\mathfrak {p}}^{\mathrm {s}}$ , resp. $N_{\mathfrak {p}}^{\mathrm {i}}$ , if $\mathfrak {p}$ is split, resp. inert. Therefore, if $\mathfrak {S},\mathfrak {S}^{\prime }$ are two distinct subsets, then the images of $Y_H(K^0_H(\mathfrak {S}))$ and $Y_H(K^0_H(\mathfrak {S}^{\prime }))$ do not intersect, which proves our first claim. The second part of the statement follows in the same way.

The interested reader might wonder what happens if we build our tower of level subgroups, starting from spherical level at p instead of $\Gamma _0(p)$ , as done in [Reference Loeffler31]. The resulting Cartesian diagrams, which are almost identical to those given above, can be used to relate the cycles with spherical level at p with the twisted cycles in the p-tower of level subgroups. While the diagram of Proposition 4.17 translates verbatim, a few small modifications are required in the $r=0$ case. We conclude this section by illustrating it. Let us denote

• • $K_{\mathrm {un},0} := K^{(p)}\cdot G({\mathbb {Z}}_p)$ , $K_{\mathrm {un},0}^{\prime } := K^{(p)}\cdot \Gamma ^0(p)$ ;

• • for any subset $\mathfrak {S}$ of $\Upsilon _{F,p}$ ,

  $$\begin{align*}K^0_{\mathrm{un},H}(\mathfrak{S}):= (K^{(p)} \cap H(\hat{{\mathbb{Z}}}^{(p)}))\cdot( u_{\mathfrak{S}}\Gamma^0(p)u_{\mathfrak{S}}^{-1} \cap H({\mathbb{Z}}_p)).\end{align*}$$

Lemma 4.24. The natural degeneracy map $\mathrm {pr}_{\mathrm {un},\mathfrak {S}}^H:Y_H(K^0_{\mathrm {un},H}(\mathfrak {S})) \to Y_H(K_{un,0} \cap H)$ has degree

$$\begin{align*}\prod_{\mathfrak{p} \not \in \mathfrak{S}} (q_{\mathfrak{p}} + 1) \cdot \prod_{\mathfrak{p} \in \mathfrak{S} \cap \Upsilon^{\mathrm{s}}_{F,p}} (q_{\mathfrak{p}}^2+q_{\mathfrak{p}}) \cdot \prod_{\mathfrak{p} \in \mathfrak{S} \cap \Upsilon^{\mathrm{i}}_{F,p}} q_{\mathfrak{p}}(q_{\mathfrak{p}}-1)\end{align*}$$

and the natural projection $\pi _{\mathrm {un},0}^G:Y_G(K_{\mathrm {un},0}^{\prime }) \to Y_G(K_{\mathrm {un},0})$ has degree $\prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {s}}_{F,p}} (q_{\mathfrak {p}}+1)^2 \times \prod _{\mathfrak {p} \in \Upsilon ^{\mathrm {i}}_{F,p}} (q^2_{\mathfrak {p}}+1)$