to replacing $\overline{U}_i$ with an affine open subscheme still containing $y$, we may and we shall assume that $a_{\overline{U}_i, E(\sigma_i)}^{\mathfrak{h}} : \overline{U}_i \to E(\sigma_i) \times_{\text{Spec}(\mathbb{Z})} C$ is induced by an étale morphism $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}} : \overline{U}_i \to \Xi(\sigma_i)$ such that, over the image of this étale morphism, $\Xi(\sigma_i)$ is isomorphic to $E(\sigma_i) \times_{\text{Spec}(\mathbb{Z})} C$ as schemes over $C$ (cf. [43, Lem. 2.3]), and such that $a_{\overline{U}_i, E(\sigma_i)}^{\mathfrak{h}}$ and $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$ differ by this last isomorphism. Then $\overline{U}_i^+ \cong a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$.

Consider the pullbacks $a_{\overline{U}_i, E(\sigma_i)}^{\mathfrak{h}} : \overline{U}_i' \to \Xi(\sigma_i)^+ := \bigcup_{\tau \in \Sigma_Z^+ : \tau \not\subset \sigma_i} \Xi_\tau \subset \Xi(\sigma_i)$ under $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$. Then $\overline{U}_i^+ \cong a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$ coincides with the pullback of $\Xi(\sigma_i)^+ := \bigcup_{\tau \in \Sigma_Z^+ : \tau \not\subset \sigma_i} \Xi_\tau \subset \Xi(\sigma_i)$ under $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$. Let $a_{\overline{U}_i, E(\sigma_i)}^{\mathfrak{h}} : \overline{U}_i \times_{\text{X}_{\overline{H}, \Sigma}^{\text{tor}}} \text{X}_{\overline{H}', \Sigma'}^{\text{tor}} \to \text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$ and $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}} : \overline{U}_i' \times_{\text{X}_{\overline{H}, \Sigma}^{\text{tor}}} \text{X}_{\overline{H}', \Sigma'}^{\text{tor}} \to \Xi(\sigma_i)^+ \to \Xi(\sigma_i)'$ of $a_{\overline{U}_i, \Xi(\sigma_i)}^{\mathfrak{h}}$, respectively, where $\sigma_i' \in \Sigma_{Z'}^+$ is some cone such that $x$ lies on stratum $\mathbb{Z}_{[\sigma_i]}$ of $\text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$. Let $\overline{U}_i' := \overline{U}_i \times_{\text{X}_{\overline{H}, \Sigma}^{\text{tor}}} \text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$, $\overline{U}_i'^{+} := \overline{U}_i^+ \times_{\text{X}_{\overline{H}, \Sigma}^{\text{tor}}} \text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$, $\overline{U}_i'' := \overline{U}_i \times_{\Xi(\sigma_i)} \Xi(\sigma_i)'$, and $\overline{U}_i''^{+} := \overline{U}_i^+ \times_{\Xi(\sigma_i)} \Xi(\sigma_i)'$. Then $\overline{U}_i'^{+}$ coincides with the pullback of $\tilde{Z}'$, and $\overline{U}_i''^{+}$ coincides with the pullback of $\Xi'(\sigma_i)' := \bigcup_{\tau' \in \Sigma_{Z'}^+ : \tau' \not\subset \sigma_i'} \Xi_{\tau'}' \subset \Xi'(\sigma_i)'$, and the induced morphisms $\overline{U}_i'^{+} \to \tilde{Z}'$ and $\overline{U}_i''^{+} \to \Xi'(\sigma_i)'$ are open immersions with the same image, the preimage of $\overline{U}_i^+$ in $\tilde{Z}'$. Therefore, $(\overline{U}_i')^{\wedge}_{\overline{U}_i'^{+}}$ and $(\overline{U}_i'')^{\wedge}_{\overline{U}_i''^{+}}$ are both canonically isomorphic to $(\text{X}_{\overline{H}', \Sigma'}^{\text{tor}})^{\wedge}_{\overline{U}_i'^{+}}$. Moreover, for each affine open formal subscheme $\mathcal{W} = \text{Spf}(R)$ of $(\text{X}_{\overline{H}', \Sigma'}^{\text{tor}})^{\wedge}_{\overline{U}_i''^{+}}$, the pullbacks to $W = \text{Spec}(R)$ of the stratifications of $\text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$ and $\Xi'(\sigma_i')$ coincide with each other.

By the same approximation argument as in the proofs of (7) of Theorem 2.3.2 and of Lemma 4.1.2, there exists a scheme $\overline{U}_i'''$ with étale morphisms $\overline{U}_i''' \to \overline{U}_i'$ and $\overline{U}_i''' \to \overline{U}_i''$ such that the induced étale morphisms $a_{\overline{U}_i'''} : \overline{U}_i''' \to \text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$ and $a_{\overline{U}_i'''} : \overline{U}_i''' \to \Xi'(\sigma_i')$ satisfy the same properties as the étale morphisms $\overline{U} \to \text{X}_{\overline{H}, \Sigma}^{\text{tor}}$ and $\overline{U} \to \Xi(\sigma)$ in (9) of Proposition 2.1.2 do, and are adapted to $\mathcal{Y}'$ in the sense that the pullback of $\mathcal{Y}'_{\Sigma'}^{\text{tor}}$ under $a_{\overline{U}_i'''}$ coincides with the pullback of $\mathcal{Y}'_{\tilde{Z}'}^{\mathfrak{h}}$ under $a_{\overline{U}_i'''}^{\mathfrak{h}}, \Xi'(\sigma_i')$; and such that the image of $a_{\overline{U}_i'''}$ contains $x$, so that $x$ is lifted to a point of $\overline{U}_i''$. By [43, Lem. 2.3, and its proof], up to replacing $\overline{U}_i'''$ with an affine open subscheme still containing $x$, we may and we shall assume that, over the image of $a_{\overline{U}_i'''}^{\mathfrak{h}}, \Xi'(\sigma_i')$, there is an isomorphism between $\Xi'(\sigma_i')$ and $E'(\sigma_i') \times_{\text{Spec}(\mathbb{Z})} C'$ as schemes over $C'$, which lifts the pullback of the isomorphism between $\Xi(\sigma_i)$ and $E(\sigma_i) \times_{\text{Spec}(\mathbb{Z})} C$ over the image of $a_{\overline{U}_i, \Xi'(\sigma_i')}^{\mathfrak{h}}$, and hence $a_{\overline{U}_i''', \Xi'(\sigma_i')}^{\mathfrak{h}}$ induces an étale morphism $a_{\overline{U}_i''', E'(\sigma_i')}^{\mathfrak{h}} : \overline{U}_i''' \to E'(\sigma_i') \times_{\text{Spec}(\mathbb{Z})} C'$ lifting $a_{\overline{U}_i, E(\sigma_i)}^{\mathfrak{h}}$. Thus, since $i \in I_Z$ and $x$ are arbitrary, by collecting (and reindexing) such $(\overline{U}_i''', a_{\overline{U}_i'''}^{\mathfrak{h}}, a_{\overline{U}_i''', E'(\sigma_i')}^{\mathfrak{h}})$, we obtain the desired collection $\mathcal{U}'_Z$, for $\text{X}_{\overline{H}', \Sigma'}^{\text{tor}}$. $\square$

*Remark 4.1.13.* In Lemma 4.1.12, without the assumptions in the last paragraph, $\mathcal{F}'$ is generally not a perverse sheaf even when $\mathcal{F}$ is.
4.2. **General properties and examples.** In this subsection, suppose $Y$ and $Y'$ are well-positioned subsets of $(X_H)_T$, such that $Y'$ is a (locally closed) subset of $Y$, with associated $Y' = \{Y_Z^{\natural} \mid Z \text{ and } Y'^{,\natural} = \{Y'_Z^{,\natural} \}Z\}$, respectively, as in Definition 2.2.1. Let $Y_{\Sigma}^{\text{tor}}$ and $Y'^{,\text{tor}}_{\Sigma}$ denote the partial toroidal compactifications of $Y$ and $Y'$, respectively, as in Definition 2.3.1. Let $J : Y_{\Sigma}^{',\text{tor}} \hookrightarrow Y_{\Sigma}^{\text{tor}}$, $J_Z : Y'_Z \hookrightarrow Y_Z$, and $J_Z : Y'_Z^{,\natural} \hookrightarrow Y_Z^{\natural}$ denote the canonical locally closed immersions, for all $Z$. Let $a_{Y_Z} : Y_Z \to Y_{\Sigma}^{\text{tor}}$, $a_{Y'_Z} : Y'_Z \to Y_{\Sigma}^{',\text{tor}}$, $a_{Y'_Z}^{\natural} : Y_Z \to Y_Z^{\natural}$, and $a_{Y'_Z}^{',\natural} : Y'_Z \to Y_Z^{',\natural}$ denote the canonical morphisms, for each $Z$.

**Lemma 4.2.1.** Suppose $\mathcal{F}$ and $\mathcal{F}'$ are well-positioned complexes over $Y_{\Sigma}^{\text{tor}}$ and $Y_{\Sigma}^{',\text{tor}}$, equipped with collections $\mathcal{F}^{\natural} = \{(\mathcal{F}_Z^{\natural}, \iota_Z)\}_Z$ and $\mathcal{F}^{',\natural} = \{(\mathcal{F}_Z^{',\natural}, \iota'_Z)\}_Z$, respectively, as in Definition 4.1.3. Then:

(1) $J_!\mathcal{F}'$ is a well-positioned complex over $Y_{\Sigma}^{\text{tor}}$, equipped with the collection $\{(J_{Z, !}\mathcal{F}_Z^{',\natural}, J_{Z, !\iota'_Z})\}_Z$. In particular, $J_!\Lambda_{Y_{\Sigma}^{',\text{tor}}}^*$ is a well-positioned sheaf over $Y_{\Sigma}^{\text{tor}}$, equipped with the collection $\{J_{Z, !}\Lambda_{Y_{Z}^{',\natural}}^*\}_Z$.

(2) $J^*\mathcal{F}$ is a well-positioned complex over $Y_{\Sigma}^{',\text{tor}}$, equipped with the collection $\{(J_Z^{\natural}\mathcal{F}_Z^{\natural}, J_Z^{*,\iota'_Z})\}_Z$.

(3) $\mathcal{H}^q(\mathcal{F})$ is a well-positioned sheaf over $Y_{\Sigma}^{\text{tor}}$, equipped with the collection $\{(\mathcal{H}^q(\mathcal{F}_Z^{\natural}), \mathcal{H}^q(\iota_Z))\}_Z$, for each $q$.

*Proof.* By Lemma 4.1.2 and Remark 4.1.7, we may assume that each collection $U_Z$ in Definition 4.1.3 is adapted to both $Y$ and $Y'$. Since the canonical morphism $Y'_Z \to Y_Z \times_{Y'_Z^{\natural}} Y_Z^{\natural}$ induces an isomorphism between the reduced subschemes, the lemma follows from the definitions, and from [2, XVII, 5.1.2] (for the assertion (1)). $\square$

**Lemma 4.2.2.** Suppose $\mathcal{F}'$ is as in Lemma 4.2.1. Suppose $T$ is just a point. Under the assumption (in (4) of Proposition 2.1.2) that $\Sigma$ is smooth, suppose moreover that the induced morphism $Y_C^{\natural} \to Y_Z^{\natural}$ is smooth (which is the case if $C \to Z$ is) for each $Z$. Then the canonical morphisms

$$ (4.2.3) \qquad a_{Y_Z^{\ast}}^* R J_* \mathcal{F}' \to R J_{Z, \ast} a_{Y'_Z^{\ast}} \mathcal{F}' $$

and

$$ (4.2.4) \qquad a_{Y_Z^{\natural}}^* R J_{Z, \ast} \mathcal{F}_Z^{',\natural} \to R J_{Z, \ast} a_{Y'_Z^{\natural}} \mathcal{F}_Z^{',\natural} $$

are isomorphisms in $D_c^b(Y_Z, \Lambda)$, and hence the isomorphism

$$ R J_{Z, \ast} \iota'_Z : R J_{Z, \ast} a_{Y'_Z^{\ast}} \mathcal{F}' \to R J_{Z, \ast} a_{Y'_Z^{\ast}} \mathcal{F}_Z^{',\natural} $$

canonically induces an isomorphism

$$ (4.2.5) \qquad \iota_Z : a_{Y_Z^{\ast}}^* R J_* \mathcal{F}' \xrightarrow{\sim} a_{Y'_Z^{\ast}}^* R J_{Z, \ast} \mathcal{F}_Z^{',\natural} $$

in $D_c^b(Y_Z, \Lambda)$, for each $Z$. Moreover, $R J_* \mathcal{F}'$ is a well-positioned complex over $Y_{\Sigma}^{\text{tor}}$, equipped with the collection $\{(R J_{Z, \ast} \mathcal{F}_Z^{',\natural}, \iota_Z)\}_Z$. In particular, $R J_* \Lambda_{Y_{\Sigma}^{',\text{tor}}}^*$ is a well-positioned complex over $Y_{\Sigma}^{\text{tor}}$, equipped with the collection $\{R J_{Z, \ast} \Lambda_{Y_{Z}^{',\natural}}^*\}_Z$.

*Proof.* For each $Z$, let $U_Z = \{(\overline{U}_i, a_{\overline{U}_i}, a_{\overline{U}_i, E(\sigma_i)}^{\natural})\}_{i \in I_Z}$ be as in Definition 4.1.3 (for $\mathcal{F}'$, adapted to $Y'$), which we may and we shall assume to be also adapted to
Since $Y_{\bar{U}_i}^+ \cong E(\sigma_i)^+ \times_{E(\sigma_i)} Y_{\bar{U}_i}$ and $Y'_{\bar{U}_i} \cong E(\sigma_i)^+ \times_{E(\sigma_i)} Y'_{\bar{U}_i}$, in order to show that (4.2.6) is an isomorphism, by the smooth base change theorem (and the smoothness assumptions) again, it suffices to show that the canonical morphism

$$ (4.2.8) \qquad (Y_{E(\sigma_i)+} \to Y_C^{\natural})^* R J_{C,*} \mathcal{F}_C^{\prime, \natural} \to R J_{E(\sigma_i)+,*} (Y'_{E(\sigma_i)+} \to Y_C^{\prime, \natural})^* \mathcal{F}_C^{\prime, \natural} $$

is an isomorphism. Since $J_{E(\sigma_i)+}$ is by definition the product $\text{Id}_{E(\sigma_i)}_{\bar{T}} \times_{\bar{T}} J_C$, the morphism (4.2.8) can be identified with the Künneth morphism

$$ (4.2.9) \qquad \Lambda_{E(\sigma_i)}_{\bar{T}}^{\natural} \boxtimes_{\bar{T}}^{L} R J_{C,*} \mathcal{F}_C^{\prime, \natural} \to R(\text{Id}_{E(\sigma_i)}_{\bar{T}} \times_{\bar{T}} J_C)_* (\Lambda_{E(\sigma_i)}_{\bar{T}}^{\natural} \boxtimes_{\bar{T}}^{L} \mathcal{F}_C^{\prime, \natural}), $$

which is an isomorphism, because $\bar{T}$ is a point, by [5, Sec. 4.2.7], as desired. $\square$

**Lemma 4.2.10.** Suppose that $\bar{T} = S$ is Henselian, that $Y' = Y_\eta$ and $Y'_Z^{\natural} = (Y_Z^{\natural})_\eta$, and that $\mathcal{F}'$ is as in Lemma 4.2.1. Consider the nearby cycle functors $R\Psi_Y$ and $R\Psi_{Y_Z^{\natural}}$, for all $Z$, defined by some choices of geometric points $\bar{\eta}$ and $\bar{s}$ above the generic and special points $\eta$ and $s$ of $S$, respectively, as in [43, Sec. 5.1]. Under the assumption (in (4) of Proposition 2.1.2) that $\Sigma$ is smooth, suppose moreover that $Y_C^{\natural} \to Y_Z^{\natural}$ is smooth for each $Z$. Then $R\Psi_{Y_\Sigma^{\text{tor}}}(\mathcal{F}')$ is a well-positioned complex over $(Y_\Sigma^{\text{tor}})_{\bar{s}}$, equipped with a collection $\{(R\Psi_{Y_Z^{\natural}}(\mathcal{F}_Z^{\prime, \natural}), \iota_Z)\}_Z$, for some canonical isomorphisms $\iota_Z$, as in Definition 4.1.3. In this case, $\iota_Z$ can be taken to be not just in $D_c^b((Y_Z^{\natural})_{\bar{s}}, \Lambda)$, but also in $D_c^b((Y_Z^{\natural})_{\bar{s}} \times_{\bar{\eta}} \Lambda)$; and the isomorphisms $\iota_{\bar{U}_i}$ can be taken to be not just in $D_c^b((Y_{\bar{U}_i})_{\bar{s}}, \Lambda)$, but also in $D_c^b((Y_{\bar{U}_i})_{\bar{s}} \times_{\bar{\eta}} \Lambda)$. (Here $D_c^b((\cdot)_{\bar{s}} \times_{\bar{\eta}} \Lambda)$ denotes the bounded derived category of $\Lambda$-étale constructible sheaves over $(\cdot)_{\bar{s}}$ with compatible continuous $\text{Gal}(k(\bar{\eta})/k(\eta))$-actions.)

Suppose that $\Lambda = \mathbb{Q}_\ell$ or $\mathbb{Q}_\ell$, and that $Y_C^{\natural} \to Y_Z^{\natural}$ is smooth of relative dimension $d_Z - d_E$, where $d_Z := d - d_Z$, $d := \dim((X_H)_\eta)$, $d_Z := \dim(Z_\eta)$, and $d_E := \dim(E_\eta)$, for each $Z$. If $\mathcal{F}'$ is a well-positioned perverse sheaf over $Y_\Sigma^{\prime, \text{tor}} = (Y_\Sigma^{\text{tor}})_\eta$, and if $\mathcal{F}_Z^{\prime, \natural}[-d_Z]$ is a perverse sheaf over $Y'_Z^{\natural} = (Y_Z^{\natural})_\eta$, for each $Z$, then $R\Psi_{Y_\Sigma^{\text{tor}}}(\mathcal{F}')$ is a well-positioned perverse sheaf over $(Y_\Sigma^{\text{tor}})_{\bar{s}}$, equipped with the collection $\{(R\Psi_{Y_Z^{\natural}}(\mathcal{F}_Z^{\prime, \natural}), \iota_Z)\}_Z$ as above, where $R\Psi_{Y_Z^{\natural}}(\mathcal{F}_Z^{\prime, \natural})[-d_Z]$ is a perverse sheaf, for each $Z$.

*Proof.* For the first paragraph of the lemma, by the same reduction steps as in the proof of Lemma 4.2.2, we are reduced to showing that the canonical morphism

$$ (4.2.11) \qquad \Lambda_{E(\sigma_i)}_{\bar{s}}^{\natural} \boxtimes_{\bar{s}}^{L} R \Psi_{Y_C^{\natural}} (\mathcal{F}_C^{\prime, \natural}) \to R \Psi_{E(\sigma_i)}_{\bar{s}}^{\natural} \times_{\bar{s}} Y_C^{\natural} (\Lambda_{E(\sigma_i)}_{\bar{\eta}}^{\natural} \boxtimes_{\bar{\eta}}^{L} \mathcal{F}_C^{\prime, \natural}) $$

(cf. (4.2.9)) in $D_c^b((E(\sigma_i)_S^+ \times_S Y_C^{\natural})_{\bar{s}} \times_{\bar{\eta}} \Lambda)$ is an isomorphism, where $\mathcal{F}_C^{\prime, \natural}$ denotes the pullback of $\mathcal{F}_Z^{\prime, \natural}$ under $Y'_C \to Y'_Z^{\natural}$, for each $Z$. Since $\sigma_i$ is smooth by assumption, $E(\sigma_i)^+$ is a relative normal crossings divisor on the smooth scheme $E(\sigma_i)$ over $\text{Spec}(Z)$. Hence, we have a canonical isomorphism $\Lambda_{E(\sigma_i)}_{\bar{s}}^{\natural} \simeq R\Psi_{E(\sigma_i)}_{\bar{s}}^+(\Lambda_{E(\sigma_i)}_{\bar{\eta}}^+)$ in $D_c^b(E(\sigma_i)_{\bar{s}}^+ \times_{\bar{\eta}} \Lambda)$, via which (4.2.11) induces the Künneth morphism

$$ (4.2.12) \qquad (R\Psi_{E(\sigma_i)}_{\bar{s}}^+(\Lambda_{E(\sigma_i)}_{\bar{\eta}}^+)) \boxtimes_{\bar{s}}^{L} (R\Psi_{Y_C^{\natural}}(\mathcal{F}_C^{\prime, \natural})) \to R\Psi_{E(\sigma_i)}_{\bar{s}}^+ \times_{\bar{s}} Y_C^{\natural} (\Lambda_{E(\sigma_i)}_{\bar{\eta}}^+ \boxtimes_{\bar{\eta}}^{L} \mathcal{F}_C^{\prime, \natural}). $$

Thus, (4.2.11) is an isomorphism because (4.2.12) is, by [25, Thm. 4.7]. The second paragraph of the lemma then follows from [25, Cor. 4.5] and [5, Prop. 4.2.8]. $\square$
**Lemma 4.2.13.** Suppose $\mathcal{T} = \text{Spec}(k)$ for some field $k$ that is either algebraically closed or finite, and that $\Lambda = \mathbb{Q}_\ell$ or $\mathbb{Q}_\ell$. Suppose $(X_H)_\mathcal{T}$ and $Z_\mathcal{T}$ are equidimensional of dimensions $d$ and $d_Z$, respectively, for each $Z$. Under the assumption (in (4) of Proposition 2.1.2) that $\Sigma$ is smooth, suppose moreover that $Y_C^\Sigma \to Y_Z^\Sigma$ is smooth of relative dimension $d_Z - d_E$, where $d_Z := d - d_Z$ and $d_E := \text{dim}(E_\mathcal{T})$, for each $Z$. Suppose $\mathcal{F}'$ is a well-positioned perverse sheaf over $Y_\Sigma^{\text{tor}}$, equipped with a collection $\mathcal{F}',\sharp = \{\mathcal{F}'_Z^{\sharp}\}_Z$, where $\mathcal{F}'_Z^{\sharp}[-d_Z]$ is a perverse sheaf over $Y'_Z^{\sharp}$, for each $Z$. Then $a_{Y'_Z}^*(\mathcal{F}'[-1])$ and $a_{Y'_Z}^{\sharp,*}(\mathcal{F}'_Z^{\sharp}[-1])$ are perverse sheaves (isomorphic to each other under $\iota'_Z[-1]$) in $D_c^b(Y'_Z, \Lambda)$; and $a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1]$ and $a_{Y'_Z}^{\sharp,*}J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z])[d_Z - 1]$ are perverse sheaves in $D_c^b(Y'_Z, \Lambda)$. Moreover, we have canonical isomorphisms

$$ (4.2.14) \qquad a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1] \xrightarrow{\sim} J_{Z,1*}(a_{Y'_Z}^*\mathcal{F}'[-1]) $$

and

$$ (4.2.15) \qquad a_{Y'_Z}^{\sharp,*}(J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z]))[d_Z - 1] \xrightarrow{\sim} J_{Z,1*}(a_{Y'_Z}^{\sharp,*}\mathcal{F}'_Z^{\sharp}[-1]) $$

in $D_c^b(Y'_Z, \Lambda)$ extending the compositions of canonical isomorphisms

$$ (4.2.16) \qquad J_Z^* a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1] \xrightarrow{\sim} a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1] \xrightarrow{\sim} a_{Y'_Z}^*\mathcal{F}'[-1] $$

and

$$ (4.2.17) \qquad J_Z^* a_{Y'_Z}^{\sharp,*}(J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z]))[d_Z - 1] \xrightarrow{\sim} a_{Y'_Z}^{\sharp,*}J_Z^*(J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z]))[d_Z - 1] \\ \qquad \qquad \xrightarrow{\sim} a_{Y'_Z}^{\sharp,*}\mathcal{F}'_Z^{\sharp}[-1], $$

respectively, in $D_c^b(Y'_Z, \Lambda)$; and hence the isomorphism

$$ J_{Z,1*}(\iota'_Z[-1])[1] : J_{Z,1*}(a_{Y'_Z}^*\mathcal{F}'[-1])[1] \xrightarrow{\sim} J_{Z,1*}(a_{Y'_Z}^{\sharp,*}\mathcal{F}'_Z^{\sharp}[-1])[1] $$

canonically induces an isomorphism

$$ (4.2.18) \qquad \iota_Z : a_{Y'_Z}^* J_{1*}\mathcal{F}' \xrightarrow{\sim} a_{Y'_Z}^{\sharp,*} J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z])[d_Z], $$

in $D_c^b(Y'_Z, \Lambda)$, for each $Z$. Moreover, $J_{1*}\mathcal{F}'$ is a well-positioned perverse sheaf over $Y_\Sigma^{\text{tor}}$, equipped with the collection $\{(J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z]))[d_Z], \iota_Z\}_Z$.

In particular, suppose $Y' = Y_{\text{sm}}$ and $Y'_Z^{\sharp} = (Y_Z^\Sigma)_{\text{sm}}$ in the above, which are valid choices by Lemma 2.2.10, so that $Y_\Sigma^{\text{tor}} = (Y_\Sigma^{\text{tor}})_{\text{sm}}$. Consider the intersection complexes $\mathcal{I}\mathcal{C}_{Y_\Sigma^{\text{tor}}} := J_{1*}(\Lambda_{Y_\Sigma^{\text{tor}}}_{\text{sm}}[d])$ and $\mathcal{I}\mathcal{C}_{Y'_Z} := J_{Z,1*}(\Lambda_{Y'_Z, \text{sm}}[d_Z])$. Then $\mathcal{I}\mathcal{C}_{Y_\Sigma^{\text{tor}}}$ is a well-positioned perverse sheaf, equipped with the collection $\{\mathcal{I}\mathcal{C}_{Y'_Z}^{\sharp}[d_Z]\}_Z$.

**Proof.** Since the assignment of middle perversity extensions is functorial and fully faithful (see [27, Cor. III.5.11]), in addition to showing that $a_{Y'_Z}^*(\mathcal{F}'[-1])$, $a_{Y'_Z}^{\sharp,*}\mathcal{F}'_Z^{\sharp}[-1]$, $a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1]$, and $a_{Y'_Z}^{\sharp,*}J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z])[d_Z - 1]$ are all perverse sheaves, in order to also show that the compositions of canonical isomorphisms (4.2.16) and (4.2.17) (necessarily uniquely) extend to the desired canonical isomorphisms (4.2.14) and (4.2.15), so that all remaining assertions of the lemma will follow, it suffices to show that the perverse sheaves $a_{Y'_Z}^*(J_{1*}\mathcal{F}')[-1]$ and $a_{Y'_Z}^{\sharp,*}J_{Z,1*}(\mathcal{F}'_Z^{\sharp}[-d_Z])[d_Z - 1]$ are isomorphic to the middle perversity extensions of their restrictions to $Y'_Z$, respectively. Essentially by definition (see also [27, the criterion on p. 148, after Lem.-Def. III.5.2]), both the perversity and this last property can be verified étale locally.
For each $\mathbb{Z}$, let $\mathcal{U}_{\mathbb{Z}} = \{(\bar{U}_i, a_{\bar{U}_i}, a_{\bar{U}_i, E(\sigma_i)}^{\natural})\}_{i \in I_{\mathbb{Z}}} \in \text{Def}(\mathcal{F}', \text{adapted to } \mathbb{Y}')$, which we may and we shall assume to be also adapted to $\mathbb{Y}$, by Lemma 4.1.2 and Remark 4.1.7. Since $\mathcal{U}_{\mathbb{Z}}$ induces an affine open covering of $\bar{\mathbb{Z}}$ by assumption, by the explanation in the previous paragraph, and by using the canonical isomorphisms induced by $i'_{\mathbb{Z}}$ and $i'_{\bar{U}_i}$, for each $i \in I_{\mathbb{Z}}$, it suffices to show that $i_{\mathbb{Y}'_{\bar{U}_i}}^*(a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural}) \mathcal{F}'_{\mathbb{Z}}[-1]$ is a perverse sheaf in $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$, and that the composition of canonical isomorphisms

$$ (4.2.19) \qquad J_{\bar{U}_i}^* i_{\bar{Y}'_{\bar{U}_i}}^* (J_{\bar{U}_i, !_*} a_{\bar{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1] \xrightarrow{\sim} i_{\bar{Y}'_{\bar{U}_i}}^* J_{\bar{U}_i}^* (J_{\bar{U}_i, !_*} a_{\bar{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1] \\ \qquad \qquad \qquad \qquad \qquad \qquad \xrightarrow{\sim} i_{\bar{Y}'_{\bar{U}_i}}^* a_{\bar{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}}[-1]. $$

in $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$ (necessarily uniquely) extends to some isomorphism

$$ (4.2.20) \qquad i_{\mathbb{Y}'_{\bar{U}_i}}^* (J_{\bar{U}_i, !_*} a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1] \xrightarrow{\sim} J_{\bar{U}_i}^* i_{\bar{Y}'_{\bar{U}_i}}^* (i_{\bar{Y}'_{\bar{U}_i}}^* a_{\bar{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1]) $$

in $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$. (Then $i_{\mathbb{Y}'_{\bar{U}_i}}^* (J_{\bar{U}_i, !_*} a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1]$ is a perverse sheaf in $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$ that is isomorphic to the middle perversity extension of its restriction to $\mathbb{Y}'_{\bar{U}_i}$.)

Under the assumption that $\Sigma$ is smooth, $E(\sigma_i)^+ = \text{a relative normal cross-ings divisor on the smooth scheme } E(\sigma_i) \text{ over Spec}(\mathbb{Z})$, and hence $\Lambda_{E(\sigma_i)}_{\mathbb{Z}}^+ [d_E - 1]$ is a perverse sheaf over $E(\sigma_i)^+_{\mathbb{Z}}$, by [27, Lem. III.6.5]. Since $\mathcal{F}'_{\mathbb{Z}}[-d^2]$ is a perverse sheaf, under the assumption that $\mathbb{Y}_C^{\natural} \to \mathbb{Y}_{\mathbb{Z}}^{\natural}$ is smooth of relative dimension $d^2 - d_E$, by the smooth base change theorem (see [2, XVI, 1.2]) and by the t-exactness of smooth morphisms up to shifting by the relative dimensions (see [5, Sec. 4.2.5]), $\mathcal{F}'_{\mathbb{Z}}[-d_E] := (\mathbb{Y}'_{\mathbb{C}}^{\natural} \to \mathbb{Y}'_{\mathbb{Z}}^{\natural})^* \mathcal{F}'_{\mathbb{Z}}[-d_E]$ is also a perverse sheaf. Since $a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural}$ and $a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural}$ are pullbacks of the composition of the étale morphism $a_{\bar{U}_i, E(\sigma_i)}^{\natural} : \bar{U}_i \to E(\sigma_i) \times_{\text{Spec}(\mathbb{Z})} C$ with the canonical morphism $E(\sigma_i) \times_{\text{Spec}(\mathbb{Z})} C \to \mathbb{Z}$, and since $i_{\mathbb{Y}'_{\bar{U}_i}}$ and $i_{\mathbb{Y}'_{\bar{U}_i}}$ are pullbacks of the closed immersion $E(\sigma_i)^+ \to E(\sigma)$, it follows that $i_{\mathbb{Y}'_{\bar{U}_i}}^* a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}}[-1]$ and $i_{\mathbb{Y}'_{\bar{U}_i}}^* (J_{\bar{U}_i, !_*} a_{\mathbb{Y}'_{\bar{U}_i}}^{\natural} \mathcal{F}'_{\mathbb{Z}})[-1]$ are perverse sheaves in $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$ and $D_c^b(\mathbb{Y}'_{\bar{U}_i}, \Lambda)$, respectively, because they are compatibly isomorphic to the pullbacks under compatible étale morphisms of the perverse sheaves $(\Lambda_{E(\sigma_i)}_{\mathbb{Z}}^+ [d_E - 1])^{\boxtimes L}(\mathcal{F}'_{\mathbb{C}}^{\natural}[-d_E])$ and $(\Lambda_{E(\sigma_i)}_{\mathbb{Z}}^+ [d_E - 1])^{\boxtimes L}(J_{C, !_*}(\mathcal{F}'_{\mathbb{C}}^{\natural}[-d_E]))$ (see [5, Prop. 4.2.8]), respectively; and that (via these compatible isomorphisms) the composition (4.2.19) can be identified with the pullback of the canonical isomorphism

$$ (4.2.21) \qquad (\text{Id}_{E(\sigma_i)}_{\mathbb{Z}}^+ \times J_C)^*(\Lambda_{E(\sigma_i)}_{\mathbb{Z}}^+ [d_E - 1])^{\boxtimes L}(J_{C, !_*}(\mathcal{F}'_{\mathbb{C}}^{\natural}[-d_E])) \\ \qquad \qquad \xrightarrow{\sim} (\Lambda_{E(\sigma_i)}_{\mathbb{Z}}^+ [d_E - 1])^{\boxtimes L}(\mathcal{F}'_{\mathbb{C}}^{\natural}[-d_E]). $$
Since (4.2.21) extends to the Künneth isomorphism

$$ (4.2.22) \qquad (\Lambda_{E(\sigma_i)_T^+} [d_E - 1])^{\boxtimes L_T} (J_{C, !*} (\mathcal{F}_C'^{\sharp, \sharp} [-d_E])) \\ \qquad \qquad \qquad \qquad \qquad \qquad \sim \to (\mathrm{Id}_{E(\sigma_i)_T^+} \times_T J_C)_{!*} ((\Lambda_{E(\sigma_i)_T^+} [d_E - 1])^{\boxtimes L_T} (\mathcal{F}_C'^{\sharp, \sharp} [-d_E])) $$

(see [5, Prop. 4.2.8] again), it follows that (4.2.19) also extends to some isomorphism (4.2.20), which can be identified with the pullback of (4.2.22), as desired. $\square$

**Lemma 4.2.23.** Suppose that $S$ is Henselian, and that $\Lambda = \mathbb{Q}_\ell$ or $\bar{\mathbb{Q}}_\ell$ for simplicity. Consider the nearby cycle functors $R\Psi_{X_{\mathcal{H},\Sigma}^{\mathrm{tor}}}$ and $R\Psi_Z$, for all $Z$, defined by some choices of geometric points $\bar{\eta}$ and $\bar{s}$ above the generic and special points $\eta$ and $s$ of $S$, respectively, as in [43, Sec. 5.1]. Under the assumption (in (4) of Proposition 2.1.2) that $\Sigma$ is smooth, suppose moreover that $C \to Z$ has connected geometric fibers and is smooth of relative dimension $d^Z - d_E$, where $d := \dim((X_{\mathcal{H}})_\eta)$, $d_Z := \dim(Z_\eta)$, $d^Z := d - d_Z$, and $d_E = \dim(E_T)$, for each $Z$, so that Assumption 3.7.7 holds. Suppose $Y \in \mathrm{Supp}(R\Psi_{X_{\mathcal{H},\Sigma}^{\mathrm{tor}}} (\Lambda[d]))$ and $G \in \mathrm{Loc}_Y(R\Psi_{X_{\mathcal{H},\Sigma}^{\mathrm{tor}}} (\Lambda[d]))$, with corresponding $Y_Z^{\sharp} \in \mathrm{Supp}(R\Psi_Z(\Lambda[d_Z]))$ and $G_Z^{\sharp} \in \mathrm{Loc}_{Y_Z^{\sharp}}(R\Psi_Z(\Lambda[d_Z]))$, for each $Z$, as in Lemma 3.7.9 and Proposition 3.7.13. Then $Y \cap (X_{\mathcal{H}})_{\bar{s}}$ is a well-positioned subset of $(X_{\mathcal{H}})_{\bar{s}}$, by Proposition 3.7.13, with toroidal compactification $Y$ as in Definition 2.3.1. Moreover, there exist smooth open subschemes $Y^0$ and $Y_Z^{\sharp,0}$ of $Y$ and $Y_Z^{\sharp}$ (with their reduced subscheme structures), respectively, for each $Z$, over which $G$ and $G_Z^{\sharp}$ are defined, such that $Y^0 \cap (X_{\mathcal{H}})_{\bar{s}}$ is a well-positioned subset of $(X_{\mathcal{H}})_{\bar{s}}$, with toroidal compactification $Y^0$ as in Definition 2.3.1. Let $J_Y : Y^0 \hookrightarrow Y$ and $J_{Y_Z^{\sharp}} : Y_Z^{\sharp,0} \hookrightarrow Y_Z^{\sharp}$ denote the canonical open immersions, for each $Z$. By [19, Appendix, Thm. 10.1] and [5, Thm. 5.3.1, and Cor. 5.3.2 and 5.3.4], $\mathcal{I}\mathcal{C}_Y(G) := J_{Y, !*}(G[\dim(Y)])$ (resp. $\mathcal{I}\mathcal{C}_{Y_Z^{\sharp}}(G_Z^{\sharp}) := J_{Y_Z^{\sharp}, !*}(G_Z^{\sharp}[\dim(Y_Z^{\sharp})])$, for each $Z$) is a pure perverse sheaf. By [5, Thm. 5.3.5 and 5.3.8], the $\mathcal{I}\mathcal{C}_Y(G)$-isotypic subquotient $\mathcal{H}$ of $R\Psi_{X_{\mathcal{H},\Sigma}^{\mathrm{tor}}} (\Lambda[d])$ (resp. $\mathcal{I}\mathcal{C}_{Y_Z^{\sharp}}(G_Z^{\sharp})[d^Z]$-isotypic subquotient $\mathcal{H}_Z^{\sharp}$ of $R\Psi_Z(\Lambda[d])$) is defined. Then $\mathcal{H}$ is a well-positioned pure perverse sheaf over $Y$, equipped with a collection $\{(\mathcal{H}_Z^{\sharp}, \iota_Z)\}_Z$, where $\iota_Z$ is induced by its analogue for nearby cycles as in Lemma 4.2.10, for each $Z$.

**Proof.** These follow from Proposition 3.7.13, from Lemmas 2.2.10, 4.2.10, and 4.2.13 and their proofs, and from the following: For each $Z$, and for each collection $U_Z = \{(\bar{U}_i, a_{\bar{U}_i}, a_{\bar{U}_i, E(\sigma_i)})\}_{i \in I_Z}$ as in Definition 4.1.3, the isomorphisms $\iota_Z$ and $\iota_{\bar{U}_i}$, for all $i \in I_Z$, are induced by their analogues for nearby cycles, because the corresponding isotypical subquotients of nearby cycles match over $\bar{U}_i$, for all $i \in I_Z$, and because the fibered category of perverse sheaves is a stack (see [5, Sec. 2.2.19]). $\square$

**Example 4.2.24.** (This is a more detailed version of a remark we made in the introduction.) Suppose we are in the setting of Section 3.4. For simplicity, assume that $\mathcal{O} \boxtimes \mathbb{Q}$ involves no factor of type D, in the sense of [38, Def. 1.2.1.15], so that $G \boxtimes \mathbb{Q}$ is connected. Suppose $\mathcal{H} = \mathcal{U}(n_0p^{2016}) := \ker(G(\hat{Z}) \to G(\mathbb{Z}/(n_0p^{2016}\mathbb{Z})))$ for some $n_0 \ge 3$. By Lemma 2.1.9, $C \to Z$ is an abelian scheme, for each $Z$. Consider any Newton stratum $(X_{\mathcal{H}})_s^b$, for some $b \in B(G \boxtimes_{\mathbb{Z}} \mathbb{Q}_p)$, as in Proposition 3.3.1, and any Ekedahl–Oort stratum $(X_{\mathcal{H}})_s^w$, for some $w \in W^{M_0}$, as in Proposition
3.5.1, which are well-positioned subschemes of $(X_{\mathcal{H}})_s$, by Propositions 3.3.9 and 3.5.5. By Lemma 2.2.3 and Proposition 2.3.12, any irreducible component $\mathcal{Y}$ of the intersection $(X_{\mathcal{H}})^b_s \cap (X_{\mathcal{H}})^w_s$ (with its reduced subscheme structure) is a well-positioned subscheme of $(X_{\mathcal{H}})_s$. Thus, the (generally infinite-dimensional) coherent cohomology of its partial toroidal compactification still satisfies Koecher's principle, by Theorem 2.5.10. Moreover, by Lemmas 4.2.1, 4.2.2, and 4.2.13, if we denote by $J : \mathcal{Y} \hookrightarrow (X_{\mathcal{H},\Sigma}^{\text{tor}})_s$ the canonical immersion, then $J_!\mathcal{Q}_\ell$ (resp. $RJ_*\mathcal{Q}_\ell$, resp. the intersection complex $\mathcal{I}\mathcal{C}_{\mathcal{Y}^{\text{tor}}}$) is a well-positioned sheaf (resp. complex, resp. perverse sheaf) over $(X_{\mathcal{H},\Sigma}^{\text{tor}})_s$, and we will have generalizations of Pink's and Morel's formulae for these in Sections 4.3 and 4.5 (see Theorems 4.3.16 and 4.5.26 below).

**4.3. Pink's formula.** The goal of this subsection is to generalize [67, Thm. 5.3.1]. We will first state its original form for automorphic étale sheaves (see Theorem 4.3.10 below), and then its generalization for tensor products with well-positioned complexes (see Theorem 4.3.16 below).

Let us start with some preparations. For simplicity, in this subsection, let us make the following assumptions (although some of them can be relaxed):

**Assumption 4.3.1.** (1) In all cases, assume that $\mathcal{H} = \mathcal{H}^{\ell,p}U_\ell(\ell^r)\mathcal{H}_p$, where $\mathcal{H}^{\ell,p} \subset G(\mathcal{Z}^{\ell,p})$ and $\mathcal{H}_p \subset G(\mathcal{Z}_p)$ are open compact subgroups, and where $U_\ell(\ell^r) := \ker(G(\mathcal{Z}_\ell) \to G(\mathcal{Z}/\ell^r\mathcal{Z}))$, for some $r \ge 1$.

(2) In Cases (Sm), (Nm), or (Spl), also assume that $\mathcal{O} \otimes_{\mathcal{Z}} \mathcal{Q}$ involves no factor of type D (as in [38, Def. 1.2.1.15]), so that $G \otimes_{\mathcal{Z}} \mathcal{Q}$ is connected.

(3) In Case (Sm), assume in addition that $\ell \notin \mathcal{D}$. (This is forced by the condition (1) above when $r > 1$.)

(4) In Cases (Nm), (Spl), and (Hdg), assume in addition that $C \to \mathcal{Z}$ is an abelian scheme torsor (of constant relative dimension) over a finite étale cover, for each $\mathcal{Z}$ (cf. Remark 2.1.8). In Cases (Nm), (Spl), this is true, for example, when we assume that $\mathcal{H}$ is a principal level, and that we are in the setting preceding Lemma 2.1.9. (See also Remark 2.1.11). In Case (Hdg), this is true, for example, when $p > 2$ and $\mathcal{H}_p$ is a maximal hyperspecial open compact subgroup of $G(\mathcal{Q}_p)$ (see [47, Sec. 4.3 and 5.3]).

**Lemma 4.3.2.** Under Assumption 4.3.1, there exists a system of open compact subgroups $\mathcal{H}^{(\ell r')} \subset \mathcal{H}$, labeled by integers $r' \ge r$, such that:

(1) For each $r' \ge r$, we have $\mathcal{H}^{(\ell r')} = \mathcal{H}^{\ell,p}U_\ell(\ell^{r'})\mathcal{H}_p$, where $\mathcal{H}^{\ell,p}$ and $\mathcal{H}_p$ are as above, and where $U_\ell(\ell^{r'}) := \ker(G(\mathcal{Z}_\ell) \to G(\mathcal{Z}/\ell^{r'}\mathcal{Z}))$.

(2) By writing $\mathcal{H}' = \mathcal{H}^{(\ell r')}$ for simplicity, the corresponding canonical morphisms $X_{\mathcal{H}'} \to X_{\mathcal{H}}$ and their boundary analogues $\Xi' \to \Xi$, $C' \to C$ and $\mathcal{Z}' \to \mathcal{Z}$ are finite étale. The morphisms $C \to \mathcal{Z}$ and $C' \to \mathcal{Z}'$ are abelian scheme torsors over finite étale covers of the base schemes, and the morphisms $\Xi \to C$ and $\Xi' \to C'$ are torus torsors. The induced morphism $C' \to C \times_{\mathcal{Z}} \mathcal{Z}'$ is étale locally over $\mathcal{Z}'$ the multiplication by $\ell^{r'-r}$ on an abelian scheme, and the canonical morphism $\Xi' \to \Xi \times_C C'$ is Zariski locally over $C'$ the multiplication by $\ell^{r'-r}$ on a torus.

*Proof.* Part (1) of the lemma is just the definition of $\mathcal{H}^{(\ell r')} \subset \mathcal{H}$, for each $r' \ge r$. It remains to verify the assertions in part (2) of the lemma. In Case (Sm), these
follow from the constructions in [38, Sec. 6.2.2–6.2.4; see also the errata]. In Cases (Nm) and (Spl), since $\mathcal{H}/\mathcal{H}' \cong \mathcal{U}_\ell(\ell^r)/\mathcal{U}_\ell(\ell^{r'})$, we can define relative moduli $C'^{,\mathrm{mod}}$ and $\Xi'^{,\mathrm{mod}}$ over $C$ and $\Xi$ parameterizing the $\ell$-power parts of the degenerations of level structures, as in [38, Sec. 6.2.4] (see also [32, Sec. 1.3.2]), which are finite étale over $C$ and $\Xi$, respectively, and have the desired property as (instead of $C'$ and $\Xi'$) in the last sentence of the lemma. Then the induced morphisms $C' \to C'^{,\mathrm{mod}}$ and $\Xi' \to \Xi'^{,\mathrm{mod}}$ are isomorphisms by Zariski’s main theorem, because they are finite morphisms between normal schemes which induce isomorphisms in characteristic zero. In Case (Hdg), these follow from the boundary descriptions in characteristic zero in [47, Sec. 2.1] (based on [3] and [66]), and in characteristic $p$ (with hyperspecial levels at $p$) in [47, Sec. 4.3 and 5.3]. □

Let $Z$ be a stratum of $X_{\mathcal{H}}^{\min}$.

In Cases (Sm), (Nm), or (Spl), $Z$ is associated with some cusp label $[(Z_{\mathcal{H}}, \Phi_{\mathcal{H}}, \delta_{\mathcal{H}})]$, which determines an $\mathcal{H}$-orbit $Z_{\mathcal{H}}$ of a fully symplectic liftable filtration $Z = \{Z_{-i}\}_{i \in Z}$ on $L \otimes_{\mathbb{Z}} \hat{Z}$ (see [38, Def. 5.2.7.1 and 5.4.2.4]). Then we fix the choices of $P$ and $V = \{V_{-i}\}_{i \in Z}$ as in Lemma 3.3.6, where $V_{-2} \otimes_{\mathbb{Q}} A^{\infty}$ lies in the $\mathcal{H}$-orbit of $Z_{-2} \otimes_{\mathbb{Z}} \mathbb{Q}$, and define various groups as in Definition 3.3.8.

In Case (Hdg), it is the consequence of the characteristic zero theory in [4] and [66] that $Z$ is associated with some parabolic subgroup $P$ of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ whose image in each simple factor of $(G \otimes_{\mathbb{Z}} \mathbb{Q})^{\mathrm{ad}}$ is either the whole factor or a proper maximal parabolic subgroup. Let $U$ and $M$ denote the unipotent radical and the Levi quotient of $P$, respectively. Then we have an exact sequence $1 \to G_h \to M \to G_l \to 1$, where $G_h$ defines the boundary analogue of $X_{\mathcal{H}}$, which is a finite cover of $Z$ in [66] and [47]. (Unlike in PEL cases, we do not have $M \cong G_l \times G_h$.) In what follows, we will not be as precise as in Cases (Sm), (Nm), and (Spl), but only state the bare minimum for the application of Pink’s arguments in [67].

**Remark 4.3.3.** The group $\Gamma$ in Proposition 2.1.2 is by definition a subgroup of $G_l(\mathbb{Q})$. Even in Cases (Sm), (Nm), or (Spl), where $M \cong G_l \times G_h$, it is generally not true that $\mathcal{H}_M$ coincides with the semi-direct product $\mathcal{H}_l \times \mathcal{H}_h$ or $\mathcal{H}'_l \times \mathcal{H}'_h$ (see [38, Def. 5.3.1.4 and 5.3.1.1, and the errata], and see [32, Def. 1.2.1.11 and 1.2.1.12]), and $\Gamma' := \Gamma \cap \mathcal{H}_l$ is a finite index subgroup of $\Gamma$ which is generally different from $\Gamma$. This discrepancy between $\Gamma$ and $\Gamma'$ defines a finite étale cover of $Z$ in characteristic zero, which is consistent with the consideration in [66, Sec. 6.3] (cf. Definition 4.3.4 below). Nevertheless, in Cases (Sm), (Nm), or (Spl), $\Gamma' = \Gamma$ when and exactly when $M \cong G_l \times G_h$ induces a direct product $\mathcal{H}_M \cong \mathcal{H}_l \times \mathcal{H}_h$, in which case $\mathcal{H}_l = \mathcal{H}'_l$ and $\mathcal{H}'_h = \mathcal{H}_{h,\Phi} = \mathcal{H}_h$. This is the case, for example, when $\mathcal{H}$ is a principal level (and when the technical assumption [38, Cond. 1.4.3.10] is satisfied).

Consider any algebraic representation $\xi$ of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ on a finite-dimensional vector space $V_\xi$ over $\mathbb{Q}_{\ell}$, with an associated étale sheaf $\mathcal{V}_\xi$ over $X_{\mathcal{H}}$, as explained in [43, Sec. 3.1] (which is consistent with [67, Sec. 1 and 4.9]). The association is functorial and also applies to complexes $V$ of algebraic representations of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ on finite-dimensional vector spaces over $\mathbb{Q}_{\ell}$, with associated complexes $\mathcal{V}$ of $\mathbb{Q}_{\ell}$-étale sheaves over $X_{\mathcal{H}}$. For each algebraic group $(\cdot)$ over $\mathbb{Q}$, let $D^b(\cdot, \mathbb{Q}_{\ell})$ denote the bounded derived category of algebraic representations of $(\cdot)$ on finite-dimensional vector
spaces over $\bar{Q}_\ell$. Then the above association induces a functor from $D^b(G \otimes_Z \mathbb{Q}, \bar{Q}_\ell)$ to $D_c^b(X_H, \bar{Q}_\ell)$ (see the beginning of Section 4.1).

Following [67, Sec. 5.2], let us introduce the following definitions:

**Definition 4.3.4.** For the group $\Gamma$ in Proposition 2.1.2 associated with $Z$, consider the Stein factorization $C^{\text{st}} := \text{Spec}_{\mathcal{O}_Z}((C \to Z)_* \mathcal{O}_C) \to Z$, with an induced action of $\Gamma$. Then we denote by $\Gamma'$ the largest (finite index) subgroup of $\Gamma$ which acts trivially on $C^{\text{st}}$. (In Cases (Sm), (Nm), and (Spl), this is the same $\Gamma'$ as in Remark 4.3.3.)

**Definition 4.3.5.** Let $u := \text{Lie } U(\mathbb{Q}_\ell)$ denote the Lie algebra of the $\ell$-adic analytic Lie group $U(\mathbb{Q}_\ell)$. For each $V \in D^b(\mathbb{P}, \bar{Q}_\ell)$, we denote by $\text{RInv}(u, V)$ the Lie algebra cohomology of $V$, which is an object of $D^b(\mathbb{M}, \bar{Q}_\ell)$.

**Definition 4.3.6.** For each object $V \in D^b(\mathbb{M}, \bar{Q}_\ell)$, we denote by $\text{RInv}(\Gamma', V)$ the group cohomology of $V$, which defines an object of $D^b(\mathbb{G}_h, \bar{Q}_\ell)$.

**Remark 4.3.7.** In Cases (Sm), (Nm), or (Spl), the boundary stratum $Z$ and its finite cover $C^{\text{st}}$ are analogues of $X_H$ associated with $G_h$ at levels $\mathcal{H}_{h,\Phi}$ and $\mathcal{H}'_{h'}$, respectively, and we have $C^{\text{st}}/(\Gamma/\Gamma') \xrightarrow{\sim} Z$. In Case (Hdg), it is only the $C^{\text{st}}$ in Definition 4.3.4 that is associated with $G_h$, but we still have $C^{\text{st}}/(\Gamma/\Gamma') \xrightarrow{\sim} Z$. In all cases, the constructions in [43, Sec. 3.1] and [67, Sec. 1 and 4.9] associate with any $\text{RInv}(\Gamma', V)$ as in Definition 4.3.6 an object of $D_c^b(C^{\text{st}}, \bar{Q}_\ell)$, and then the action of $\Gamma/\Gamma'$ on it defines an object of $D_c^b(Z, \bar{Q}_\ell)$ (cf. [66, (1.10) and (4.7)]) because $C^{\text{st}} \to Z$ is étale by assumption (see (4) of Assumption 4.3.1).

**Definition 4.3.8.** For each object $V \in D^b(G \otimes_Z \mathbb{Q}, \bar{Q}_\ell)$, we define

$$ (4.3.9) \qquad V_Z^{\sharp} := \text{RInv}(\Gamma', \text{RInv}(u, V|_P)) $$

as in Definition 4.3.6, with an associated object $V_Z^{\sharp}$ of $D_c^b(Z, \bar{Q}_\ell)$ as in Remark 4.3.7.

When $V$ is represented by an algebraic representation $V_\xi$ of $G \otimes_Z \mathbb{Q}$ on a finite-dimensional vector space over $\bar{Q}_\ell$, we denote by $V_{\xi,Z}^{\sharp}$ the unique representative of $V_Z^{\sharp}$ which is a direct sum of shifts of algebraic representations of $G_h$ on finite-dimensional vector spaces over $\bar{Q}_\ell$ (which exists because $G_h$ is reductive), and by $V_{\xi,Z}^{\sharp}$ the associated direct sum of shifts of $\bar{Q}_\ell$-étale sheaves over $Z$ representing $V_Z^{\sharp}$. As usual, we shall also denote by $V_\xi$, $V_\xi$, $V_{\xi,Z}^{\sharp}$, and $V_{\xi,Z}^{\sharp}$ the induced objects in the derived categories.

Suppose $\mathcal{T}$ is just a point. Let $i^{\text{min}}: \mathcal{Z}_{\mathcal{T}} \to (X_H^{\text{min}})_{\mathcal{T}}$ and $i^{\text{tor}}: \tilde{\mathcal{Z}}_{\mathcal{T}} \to (X_{H,\Sigma}^{\text{tor}})_{\mathcal{T}}$ (resp. $j^{\text{min}}: (X_H^{\text{min}})_{\mathcal{T}} \hookrightarrow (X_H^{\text{min}})_{\mathcal{T}}$ and $j^{\text{tor}}: (X_H^{\text{tor}})_{\mathcal{T}} \hookrightarrow (X_{H,\Sigma}^{\text{tor}})_{\mathcal{T}}$) denote the canonical locally closed (resp. open) immersions, which satisfy $i^{\text{min}} = \mathcal{f} \circ i^{\text{tor}}$ and $j^{\text{min}} = \mathcal{f} \circ j^{\text{tor}}$, where $\mathcal{f} := (\mathcal{f}_{H,\Sigma})_{\mathcal{T}}: (X_{H,\Sigma}^{\text{tor}})_{\mathcal{T}} \to (X_H^{\text{min}})_{\mathcal{T}}$ is as in (1) of Theorem 2.3.2.

**Theorem 4.3.10 (Pink).** Let $V_\xi$ be an algebraic representation of $G \otimes_Z \mathbb{Q}$ on a finite-dimensional vector space over $\bar{Q}_\ell$, which defines $V_\xi$ etc as in Definition 4.3.8. For simplicity, we shall denote by the same symbols the pullbacks of $V_\xi$ and $V_{\xi,Z}^{\sharp}$ under $\mathcal{T} \to \mathcal{S}$. Then, under Assumption 4.3.1, there exists a canonical isomorphism

$$ (4.3.11) \qquad i^{\text{min},*} R_j^{\text{min}} V_\xi \xrightarrow{\sim} V_{\xi,Z}^{\sharp} $$
in $D_c^b(Z_T, \bar{\mathbb{Q}}_\ell)$. More precisely, by the proper base change theorem (see [2, XII, 5.1]), we have a canonical isomorphism

$$ (4.3.12) \qquad i^{\min,*} R_j^{\min} \mathcal{V}_\xi \xrightarrow{\sim} R \oint_{Z,*} i^{\text{tor},*} R_j^{\text{tor}} \mathcal{V}_\xi, $$

where $\oint_Z : \tilde{Z}_T \to Z_T$ is the pullback of $\oint$. Then the upshot is the existence of an isomorphism

$$ (4.3.13) \qquad R \oint_{Z,*} i^{\text{tor},*} R_j^{\text{tor}} \mathcal{V}_\xi \xrightarrow{\sim} \mathcal{V}_{\xi,Z}^{\natural}, $$

which depends only on $V_\xi|_P$, such that (4.3.11) is the composition

$$ (4.3.14) \qquad i^{\min,*} R_j^{\min} \mathcal{V}_\xi \xrightarrow{\sim} R \oint_{Z,*} i^{\text{tor},*} R_j^{\text{tor}} \mathcal{V}_\xi \xrightarrow{\sim} \mathcal{V}_{\xi,Z}^{\natural}. $$

As level varies, such isomorphisms are compatible with Hecke actions of elements of $P(\mathbb{A}^\infty)$ (when defined). (In characteristic $p$, we usually cannot expect the Hecke actions to be defined for all elements of $P(\mathbb{A}^\infty)$.)

**Proof.** The existence of the canonical isomorphism (4.3.11) follows from [67, Thm. 4.2.1 and Prop 5.2.1], while the later explanatory statements are based on the proofs there. (We could have referred to [67, Thm. 5.3.1] when the simplifying assumption $H_Q = H_C K_P$ there is satisfied, which corresponds to the assumption that $\Gamma = \Gamma'$ in our setting; cf. [67, paragraph after (3.7.4)], Remark 4.3.3, and Definition 4.3.4.) Although they were stated only over $\text{Spec}(F_0)$, the same arguments there work as long as there are systems of minimal and toroidal compactifications with the same axiomatic properties as in [67, Sec. 3.7–3.11; see also Sec. 4.9], which is the case here, by considering higher principal levels at $\ell$, thanks to Lemma 4.3.2. $\square$

**Remark 4.3.15.** The proof of Theorem 4.3.10 in [67] is consistent with the one here: In Cases (Sm), (Nm), and (Spl), this is because of the comparisons in [36, Thm. 4.1.1 and 5.1.1; see also the errata] and in [39]. In Case (Hdg), this is because [47] and [67] were based on the same characteristic zero results in [66].

Let $Y$ be a well-positioned subset of $(X_H)_T$, with associated $Y^{\natural} = \{Y_Z^{\natural}\}_Z$ as in Definition 2.2.1, and with partial minimal and toroidal compactifications $Y^{\min}$ and $Y_{\Sigma}^{\text{tor}}$ as in Definition 2.3.1 and Theorem 2.3.2. Let $i_Y^{\min}: Y_Z \to Y^{\min}$ and $i_Y^{\text{tor}}: Y_{\bar{Z}} \to Y_{\Sigma}^{\text{tor}}$ (resp. $j_Y^{\min}: Y \hookrightarrow Y^{\min}$ and $j_Y^{\text{tor}}: Y \hookrightarrow Y_{\Sigma}^{\text{tor}}$) denote the canonical locally closed (resp. open) immersions, which satisfy $i_Y^{\min} = \oint_Y \circ i_Y^{\text{tor}}$ and $j_Y^{\min} = \oint_Y \circ j_Y^{\text{tor}}$, where $\oint_Y := \oint_{Y,\Sigma}: Y_{\Sigma}^{\text{tor}} \to Y^{\min}$ is the canonical proper surjection as in (1) of Proposition 2.1.2. Now we can formulate our generalization of Theorem 4.3.10:

**Theorem 4.3.16.** Let $\mathcal{F}$ be a well-positioned complex over $Y_{\Sigma}^{\text{tor}}$, equipped with a collection $\mathcal{F}^{\natural} = \{(\mathcal{F}_Z^{\natural}, \iota_Z)\}_Z$ as in Definition 4.1.3. Let $V_\xi$ be an algebraic representation of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ on a finite-dimensional vector space over $\bar{\mathbb{Q}}_\ell$. Let $\mathcal{F}_Y, \mathcal{V}_{\xi,Y}$, and $\mathcal{V}_{\xi,Y_Z}^{\natural}$ denote the pullbacks of $\mathcal{F}$, $\mathcal{V}_\xi$, and $\mathcal{V}_{\xi,Z}^{\natural}$ (see Definition 4.3.8) to $Y, Y$, and $Y_Z$, respectively. Then, under Assumption 4.3.1, there exists a canonical isomorphism

$$ (4.3.17) \qquad i_Y^{\min,*} R_{Y,*}^{\min} (\mathcal{V}_{\xi,Y} \otimes_{\mathbb{Z}}^{L} \mathcal{F}_Y) \xrightarrow{\sim} \mathcal{V}_{\xi,Y_Z}^{\natural} \otimes_{\mathbb{Z}}^{L} \mathcal{F}_Z^{\natural} $$

in $D_c^b(Y_Z, \bar{\mathbb{Q}}_\ell)$. As level varies, such isomorphisms are compatible with actions of elements of $P(\mathbb{A}^\infty)$ when they are defined on some collections of $Y$ and $\mathcal{F}_Z^{\natural}$ (which is possible by Lemma 4.1.12).
In order to prove Theorem 4.3.16, we need some further preparations.

**Lemma 4.3.18.** *There is a canonical isomorphism*

$$ (4.3.19) \qquad i_Y^{\min,*} R_{Y,*} (\mathcal{V}_{\xi,Y} \otimes_L \mathcal{F}_Y) \xrightarrow{\sim} R \oint_{Y,Z,*} i_Y^{\text{tor},*} R_{Y,*} (\mathcal{V}_{\xi,Y} \otimes_L \mathcal{F}_Y), $$
where $\oint_{Y,Z} : Y_Z \to Y_Z$ is the pullback of $\oint_Y$.

*Proof.* This follows from the proper base change theorem (see [2, XII, 5.1]). □

**Lemma 4.3.20.** Let $\Lambda$ be as in the beginning of Section 4.1. Suppose $f: T \to T'$ is a morphism between schemes of residue characteristics prime to $\ell$, which we assume to be separated and of finite type over some regular scheme of dimension $\le 1$. Suppose $\mathcal{G} \in D_c^b(T, \Lambda)$ and $\mathcal{H} \in D_c^b(T', \Lambda)$. Then there is a canonical morphism

$$ (4.3.21) \qquad (Rf_*\mathcal{G})^L \otimes \mathcal{H} \to Rf_*(\mathcal{G}^L \otimes (f^*\mathcal{H})) $$
in $D_c^b(T', \Lambda)$.

*Proof.* By adjunction, we have a canonical morphism $f^*Rf_*(\mathcal{G}) \to \mathcal{G}$, which induces a canonical morphism $f^*(Rf_*(\mathcal{G})^L \otimes \mathcal{H}) \xrightarrow{\sim} (f^*Rf_*(\mathcal{G}))^L \otimes (f^*\mathcal{H}) \to \mathcal{G}^L \otimes (f^*\mathcal{H})$, which in turn induces the desired morphism (4.3.21). □

Consequently, we have a canonical morphism

$$ (4.3.22) \qquad (Rj_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y})^L \otimes \mathcal{F} \to Rj_{Y,*}^{\text{tor}} (\mathcal{V}_{\xi,Y} \otimes_L \mathcal{F}_Y). $$

**Lemma 4.3.23.** In Lemma 4.3.20, suppose that $f: T \to T'$ is a morphism between schemes of finite type over $S_0 = \text{Spec}(k)$ for some field $k$, which is étale locally of the form $f = g \times_{S_0} \text{Id}_S : T = E \times_S S \hookrightarrow T' = E'_S \times_S S$, where $g: E \to E'$ is a morphism between schemes separated and of finite type over $S_0$, and where $S$ is of finite type over $S_0$. Suppose moreover that such an étale local factorization of $f$ can be chosen such that $\mathcal{G}$ and $\mathcal{H}$ are the pullbacks of some $\mathcal{L} \in D_c^b(E, \Lambda)$ and $\mathcal{M} \in D_c^b(S, \Lambda)$, respectively. Then (4.3.21) is an isomorphism.

*Proof.* Up to étale localization, we may identify (4.3.21) with the Künneth morphism $(Rg_*\mathcal{L})^{\otimes L}_{S_0} \mathcal{M} \to Rf_*(\mathcal{L}^{\otimes L}_{S_0} \mathcal{M})$, which is an isomorphism by [5, Sec. 4.2.7]. □

*Remark 4.3.24.* In Lemma 4.3.20, when $f$ is compactifiable, we have a canonical isomorphism $(Rf_!\mathcal{G})^L \otimes \mathcal{H} \xrightarrow{\sim} Rf_!(\mathcal{G}^L \otimes (f^*\mathcal{H}))$, which is the well-known projection formula (see [2, XVII, 5.2.9]), by the proper base change theorem (see [2, XVII, 5.2.2]). However, such a formula is false in general for $Rf_*$ instead of $Rf_!$.

**Corollary 4.3.25.** *The canonical morphism (4.3.22) is an isomorphism.*

*Proof.* For each $Z$, let $\mathcal{U}_Z = \{(\bar{U}_i, a_{\bar{U}_i}, a_{\bar{U}_i, E(\sigma_i)}^\natural)\}_{i \in I_Z}$ be as in Definition 4.1.3. Let $U_i$ denote the common preimage of $X_\mathcal{H}$ and $E$ in $\bar{U}_i$ (see Corollary 2.1.7), and let $a_{\bar{U}_i} : \bar{Y}_{U_i} := U_i \times_{X_\mathcal{H}} Y \to Y$, $a_{\bar{U}_i} : \bar{Y}_{\bar{U}_i} := \bar{U}_i \times_{X_{\mathcal{H},\Sigma}^{\text{tor}}} Y_\Sigma^{\text{tor}} \to Y_\Sigma^{\text{tor}}$, $J_{\bar{U}_i} : \bar{Y}_{U_i} \hookrightarrow \bar{Y}_{\bar{U}_i}$, and $a_{\bar{Y}_{U_i}}^\natural : \bar{Y}_{U_i} \to \bar{Y}_Z^\natural$ denote the induced canonical morphisms. In order to show
that (4.3.22) is an isomorphism, by étale localization, it suffices to show that, for each $Z$ and each $i \in I_Z$ as above, the induced morphism

$$ (4.3.26) \qquad (RJ_{\bar{U}_{i,*}} a_{\bar{Y}_{U_i}}^* \mathcal{V}_{\xi,Y})^L \otimes (a_{\bar{Y}_{\bar{U}_i}}^* \mathcal{F}) \to RJ_{\bar{U}_{i,*}} a_{\bar{Y}_{U_i}}^* (\mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Y $$

is an isomorphism, for each $i \in I$. Since $\mathcal{V}_{\xi}$ is lisse, by using the isomorphism $t_{\bar{U}_i} : a_{\bar{Y}_{\bar{U}_i}}^* \mathcal{F} \simeq a_{\bar{Y}_{\bar{U}_i}}^{\sharp,*} \mathcal{F}_Z^{\sharp}$ as in (4.1.5), this follows from Lemma 4.3.23, as desired. $\square$

**Lemma 4.3.27.** It suffices to prove Theorem 4.3.16 when $Y = (X_H)_T$.

*Proof.* Let $J : Y \to (X_H)_T$ and $J_Z : Y_Z \to Z_T$ denote the canonical morphisms. By [2, XVII, 5.1.2 and 5.1.6], $J_Z,!i_Y^{\min,*}Rj_{Y,*}^{\min}(\mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Y) \cong i^{\min,*}Rj_{*}^{\min}(J_!(\mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Y)$. Since $J_!(\mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Y) \cong \mathcal{V}_{\xi}^L \otimes (J_!(\mathcal{F}_Y))$ and $J_Z,!(\mathcal{V}_{\xi,Z}^{\sharp})^L \otimes \mathcal{F}_Z^{\sharp}) \cong \mathcal{V}_{\xi,Z}^{\sharp} \otimes (J_Z,!(\mathcal{F}_Z^{\sharp}))$ because $\mathcal{V}_{\xi}$ and $\mathcal{V}_{\xi,Z}^{\sharp}$ are direct sums of shifted lisse sheaves, we are reduced to constructing (4.3.17) when $Y, \mathcal{F}$, and $\mathcal{F}_Z^{\sharp}$ are replaced with $(X_H)_T, J_!\mathcal{F}$, and $J_Z,!(\mathcal{F}_Z^{\sharp})$, respectively. Hence, this lemma follows from (1) of Lemma 4.2.1. $\square$

*Proof of Theorem 4.3.16.* By Lemma 4.3.27, we may and we shall assume that $Y = (X_H)_T$. In this case, the desired (4.3.17) is the following composition

$$ \begin{align} & i_Y^{\min,*} R j_{Y,*}^{\min} (\mathcal{V}_{\xi,Y} \otimes \mathcal{F}_Y) \quad (4.3.19) \nonumber \\ & \xrightarrow{\sim} R \oint_{Y_{Z,*}} i_Y^{\text{tor},*} R j_{Y,*}^{\text{tor}} (\mathcal{V}_{\xi,Y} \otimes \mathcal{F}_Y) \nonumber \\ & R \oint_{Y_{Z,*}} i_Y^{\text{tor},*} (4.3.22)^{-1} \xrightarrow{\sim} R \oint_{Y_{Z,*}} i_Y^{\text{tor},*} ((R j_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}) \nonumber \\ & \xrightarrow{\text{can.}} R \oint_{Y_{Z,*}} ((i_Y^{\text{tor},*} R j_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y})^L \otimes (i_Y^{\text{tor},*} \mathcal{F})) \tag{4.3.28} \\ & R \oint_{Y_{Z,*}} (\text{Id} \otimes \iota_Z) \xrightarrow{\sim} R \oint_{Y_{Z,*}} ((i_Y^{\text{tor},*} R j_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y})^L \otimes (\oint_{Y_Z}^* \mathcal{F}_Z^{\sharp})) \nonumber \\ & \xrightarrow{\text{proj.}} (R \oint_{Y_{Z,*}} i_Y^{\text{tor},*} R j_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Z^{\sharp} \nonumber \\ & \xrightarrow{(4.3.13)} \mathcal{V}_{\xi,Y_Z}^{\sharp} \otimes \mathcal{F}_Z^{\sharp}, \nonumber \end{align} $$

where (4.3.13) is applicable because $Y = (X_H)_T$ and so $\mathcal{V}_Z^{\sharp} = Z_T$, and where the isomorphism denoted “proj.” is based on the well-known projection formula (see [2, XVII, 5.2.9] and Remark 4.3.24), which is applicable because $\oint_{Y_{Z,*}}$ is proper. $\square$

*Remark 4.3.29.* Instead of using Lemma 4.3.27 and (4.3.13), we could have directly established the isomorphism $R \oint_{Y_{Z,*}} i_Y^{\text{tor},*} R j_{Y,*}^{\text{tor}} \mathcal{V}_{\xi,Y} \xrightarrow{\sim} \mathcal{V}_{\xi,Y_Z}^{\sharp}$ needed in the last step of (4.3.28), without assuming $Y = (X_H)_T$, by the same argument as in the proof of Theorem 4.3.10, thanks to Theorem 2.3.2, Proposition 2.4.2, and Lemma 4.3.2.

*Remark 4.3.30.* Theorems 4.3.10 and 4.3.16 have analogues over $Z_\ell$—we then need to replace the Lie algebra cohomology in Definitions 4.3.5 and 4.3.8 by continuous group cohomology. (See [67, Thm. 4.2.1].)

*Remark 4.3.31.* We have seen in Theorem 4.3.16 that $i_Y^{\min,*} R j_{Y,*}^{\min}(\mathcal{V}_{\xi,Y})^L \otimes \mathcal{F}_Y)$, for any well-positioned complex $\mathcal{F}$, is canonically isomorphic to the (derived) tensor product of $\mathcal{V}_{\xi,Y_Z}^{\sharp}$ and $\mathcal{F}_Z^{\sharp}$, where $\mathcal{V}_{\xi,Y_Z}^{\sharp}$ is the pullback of the complex $\mathcal{V}_{\xi,Z}^{\sharp}$ essentially
defined in [67]. This is intriguing because the minimal compactification is far from being a product near the boundary. As a consequence of this sheaf-theoretic product structure, we see that the following problems are, at least intuitively, unrelated:

(1) Studying the bad reduction of a Shimura variety at $p$, or studying its nearby cycles, or studying the constant sheaves or intersection complexes on natural stratifications such as the Newton or the Ekedahl-Oort ones.

(2) Studying the behavior of such complexes near the boundary strata of the minimal compactification.

**Remark 4.3.32.** By Theorem 4.3.16, under Assumption 4.3.1, $V_{\xi, Y_Z}^{\natural} \otimes_F \mathcal{F}_Z^{\natural}$ is uniquely determined by $\mathcal{F}$ and $V_\xi$. When $V_\xi = \bar{Q}_\ell$ is the trivial representation, $V_{\xi, Z}^{\natural}$ has a direct summand $\bar{Q}_\ell$ in degree zero, by “Lieberman’s trick”. (This is a representation-theoretic statement independent of the consideration of integral models.) Therefore, $\mathcal{F}_Z^{\natural}$ is uniquely determined by $V_{\xi, Y_Z}^{\natural} \otimes_F \mathcal{F}_Z^{\natural}$ in this case, and also by $\mathcal{F}$ (cf. Remark 4.1.10).

**4.4. Mantovan's formula with boundary terms.** The goal of this subsection is to present a generalization of [48, Sec. 8, Thm. 22] and [49, Thm. 3.1] in a way different from our previous one [43, Thm. 6.32], which also shows certain boundary terms. (See Theorem 4.4.7 and Remark 4.4.13 below.)

Let us resume the context of Section 3.3. Assume moreover that $p$ is a good prime for the integral PEL datum $(\mathcal{O}, \star, L, \langle \cdot, \cdot \rangle, h_0)$ as in [38, Def. 1.4.1.1]; that we are in Case (Nm) with the trivial collection $J = \{j_0\}$ with $\{(g_{j_0}, L_{j_0}, \langle \cdot, \cdot \rangle_{j_0}) = \{(1, L, \langle \cdot, \cdot \rangle)\}$, as in [40, Ex. 2.3]; and that $\mathcal{O} \otimes_{\mathbb{Z}} \mathbb{Q}$ is simple and involves no factor of type D, in the sense of [38, Def. 1.2.1.15]. (This is the setting in [43, Sec. 6.3].) In this case, $(G \otimes_{\mathbb{Z}} \mathbb{Q})^{\text{ad}}$ is also simple over $\mathbb{Q}$. Let us also import Assumption 4.3.1 from Section 4.3. For the sake of concreteness, let us fulfil the requirement in Assumption 4.3.1 that $C \to Z$ is an abelian scheme torsor for each $Z$ by assuming that $\mathcal{H}$ is a principal level, by Lemma 2.1.9. This will be harmless for our purpose in this subsection, because principal levels are cofinal among all choices of levels.

Suppose $S = \text{Spec}(\mathcal{O}_K)$, where $K$ is the $v$-adic completion of $F_0$ at a place $v|p$. Consider the nearby cycle functors $R\Psi_{X_H}$ etc defined by some choices of geometric points $\bar{\eta}$ and $\bar{s}$ above the generic and special points $\eta$ and $s$ of $S$, respectively, as in [43, Sec. 5.1]. For each $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$, and for each irreducible algebraic representation $\xi$ of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ on a finite-dimensional vector space $V_\xi$ over $\bar{Q}_\ell$, recall that Mantovan proved the following formula (see [43, Thm. 6.26], with $\mathcal{H}$ running over a cofinal system of higher levels, such as all higher principal levels)

$$ (4.4.1) \quad \sum_i (-1)^i [H_{\bar{\eta}, c}^i(X_{\bar{s}}^b, (R\Psi\chi(V_\xi))|_{X_{\bar{s}}^b})] = \sum_j (-1)^j \mathcal{E}_b([H_{\bar{\eta}, c}^j(|\mathbf{g}^b, V_\xi)]) $$

between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$, where $G(\mathbb{Q}_p)^+ := \{g \in G(\mathbb{Q}_p) : g^{-1}(L \otimes_{\mathbb{Z}} \mathbb{Z}_p) \subset L \otimes_{\mathbb{Z}} \mathbb{Z}_p\}$ (cf. [48, p. 599]), and where $\text{lg}^b$ is the limit of the so-called *Igusa varieties* over certain central leaf of $X_{\bar{s}}^b$, and where $\mathcal{E}_b$ is the so-called *Mantovan functor*. (For more details, see [43, Sec. 6.3] and the references there to [48], [49], and other works.)
Let $\mathbb{Z}$ be a stratum of $X_{\mathcal{H}}^{\min}$, which is associated with some parabolic subgroup $P$ of $G \otimes_{\mathbb{Z}} \mathbb{Q}$, as in Lemma 3.3.6, with its various quotients of subgroups defined in Definition 3.3.8. Since $O \otimes_{\mathbb{Z}} \mathbb{Q}$ is simple by assumption, $P$ is either all of $G \otimes_{\mathbb{Z}} \mathbb{Q}$, or a proper maximal parabolic subgroup of $G \otimes_{\mathbb{Z}} \mathbb{Q}$. For each $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$, by Proposition 3.3.9 and its proof, $Y := (X_{\mathcal{H}})_s^b$ is a well-positioned subset of $(X_{\mathcal{H}})_s$, associated with the collection $Y^\natural := \{Y^\natural_Z := Z_s^b\}_{Z}$, where $Z_s^b$ is either the empty subset or the Newton stratum $Z_s^{b_h}$ of $Z_s$. In the latter case, $(X_{\mathcal{H}})_s^b \neq \emptyset$, and $b$ is the image of some $b_P \in B(P \otimes_{\mathbb{Z}} \mathbb{Q}_p)$, with image $b_M = (b_l, b_h)$ in $B(M \otimes_{\mathbb{Z}} \mathbb{Q}_p) \simeq B(G_l \otimes_{\mathbb{Z}} \mathbb{Q}_p) \times B(G_h \otimes_{\mathbb{Z}} \mathbb{Q}_p)$. Conversely, by the construction of Mumford families, every $b_h \in B(G_h \otimes_{\mathbb{Z}} \mathbb{Q}_p)$ such that $Z_s^{b_h}$ is nonempty comes from some $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$ as above. For such $Y$ and $Y^\natural$, let $(X_{\mathcal{H},\Sigma}^{\text{tor}})_s^b := Y_{\Sigma}^{\text{tor}}$ be defined as in Definition 2.3.1.

By Lemmas 4.2.1 and 4.2.10, $R\Psi_{X_{\mathcal{H},\Sigma}^{\text{tor}}}(\bar{Q}_{\ell})|_{(X_{\mathcal{H},\Sigma}^{\text{tor}})_s^b}$ is a well-positioned complex over $(X_{\mathcal{H},\Sigma}^{\text{tor}})_{\bar{s}}$, equipped with the collection $\{R\Psi_Z(\bar{Q}_{\ell})|_{Z_s^b}\}_{Z}$. By abuse of notation, we shall still denote their extensions by zero to the whole schemes $(X_{\mathcal{H},\Sigma}^{\text{tor}})_{\bar{s}}$ and $Z_{\bar{s}}$, respectively, by the same symbols. Therefore, by Theorem 4.3.16, we have an isomorphism

$$ (4.4.2) \qquad i_{\bar{s}}^{\min,*} R j_{\bar{s},*}^{\min} (V_{\xi}^{L} \otimes R \Psi_{X_{\mathcal{H}}} (\bar{Q}_{\ell}) |_{(X_{\mathcal{H}})_{\bar{s}}^{b}}) \xrightarrow{\sim} V_{\xi,Z}^{\natural} \otimes_{\mathbb{Z}}^{L} R \Psi_Z (\bar{Q}_{\ell}) |_{Z_{\bar{s}}^{b}} $$

in $D_c^b(Z_{\bar{s}}, \bar{Q}_{\ell})$, where $V_{\xi}$ and $V_{\xi,Z}^{\natural}$ also denote their pullbacks to $(X_{\mathcal{H}})_{\bar{s}}$ and $Z_{\bar{s}}$, respectively. Note that (4.4.2) is not just an isomorphism in $D_c^b(Z_{\bar{s}}, \bar{Q}_{\ell})$, but also an isomorphism in $D_c^b(Z_{\bar{s}} \times \bar{\eta}, \bar{Q}_{\ell})$, because the proof of Theorem 4.3.16 only uses canonical isomorphisms which are compatible with the actions of $\text{Gal}(k(\bar{\eta})/k(\eta)) = \text{Gal}(\bar{K}/K)$ (cf. Lemma 4.2.10 and its proof).

Since $V_{\xi}$ is lisse, the canonical morphism $V_{\xi}^{L} \otimes R\Psi_{X_{\mathcal{H}}} (\bar{Q}_{\ell}) \to R\Psi_{X_{\mathcal{H}}} (V_{\xi})$ is an isomorphism in $D_c^b((X_{\mathcal{H}})_{\bar{s}} \times \bar{\eta}, \bar{Q}_{\ell})$. Similarly, since the complex $V_{\xi,Z}^{\natural}$ is a direct sum of shifted lisse sheaves (see Definition 4.3.8), the canonical morphism $V_{\xi,Z}^{\natural} \otimes_{\mathbb{Z}}^{L} R\Psi_Z(\bar{Q}_{\ell}) \to R\Psi_Z(V_{\xi,Z}^{\natural})$ is an isomorphism in $D_c^b(Z_{\bar{s}} \times \bar{\eta}, \bar{Q}_{\ell})$. Hence, (4.4.2) can be rewritten as an isomorphism

$$ (4.4.3) \qquad i_{\bar{s}}^{\min,*} R j_{\bar{s},*}^{\min} (R \Psi_{X_{\mathcal{H}}} (V_{\xi}) |_{(X_{\mathcal{H}})_{\bar{s}}^{b}}) \xrightarrow{\sim} R \Psi_Z (V_{\xi,Z}^{\natural}) |_{Z_{\bar{s}}^{b}} $$

in $D_c^b(Z_{\bar{s}}, \bar{Q}_{\ell})$, where $R\Psi_{X_{\mathcal{H}}} (V_{\xi})|_{(X_{\mathcal{H}})_{\bar{s}}^b}$ and $R\Psi_Z(V_{\xi,Z}^{\natural})|_{Z_{\bar{s}}^b}$ abusively denote their extensions by zero to the whole schemes $(X_{\mathcal{H}})_{\bar{s}}$ and $Z_{\bar{s}}$.

As $\mathcal{H} \subset G(\hat{\mathbb{Z}})$ varies among principal levels, the associated $\mathcal{H}_P := \mathcal{H} \cap P(\mathbb{A}^{\infty})$, $\mathcal{H}_U := \mathcal{H} \cap U(\mathbb{A}^{\infty})$, $\mathcal{H}_M := \mathcal{H}_P/\mathcal{H}_U$, and $\mathcal{H}_h := (M(\mathbb{A}^{\infty}) \to G_h(\mathbb{A}^{\infty}))(\mathcal{H}_M)$ (see Definition 3.3.8 and Remark 4.3.3) vary accordingly among principal levels in the respective ambient groups. We can also consider a projective system of higher level analogues of $Z$, as $\mathcal{H}'_h = \mathcal{H}_{h,\Phi} = \mathcal{H}_h \subset G_h(\mathbb{A}^{\infty})$ vary among principal levels, which we abusively denote as $\hat{Z}$.

However, to formulate the boundary version of (4.4.1), it is better to consider the $G_l(\mathbb{A}^{\infty})$-orbit of $\hat{Z}$ (which is a disjoint union of locally closed subsets in the limit of the projective system $X^{\min} = \{X_{\mathcal{H}}^{\min}\}_{\mathcal{H}}$ parameterized by principal levels $\mathcal{H}$), which we abusively denote as $X_P$. The elements of $G_l(\mathbb{A}^{\infty})$ act by twisting the torus arguments in the cusp labels, without changing the associated filtration
Z. The finite level object $X_{P,\mathcal{H}}$ is the disjoint union of all $Z$ (the strata of $X_{\mathcal{H}}^{\min}$) parameterized by cusp labels $[(Z_{\mathcal{H}}, \Phi_{\mathcal{H}}, \delta_{\mathcal{H}})]$ with the same underlying $\mathcal{H}$-orbit $Z_{\mathcal{H}}$, where $Z_{\mathcal{H}} \otimes_{Z} \mathbb{Q}$ has stabilizer $P \otimes A^{\infty}$. Such cusp labels $[(Z_{\mathcal{H}}, \Phi_{\mathcal{H}}, \delta_{\mathcal{H}})]$ are in bijection with $G_l(\mathbb{Q}) \backslash G_l(A^{\infty}) / \mathcal{H}'_l = G_l(\mathbb{Q}) \backslash G_l(A^{\infty}) / \mathcal{H}_l$. The generic fiber $(X_P)_\eta$ admits an action of $P(A^{\infty})$ with trivial restriction to $U(A^{\infty})$, and the induced action of $M(A^{\infty}) \cong P(A^{\infty}) / U(A^{\infty}) \cong G_l(A^{\infty}) \times G_h(A^{\infty})$ is the combination of the action of $G_l(A^{\infty})$ as above and the action of $G_h(A^{\infty})$ on $\hat{Z}_\eta$.

Since $\mathbf{P}^+ / \mathbb{R}_{>0}^\times \cong G_l(\mathbb{R}) / \mathcal{U}_{l,\infty}$ (where $\mathcal{U}_{l,\infty}$ is the stabilizer of some element of $\mathbf{P}^+$) is isomorphic to the Riemannian symmetric space of $G_l(\mathbb{R})$, the group cohomology of $\Gamma$ can be computed using the cohomology of the manifold $\Gamma \backslash \mathbf{P}^+ / \mathbb{R}_{>0}^\times$. By putting together the $\mathcal{V}_{\xi,Z}^{\natural}$ over each $Z$ at level $\mathcal{H}$, each of which being a cohomology complex of $\Gamma$, we obtain the cohomology complex over $G_l(\mathbb{Q}) \backslash G_l(\mathcal{A}) / \mathcal{U}_{l,\infty} \mathcal{H}_l$. Thus, by taking limit over all principal levels $\mathcal{H}$, we obtain a limit $\mathcal{V}_{\xi,P}^{\natural}$ over $X_P$, which admits an admissible action of $G_l(A^{\infty})$.

Suppose that $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$ is the image of some $b_P \in B(P \otimes_{\mathbb{Q}} \mathbb{Q}_p)$ with image $b_M = (b_l, b_h)$ in $B(M \otimes_{\mathbb{Q}} \mathbb{Q}_p) \cong B(G_l \otimes_{\mathbb{Q}} \mathbb{Q}_p) \times B(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)$. Then we can define $(X_P)_s^{b_h}$ to be the $G_l(A^{\infty})$-orbit of $\hat{Z}_s^{b_h}$, the preimage of $Z_s^{b_h}$, and write symbolically

$$ (4.4.4) \qquad i_{P, \bar{s}}^{\min, *} R j_{\bar{s}, *}^{\min} (R \Psi_X(\mathcal{V}_{\xi}) |_{X_{\bar{s}}^b}) \xrightarrow{\sim} R \Psi_{X_P}(\mathcal{V}_{\xi, P}^{\natural}) |_{(X_P)_s^{b_h}}. $$

By applying the analogue of (4.4.1) to $X_P$, we obtain the following equality

$$ (4.4.5) \qquad \sum_i (-1)^i [H_{\acute{e}t,c}^i((X_P)_{\bar{s}}, i_{P, \bar{s}}^{\min,*} R j_{\bar{s},*}^{\min} (R \Psi_X(\mathcal{V}_{\xi})|_{X_{\bar{s}}^b})] \\ = \sum_j (-1)^j \mathcal{E}_{b_h} ([H_{\acute{e}t,c}^j(\mathbf{I}_{g_P^{b_h}}, \mathcal{V}_{\xi,P}^{\natural})]) $$

between virtual representations of $G_l(A^{\infty}) \times G_h(A^{\infty,p}) \times G_h(\mathbb{Q}_p)^+ \times W_K$, where $G_h(\mathbb{Q}_p)^+$ is the sub-monoid of $G_h(\mathbb{Q}_p)$ analogous to the sub-monoid $G(\mathbb{Q}_p)^+$ of $G(\mathbb{Q}_p)$, and where $\mathbf{I}_{g_P^{b_h}}$ is the analogous limit of Igusa varieties over $(X_P)_s^{b_h}$ (whose pullback to $\hat{Z}_s$ is the usual one associated with $G_h$, over some chosen central leaf at the bottom level), where each $H_{\acute{e}t,c}^j(\mathbf{I}_{g_P^{b_h}}, \mathcal{V}_{\xi,P}^{\natural})$ admits an admissible/continuous action of $G_l(A^{\infty}) \times G_h(A^{\infty,p}) \times J_{b_h}(\mathbb{Q}_p) \times W_K$, and where

$$ (4.4.6) \qquad \mathcal{E}_{b_h} : \text{Groth}(G_l(A^{\infty}) \times G_h(A^{\infty,p}) \times J_{b_h}(\mathbb{Q}_p)) \to \text{Groth}(G_l(A^{\infty}) \times G_h(A^{\infty}) \times W_K). $$

is the combination of the identity functor on Groth($G_l(A^{\infty})$) and the corresponding Mantovan functor for $G_h$ (cf. [43, (6.20)]).

**Theorem 4.4.7** (cf. [48, Sec. 8, Thm. 22] and [49, Thm. 3.1], and also [43, Thm. 6.26 and 6.32]). For each $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$, we have an equality

$$ (4.4.8) \qquad \sum_i (-1)^i [H_{\acute{e}t}^i(X_{\bar{s}}, (R\Psi_X(\mathcal{V}_{\xi})))|_{X_{\bar{s}}^b}] \\ = \sum_P \mathrm{Ind}_{P(A^{\infty})}^{G(A^{\infty})} \left( \sum_j (-1)^j \mathcal{E}_{b_h} ([H_{\acute{e}t,c}^j(\mathbf{I}_{g_P^{b_h}}, \mathcal{V}_{\xi,P}^{\natural})]) \right) $$

between virtual representations of $G(A^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$, where the sum over $P$ runs over a complete (finite) set of representatives of conjugacy classes of (rational)
parabolic subgroups of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ that is either (proper) maximal or $G \otimes_{\mathbb{Z}} \mathbb{Q}$ itself such that $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$ is the image of some $b_P \in B(P \otimes_{\mathbb{Q}} \mathbb{Q}_p)$ with image $b_M = (b_l, b_h)$ in $B(M \otimes_{\mathbb{Q}} \mathbb{Q}_p) \cong B(G_l \otimes_{\mathbb{Q}} \mathbb{Q}_p) \times B(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)$. (Note that $b_h$ depends on P.)

Overall, we have an equality

$$ (4.4.9) \qquad \sum_i (-1)^i [H_{\hat{\mathfrak{e}}t}^i(X_{\bar{s}}, R\Psi_X(\mathcal{V}_{\xi}))] \\ = \sum_P \mathrm{Ind}_{P(\mathbb{A}^{\infty})}^{G(\mathbb{A}^{\infty})} \left( \sum_{b_h \in B(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)} \sum_j (-1)^j \mathcal{E}_{b_h}([H_{\hat{\mathfrak{e}}t,c}^j(\mathbf{lg}_P^{b_h}, \mathcal{V}_{\xi,P}^b)]) \right) $$

between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$, and an equality

$$ (4.4.10) \qquad \sum_i (-1)^i [H_{\hat{\mathfrak{e}}t}^i(X_{\bar{\eta}}, \mathcal{V}_{\xi})] \\ = \sum_P \mathrm{Ind}_{P(\mathbb{A}^{\infty})}^{G(\mathbb{A}^{\infty})} \left( \sum_{b_h \in B(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)} \sum_j (-1)^j \mathcal{E}_{b_h}([H_{\hat{\mathfrak{e}}t,c}^j(\mathbf{lg}_P^{b_h}, \mathcal{V}_{\xi,P}^b)]) \right) $$

between virtual representations of $G(\mathbb{A}^{\infty}) \times W_K$, where the sums over $P$ run over a complete (finite) set of representatives of conjugacy classes of (rational) parabolic subgroups of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ that is either (proper) maximal or $G \otimes_{\mathbb{Z}} \mathbb{Q}$ itself.

*Proof.* Let $X^{\min}$ denote the limit of $X_H^{\min}$ over principal levels $\mathcal{H} \subset G(\hat{\mathbb{Z}})$. By [39, Prop. A.5.8 and A.5.9], we have a stratification $\coprod_P (G(\mathbb{A}^{\infty}) \cdot X_P)$ of $X^{\min}$, where the disjoint union is over the same $P$'s as in the second paragraph of the statement of the theorem, and where $P(\mathbb{A}^{\infty})$ is the stabilizer of $X_P$ in $G(\mathbb{A}^{\infty})$. By (2) of Proposition 2.1.2, the same are true over the fibers over $\bar{s}$. By Proposition 3.3.9, $(X_P)_{\bar{s}}$ contains some nonempty $Z_{\bar{s}}^b$ only when $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$ is the image of some $b_P \in B(P \otimes_{\mathbb{Q}} \mathbb{Q}_p)$. Thus, we obtain the equality (4.4.8) between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$ by summing the parabolic inductions of (4.4.5) over the same $P$'s as in the first paragraph of the statement of the theorem.

As in [43, (6.24)], we have an equality

$$ (4.4.11) \qquad \sum_i (-1)^i [H_{\hat{\mathfrak{e}}t}^i(X_{\bar{s}}, R\Psi_X(\mathcal{V}_{\xi}))] \\ = \sum_{b \in B(G \otimes_{\mathbb{Q}} \mathbb{Q}_p)} \sum_i (-1)^i [H_{\hat{\mathfrak{e}}t}^i(X_{\bar{s}}^b, (R\Psi_X(\mathcal{V}_{\xi}))|_{X_{\bar{s}}^b})] $$

between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$. By combining the contributions from all $b \in B(G \otimes_{\mathbb{Z}} \mathbb{Q}_p)$, we obtain the equality (4.4.9) between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$. Then we obtain (4.4.10) as an equality between virtual representations of $G(\mathbb{A}^{\infty,p}) \times G(\mathbb{Q}_p)^+ \times W_K$, by [43, Cor. 5.20 and Rem. 5.35], which extends to the desired equality between virtual representations of $G(\mathbb{A}^{\infty}) \times W_K$, by the same explanation as in [48, Sec. 8]. $\square$

*Remark* 4.4.12. The term $H_{\hat{\mathfrak{e}}t}^i(X_{\bar{s}}, (R\Psi_X(\mathcal{V}_{\xi}))|_{X_{\bar{s}}^b})$ in (4.4.9) differs from both $H_{\hat{\mathfrak{e}}t}^i(X_{\bar{s}}, (R\Psi_X(\mathcal{V}_{\xi}))|_{X_{\bar{s}}^b})$ and $H_{\hat{\mathfrak{e}}t,c}^i(X_{\bar{s}}, (R\Psi_X(\mathcal{V}_{\xi}))|_{X_{\bar{s}}^b})$ in general.
**Remark 4.4.13.** Compared with its analogue for $\sum_i (-1)^i [H_{\tilde{\mathfrak{e}}t,c}^i(\mathcal{X}_{\bar{\eta}}, \mathcal{V}_{\xi})]$ in [43, (6.33)], the formula (4.4.10) is more elaborate and contains the additional (boundary cohomology) terms parameterized by all proper maximal parabolic subgroups P of $G \otimes_{\mathbb{Z}} \mathbb{Q}$, although (in theory) these two formulae imply each other by duality.

**Remark 4.4.14.** Here is an alternative argument for establishing the equality (4.4.10) (which does not rely on Proposition 3.3.9 and Theorem 4.3.16): By applying the analogues of [43, Cor. 5.20, Rem. 5.35, and (6.24)] and (4.4.1) for $X_P$, we obtain an equality

$$ (4.4.15) \qquad \sum_i (-1)^i [H_{\tilde{\mathfrak{e}}t,c}^i((X_P)_{\bar{\eta}}, \mathcal{V}_{\xi,P}^{\bar{\eta}})] \\ = \sum_{b_h \in \mathbb{B}(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)} \sum_j (-1)^j \mathcal{E}_{b_h}([H_{\tilde{\mathfrak{e}}t,c}^j(\mathbf{\Gamma}_{P}^{b_h}, \mathcal{V}_{\xi,P}^{\bar{\eta}})]) $$

between virtual representations of $G_l(\mathbb{A}^{\infty}) \times G_h(\mathbb{A}^{\infty,p}) \times G_h(\mathbb{Q}_p)^+ \times W_K$, which extends to an equality between virtual representations of $G_l(\mathbb{A}^{\infty}) \times G_h(\mathbb{A}^{\infty}) \times W_K$, by the same explanation as in [48, Sec. 8]. By Theorem 4.3.10 (the original theorem of Pink's), we can rewrite (4.4.15) as an equality

$$ (4.4.16) \qquad \sum_i (-1)^i [H_{\tilde{\mathfrak{e}}t,c}^i((X_P)_{\bar{\eta}}, i_{P,\bar{\eta}}^{\min,*} R j_{\bar{\eta},*}^{\min} \mathcal{V}_{\xi})] \\ = \sum_{b_h \in \mathbb{B}(G_h \otimes_{\mathbb{Q}} \mathbb{Q}_p)} \sum_j (-1)^j \mathcal{E}_{b_h}([H_{\tilde{\mathfrak{e}}t,c}^j(\mathbf{\Gamma}_{P}^{b_h}, \mathcal{V}_{\xi,P}^{\bar{\eta}})]) $$

between virtual representations of $G_l(\mathbb{A}^{\infty}) \times G_h(\mathbb{A}^{\infty}) \times W_K$. Thus, we obtain the desired equality (4.4.10) by summing the parabolic inductions of (4.4.16) over the same P's as in the second paragraph of Theorem 4.4.7. However, such an argument cannot isolate the contribution of a single Newton stratum as in (4.4.8).

**4.5. Morel's formula.** The goal of this subsection is to generalize Morel's results concerning restrictions to boundary strata of intersection complexes of automorphic étale sheaves in [55, Sec. 5.2], [56, Sec. 4.2], and [58, Sec. 1.4] (see also [57]), and their variants in [75, Sec. 9], [76], and [77], to the case of general tensor products of automorphic étale sheaves with well-positioned pure perverse sheaves. (See Theorems 4.5.26 and 4.5.37 below.)

We shall resume the context of Section 4.3, with Assumption 4.3.1, because our generalization of Morel's results depends on our generalization of Pink's formula there, just as the original results of Morel's depend on the original formula of Pink's. For simplicity, we shall make the following assumption in this subsection:

**Assumption 4.5.1.** We are in Cases (Sm), (Nm), or (Spl), and $C \to Z$ is an abelian scheme torsor for each Z.

Recall that, in Section 4.3, we have extended the construction in [43, Prop. 3.2] to a functor from $D^b(G \otimes_{\mathbb{Z}} \mathbb{Q}, \bar{\mathbb{Q}}_\ell)$ to $D_c^b(X_H, \bar{\mathbb{Q}}_\ell)$, and we have similar constructions over each stratum Z of $X_H^{\min}$.

Let $\bar{\mathbb{Q}}$ denote the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$, and fix a choice of a field homomorphism $\bar{\mathbb{Q}} \hookrightarrow \bar{\mathbb{Q}}_\ell$. Since we will be working with weights, it is more convenient
to work with $D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$, which is isomorphic to $D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$ (via the above choice of $\bar{Q} \hookrightarrow \bar{Q}_\ell$).

The $h_0 : \mathbb{C} \to \text{End}_{\mathbb{O}\otimes_Z \mathbb{R}}(L \otimes_Z \mathbb{R})$ in Assumption 2.1.1 (cf. [38, Def. 1.2.1.3]) induces by restriction a central cocharacter $\mathbf{G}_m \otimes_Z \mathbb{R} \to G \otimes_Z \mathbb{R}$, whose reciprocal descends to a cocharacter $w : \mathbf{G}_m \otimes_Z \bar{Q} \to G \otimes_Z \bar{Q}$ (cf. [13, Sec. 1.1.1] for the general definition). For simplicity, we shall also denote by $w$ the base change $\mathbf{G}_m \otimes_Z \bar{Q} \to G \otimes_Z \bar{Q}$.

**Definition 4.5.2.** We say a complex $V \in D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$ is **pure of weight** $-a \in \mathbb{Z}$ if, by pullback under the cocharacter $w$, the group $\mathbf{G}_m \otimes_Z \bar{Q}$ acts on $\mathcal{H}^n(V)$ via the character $x \mapsto x^{a+n}$, for each $n \in \mathbb{Z}$.

**Example 4.5.3.** The representation $L \otimes_Z \bar{Q}_\ell$ of $G \otimes_Z \bar{Q}$ (with its tautological action, placed in degree zero) is pure of weight 1.

**Lemma 4.5.4.** Suppose $V \in D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$ is pure of weight $-a \in \mathbb{Z}$ as in Definition 4.5.2, with associated complex $\mathcal{V} \in D_c^b(X_H, \bar{Q}_\ell)$. For each functorial point $\bar{s} = \text{Spec}(k) \to S$, where $k$ is a perfect field of characteristic $p$, the pullback $\mathcal{V}_{\bar{s}}$ of $\mathcal{V}$ to $(X_H)_{\bar{s}}$ is mixed of weight $a$ (cf. [14, Def. 6.2.2]). More precisely, $\mathcal{H}^n(\mathcal{V}_{\bar{s}})$ is lisse and pointwise pure of weight $a+n$, for each $n \in \mathbb{Z}$ (see [14, Def. 1.2.2]).

*Proof.* This follows from [43, Prop. 3.2] and [67, Prop. 5.6.2 and Lem. 5.6.6]. □

**Remark 4.5.5.** Beware that in general $\mathcal{V}$ is not pure (as in [14, Def. 6.2.4]) when $(X_H)_{\bar{s}}$ is not smooth (over $\bar{s}$). Nevertheless, the restriction of $\mathcal{V}_{\bar{s}}$ to every smooth subscheme of $(X_H)_{\bar{s}}$ is pure of weight $a$ (see [14, Ex. 6.2.5(b)]).

**Lemma 4.5.6.** Consider any $t = \text{Spec}(k) \to S$, where $k$ is a field of characteristic $p$ that is either algebraically closed or finite. Let $V \in D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$ be pure of weight $-a \in \mathbb{Z}$, with associated $\mathcal{V} \in D_c^b(X_H, \bar{Q}_\ell)$, whose pullback to $(X_H)_t$ we abusively denote by the same symbols. Let $\mathcal{F}$ be a pure shifted perverse sheaf of weight $b \in \mathbb{Z}$ over $(X_H)_t$. Then $\mathcal{V} \otimes_t \mathcal{F}$ is pure of weight $a+b$, and is isomorphic to a direct sum of shifted perverse sheaves. If $V$ is concentrated in degree zero, and if $\mathcal{F}$ is a perverse sheaf, then $\mathcal{V} \otimes_t \mathcal{F}$ is also a perverse sheaf.

*Proof.* The tensor product $\mathcal{V} \otimes_t \mathcal{F}$ is isomorphic to a direct sum of shifted perverse sheaves because the fibered category of perverse sheaves is a category module over the symmetric monoidal fibered category of lisse sheaves. Consider $V^\vee := \text{RHom}(V, \bar{Q}_\ell)$ in $D^b(G \otimes_Z \bar{Q}, \bar{Q}_\ell)$, with associated $V^\vee := \text{RHom}(\mathcal{V}, \bar{Q}_\ell)$ in $D_c^b(X_H, \bar{Q}_\ell)$. In order to show that $\mathcal{V} \otimes_t \mathcal{F}$ is pure of weight $a+b$, it suffices to note that $\mathcal{V} \otimes_t \mathcal{F}$ is mixed of weight $\le a+b$ (by Lemma 4.5.4), and that $D_{(X_H)_t}(\mathcal{V} \otimes_t \mathcal{F}) \cong \text{RHom}(\mathcal{V} \otimes_t \mathcal{F}, \mathcal{K}_{(X_H)_t}) \cong \mathcal{V}^\vee \otimes_t \text{RHom}(\mathcal{F}, \mathcal{K}_{(X_H)_t}) \cong \mathcal{V}^\vee \otimes_t D_{(X_H)_t}(\mathcal{F})$ is mixed of weight $\le -a-b$. (See [5, Sec. 5.1.13 and Stabilités 5.1.14].) If $V$ is concentrated in degree zero, then $\mathcal{V}$ is a lisse sheaf concentrated in degree zero, and hence $\mathcal{V} \otimes_t \mathcal{F}$ is a perverse sheaf if $\mathcal{F}$ is. □
Let $Z$ be a stratum of $X_H^{\min}$, which is associated with some (rational) parabolic subgroup $P$ of $G \otimes_Z \mathbb{Q}$, as in Lemma 3.3.6, which is the stabilizer of a symplectic filtration $V = \{V_{-i}\}_{i \in Z}$ of $L \otimes_Z \mathbb{Q}$ such that $V_{-2} \otimes_Z A^{\infty}$ lies in the $\mathcal{H}$-orbit of $Z_{-2} \otimes_Z \mathbb{Q}$ for some representative $Z = \{Z_{-i}\}_{i \in Z}$ of $Z_H$.

**Definition 4.5.7.** Consider a collection $\mathcal{C} = \{Z^{(c)}\}_c$ of strata of $X_H^{\min}$ containing the open stratum $X_H$. Then the incidence relation among the strata of $X_H^{\min}$ induces a partial order on the indices $c$ by declaring that $c < c'$ whenever $\bar{Z}^{(c)} \subsetneq \bar{Z}^{(c')}$. We say that $\mathcal{C}$ is a **chain** if such an induced partial ordering is a total ordering. We say that a **chain** $\mathcal{C}$ ends with $Z$, in which case we write $\mathcal{C} \triangleright Z$, if $Z$ is the (unique) minimal element in $\mathcal{C}$.

**Lemma 4.5.8.** Suppose $V_{-2}$ and $Z_{-2}$ are as above, and suppose $\mathcal{C} = \{Z^{(c)}\}_c \triangleright Z$ as in Definition 4.5.7. Let $c_0 > c_1 > \dots > c_r$ be a total ordering of all the indices of $\mathcal{C}$, so that $X_H = Z^{(c_0)}$ and $Z = Z^{(c_r)}$. Then there exists a filtration

$$ (4.5.9) \qquad 0 = V_{-2}^{(c_0)} \subsetneq V_{-2}^{(c_1)} \subsetneq \dots \subsetneq V_{-2}^{(c_r)} = V_{-2}, $$

which induces by base extension a filtration

$$ (4.5.10) \quad 0 = V_{-2}^{(c_0)} \otimes_Z A^{\infty} \subsetneq V_{-2}^{(c_1)} \otimes_Z A^{\infty} \subsetneq \dots \subsetneq V_{-2}^{(c_r)} \otimes_Z A^{\infty} = V_{-2} \otimes_Z A^{\infty} = Z_{-2} \otimes_Z \mathbb{Q}, $$
whose $\mathcal{H}$-orbit depends only on $\mathcal{C}$. The stabilizer $\mathcal{Q}$ of (4.5.9) in $G \otimes_Z \mathbb{Q}$ is a parabolic subgroup satisfying the following conditions (cf. Definition 3.3.8):

(1) $U \subset \mathcal{Q} \subset P$, and so the unipotent radical $U_{\mathcal{Q}}$ of $\mathcal{Q}$ contains $U$, and the Levi quotient $M_{\mathcal{Q}} := \mathcal{Q}/U_{\mathcal{Q}}$ can be identified with the Levi quotient of the parabolic subgroup $\mathcal{Q}/U$ of $M$.

(2) There is a canonical homomorphism $\mathcal{Q}/U \to G_h$, which is surjective and canonically splits.

(3) $\mathcal{Q}_l := \ker((\mathcal{Q}/U) \to G_h)$ is a (rational) parabolic subgroup of $G_l$, with unipotent radical $U_{\mathcal{Q}_l} \cong \mathcal{U}_{\mathcal{Q}}/U$ and Levi quotient $M_{\mathcal{Q}_l} := \mathcal{Q}_l/\mathcal{U}_{\mathcal{Q}_l}$.

Conversely, any (rational) parabolic subgroup of $G \otimes_Z \mathbb{Q}$ satisfying these conditions define a chain linking $X_H$ and $Z$ as in Definition 4.5.7.

**Proof.** Since the incidence relations among the strata of $Z^{\min}$ are the same as their restrictions to the characteristic zero fibers, the existence of the filtrations (4.5.9) and (4.5.10) follows from [38, Thm. 7.2.4.1(4) and Lem. 5.4.2.11]. The bijection between the filtrations as in (4.5.9) and the parabolic subgroups $\mathcal{Q}$ of $G \otimes_Z \mathbb{Q}$ satisfying the above conditions then follows from [39, Prop. A.5.8 and Lem. A.4.3]. $\square$

**Remark 4.5.11.** By Lemma 4.5.8, such chains correspond to the strata of the reductive Borel–Serre compactification of (the analytification of) $X_{\mathcal{H}} \otimes_{R_0} \mathbb{C}$ above the stratum $Z \otimes_{R_0} \mathbb{C}$ of $X_H^{\min} \otimes \mathbb{C}$, where the base changes are defined by any ring homomorphism $R_0 \to \mathbb{C}$. (See [8, Sec. III.6 and III.10, and Prop. III.15.2 and III.15.4].)

**Definition 4.5.12.** For each $\mathcal{C}$ and $\mathcal{Q}$ as in Lemma 4.5.8, we define

$$ (4.5.13) \qquad r(\mathcal{Q}) := r(\mathcal{C}) := (\#\mathcal{C}) - 1. $$

We also define $\Gamma_{\mathcal{Q}}$ to be the image of $\Gamma \cap \mathcal{Q}_l(\mathcal{Q})$ (see Definition 4.3.4) under the canonical morphism $\mathcal{Q}_l(\mathcal{Q}) \to M_{\mathcal{Q}_l}(\mathcal{Q})$, and $u_{\mathcal{Q}} := \text{Lie } U_{\mathcal{Q}}(\mathcal{Q}_{\ell})$.
**Lemma 4.5.14.** For any other choice $Q'$ instead of $Q$ as in Lemma 4.5.8, with $\Gamma_Q'$ and $u_Q'$ defined as in Definition 4.5.12, there are isomorphisms $\Gamma_Q \to \Gamma_{Q'}$ and $u_Q \to u_{Q'}$ of groups and $\ell$-adic analytic Lie groups, respectively, inducing isomorphisms between their cohomology.

*Proof.* This is because (4.5.9) determines and is determined by (4.5.10), and the $\mathcal{H}$-orbit of the latter depends only on $\mathcal{C}$. $\square$

Let $\mathcal{C}$ and $Q$ be as in Lemma 4.5.8. For each $0 \le i \le r(Q)$, the stabilizer $P^{(i)}$ of $V_{-2}^{(i)}$ in $G \otimes_{\mathbb{Z}} \mathbb{Q}$ is a parabolic subgroup associated with $Z^{(c_i)}$ as in Lemma 3.3.6. By definition, we have $Q = \bigcap_{0 \le i \le r(Q)} P^{(i)}$. Moreover, there exists a homomorphism

$$ (4.5.15) \qquad s^Q: \prod_{0 \le i \le r(Q)} (G_m \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}) \to Q \otimes_{\mathbb{Q}} \bar{\mathbb{Q}} $$

which induces a cocharacter

$$ (4.5.16) \qquad s^{Q,(i)}: G_m \otimes_{\mathbb{Z}} \bar{\mathbb{Q}} \to Q \otimes_{\mathbb{Q}} \bar{\mathbb{Q}} $$

from the $i$-th factor such that $s^{Q,(i)}$ acts

(1) on $V_{-2}^{(c_i)} \otimes_{\mathbb{Q}} \bar{\mathbb{Q}}$ by $x \mapsto x^2$;

(2) on $((V_{-2}^{(c_i)})^{\perp}/V_{-2}^{(c_i)}) \otimes_{\mathbb{Q}} \bar{\mathbb{Q}}$ by $x \mapsto x$; and

(3) on $((L \otimes_{\mathbb{Z}} \bar{\mathbb{Q}})/(V_{-2}^{(c_i)})^{\perp}) \otimes_{\mathbb{Q}} \bar{\mathbb{Q}}$ by $x \mapsto 1$;

for each $0 \le i \le r(Q)$. (These are base changes of the graded pieces for the filtration $0 = V_{-3}^{(c_i)} \subset V_{-2}^{(c_i)} \subset (V_{-2}^{(c_i)})^{\perp} = V_{-1}^{(c_i)} \subset V_0^{(c_i)} = L \otimes_{\mathbb{Z}} \mathbb{Q}$.) Such homomorphisms (4.5.15) and (4.5.16) exist because the filtration (4.5.9) splits. For each $a \in \mathbb{Z}$, for each $0 \le i \le r(Q)$, and for each $V \in D^b(G \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell)$, we define

$$ (4.5.17) \qquad w_{\ge a}^{Q,(i)}(V) := \sum_{a' \ge a} \text{ (subcomplex of } V \text{ on which } s^{Q,(i)} \text{ acts by } x \mapsto x^{a'} \text{)} $$

and

$$ (4.5.18) \qquad w_{<a}^{Q,(i)}(V) := \sum_{a' < a} \text{ (subcomplex of } V \text{ on which } s^{Q,(i)} \text{ acts by } x \mapsto x^{a'} \text{)}. $$

Since (4.5.15) is the product of (4.5.16) as $i$ varies, the truncation operators as in (4.5.17) and (4.5.18) also commute with each other as $i$ varies.

For each $0 \le i \le r(Q)$, consider

$$ (4.5.19) \qquad d_{Q,(i)} := \dim(Z_{\eta}^{(c_i)}) $$

(which does not depend on the choice of $S$). For each $a \in \mathbb{Z}$, and for each $V \in D^b(M_Q \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell)$ (see Lemma 4.5.8), consider the subcomplex

$$ (4.5.20) \qquad V_{\bar{\mathbb{Q}}}^{\le a} := w_{\ge d_{Q,(r(Q))}-a}^{Q,(r(Q))} w_{< d_{Q,(r(Q))}-a}^{Q,(r(Q)-1)} \cdots w_{< d_{Q,(2)}-a}^{Q,(2)} w_{< d_{Q,(1)}-a}^{Q,(1)}(V) $$

in $D^b(M_Q \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell)$. The assignment of $V_{\bar{\mathbb{Q}}}^{\le a}$ to $V$ defines an exact functor

$$ (4.5.21) \qquad (\cdot)_{\bar{\mathbb{Q}}}^{\le a} : D^b(M_Q \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell) \to D^b(M_Q \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell). $$
**Definition 4.5.22.** Let $\mathcal{C}$ and $\mathbb{Q}$ be as in Lemma 4.5.8. For each $a \in \mathbb{Z}$, and for each object $V$ in $\mathcal{D}^b(G \otimes_{\mathbb{Z}} \mathbb{Q}, \bar{\mathbb{Q}}_\ell)$, we define

$$ (4.5.23) \qquad V_{\mathcal{C}}^{\geq a} := \mathrm{RInv}(\Gamma_{\mathbb{Q}}, (\mathrm{RInv}(u_{\mathbb{Q}}, V|_{\mathbb{Q}}))^{\leq a}_{\bar{\mathbb{Q}}}), $$

which is a complex in $\mathcal{D}^b(G_h \otimes_{\mathbb{Z}} \bar{\mathbb{Q}}, \bar{\mathbb{Q}}_\ell)$ (because $G_{h,\mathbb{Q}} \cong G_h$, as in Lemma 4.5.8), whose isomorphism class depends only on $\mathcal{C}$, but not on the choice of $\mathbb{Q}$, by Lemma 4.5.14. Then the constructions in [43, Sec. 3.1] and [67, Sec. 1 and 4.9] associated with $V_{\mathcal{C}}^{\geq a}$ a complex $V_{\mathcal{C}}^{\geq a}$ in $\mathcal{D}^b(\mathbb{Z}, \bar{\mathbb{Q}}_\ell)$ (cf. Remark 4.3.7 and Definition 4.3.8).

We need one last technical preparation:

**Lemma 4.5.24.** Suppose $\mathcal{T} = \mathrm{Spec}(k)$, where $k$ is a field that is either algebraically closed or finite. Let $\mathcal{Y}$ be a well-positioned subset of $(X_{\mathcal{H}})_{\mathcal{T}}$, with associated $\mathcal{Y}^a = \{Y_{\mathcal{Z}}^a\}_{\mathcal{Z}}$ as in Definition 2.2.1, and with partial minimal and toroidal compactifications $\mathcal{Y}^{\mathrm{min}}$ and $\mathcal{Y}_{\Sigma}^{\mathrm{tor}}$ as in Definition 2.3.1 and Theorem 2.3.2. Suppose that $\mathcal{F}$ is a perverse sheaf over $\mathcal{Y}_{\Sigma}^{\mathrm{tor}}$ equipped with a collection $\mathcal{F}^a = \{(\mathcal{F}_{\mathcal{Z}}^a, \iota_{\mathcal{Z}})\}_{\mathcal{Z}}$ as in Definition 4.1.3, such that $\mathcal{F}_{\mathcal{Z}}^a[d^2]$ is a perverse sheaf over $\mathcal{Z}$, where $d^2 := \mathrm{dim}((X_{\mathcal{H}})_{\mathcal{T}}) - \mathrm{dim}(\mathcal{Z}_{\mathcal{T}})$, for each $\mathcal{Z}$.

Now let us fix the choice of $\mathcal{Z}$. Suppose we have the morphisms $\mathcal{Z}^{\mathrm{tor}} \to \mathcal{Z}^{\mathrm{min}} \xrightarrow{\sim} \bar{\mathcal{Z}} \to X_{\mathcal{H}}^{\mathrm{min}}$, which induce $\mathcal{Y}_{\mathcal{Z}}^{\mathrm{tor}} \to \mathcal{Y}_{\mathcal{Z}}^{\mathrm{min}} \xrightarrow{\sim} \bar{\mathcal{Y}}_{\mathcal{Z}} \to \mathcal{Y}^{\mathrm{min}}$, as in Lemma 2.3.16 and its proof. Then, under the assumption (see Assumptions 4.3.1 and 4.5.1) that $\mathcal{Y}_{\mathcal{C}}^a \to \mathcal{Y}_{\mathcal{Z}}^a$ is an abelian scheme torsor (of constant relative dimension) for each stratum $\mathcal{Z}'$ of $\mathcal{Z}^{\mathrm{min}}$, there exists a well-positioned perverse sheaf $\mathcal{G}$ over $\mathcal{Z}^{\mathrm{tor}}$, equipped with a collection $\mathcal{G}^a = \{(\mathcal{G}_{\mathcal{Z}', \iota_{\mathcal{Z}'}}^a)\}_{\mathcal{Z}'}$ indexed by the strata $\mathcal{Z}'$ in $\bar{\mathcal{Z}}$, such that $\mathcal{G}_{\mathcal{Z}', \mathcal{Z}}^a = \mathcal{F}_{\mathcal{Z}', [d^2]}^a$ for each $\mathcal{Z}'$, and such that the pullbacks of $\iota_{\mathcal{Z}'}[d^2]$ and $\iota_{\mathcal{Z}'}$ to $\mathcal{Z}' \times_{\bar{\mathcal{Z}}} \mathcal{Z}_{[\sigma]}^{\mathrm{tor}}$, for some top-dimensional $\sigma \in \Sigma_{\mathcal{Z}}^+$ as in the proof of Lemma 2.3.16, coincide with each other. Proof. As in the proof of Lemma 4.2.13, since the fibered category of perverse sheaves is a stack (see [5, Sec. 2.2.19]), it suffices to construct $\mathcal{G}$ and $\iota_{\mathcal{Z}'}$ over the étale boundary charts of $\mathcal{Z}^{\mathrm{tor}}$. Let $\mathcal{Z}'$ be any stratum of $\bar{\mathcal{Z}}$. Suppose that we have already defined $\mathcal{G}$ over the largest open subscheme of $\mathcal{Z}^{\mathrm{tor}}$ containing all stratum $\mathcal{Z}''$ of $\bar{\mathcal{Z}}$ that contain $\mathcal{Z}'$ in its closure but differ from $\mathcal{Z}'$, and defined $\iota_{\mathcal{Z}''}$ and its extensions over some $\mathcal{U}_{\mathcal{Z}''}$ as in Definition 4.1.3, for all such $\mathcal{Z}''$. (When $\mathcal{Z}' = \mathcal{Z}$, we simply have $\mathcal{G}|_{\mathcal{Z}} = \mathcal{F}_{\mathcal{Z}}^a[d^2]$, with $\iota_{\mathcal{Z}'}$ and its extensions being the identity morphisms.)

Let $x$ be a point of a stratum $\mathcal{Z}'_{[\tau'']}$ of $\mathcal{Z}_{[\sigma]}^{\mathrm{tor}}$, which lies above a point $y$ of a stratum $\mathcal{Z}'_{[\tau']}$ of $\mathcal{Z}^{\mathrm{tor}}$ above $\mathcal{Z}'$. By using nested approximation as in the proof of [43, Prop. 4.3], there exist a torus homomorphism $E'' \to E'$ and an abelian scheme torsor $C'' \to C'$, and representatives $\tau''$ and $\tau'$ of $[\tau'']$ and $[\tau']$, respectively, such that there exists a commutative diagram

$$ (4.5.25) \qquad \begin{tikzcd} \mathcal{Z}^{\mathrm{tor}}_{[\sigma]} \arrow[r] \arrow[d] & \bar{U}'' \arrow[r] \arrow[d] & E''(\tau'') \arrow[r, "\times\, C''", "\mathrm{Spec}(\mathbb{Z})"] & C'' \\ \mathcal{Z}^{\mathrm{tor}} \arrow[r] \arrow[d] & \bar{U}' \arrow[r] \arrow[d] & E'(\tau') \arrow[r, "\times\, C'", "\mathrm{Spec}(\mathbb{Z})"] & C' \end{tikzcd} $$

in which the squares are all Cartesian, and in which the horizontal morphisms define compatible étale neighborhoods of $x$ and $y$ such that the preimages of $\mathcal{Z}$ and
$E' \times_{\text{Spec}(\mathbb{Z})} C'$ coincide as an open subscheme $U'$ of $\bar{U}'$; such that the preimages of $Z_{[\sigma]}$, $E''_{\text{Spec}(\mathbb{Z})} \times_{\text{Spec}(\mathbb{Z})} C''$, and $U'$ coincide as an open subscheme $U''$ of $\bar{U}''$; and such that the pullbacks of the horizontal étale morphisms under the canonical morphisms $Z' \times_{\bar{\mathbb{Z}}} Z^{\text{tor}} \to Z^{\text{tor}}$ and $E'(\tau')^+ \times_{\text{Spec}(\mathbb{Z})} C' \to E'(\tau') \times_{\text{Spec}(\mathbb{Z})} C'$ (when defined) are open immersions. Up to replacing $\bar{U}'$ and $\bar{U}''$ with open subschemes, we may assume that all étale morphisms above have connected geometric fibers. Then there exists an étale morphism $\bar{U}''' \to \bar{U}''$ with connected geometric fibers such that the pullback of $\iota_{U'''} : (Y_{U'''} \to Y_{\Sigma}^{\text{tor}})^* \mathcal{F} \xrightarrow{\sim} (Y_{U'''} \to Y_{\bar{Z}}^{\sharp})^* \mathcal{F}_{\bar{Z}}^{\sharp}$ and $\iota_{\bar{U}'''} : (Y_{\bar{U}'''} \to Y_{\Sigma}^{\text{tor}})^* \mathcal{F} \xrightarrow{\sim} (Y_{\bar{U}'''} \to Y_{\bar{Z}'}^{\sharp})^* \mathcal{F}_{\bar{Z}'}^{\sharp}$ (see Definition 4.1.3), where $Y_{U'''} := U''' \times_{X_{\mathcal{H},\Sigma}^{\text{tor}}} Y_{\Sigma}^{\text{tor}}$ and $Y_{\bar{U}'''} := \bar{U}''' \times_{X_{\mathcal{H},\Sigma}^{\text{tor}}} Y_{\Sigma}^{\text{tor}}$.

Let $\mathcal{G}_{Z',U'}^{\sharp}$ denote the pullback of $\mathcal{G}_{Z',U'}^{\sharp} = \mathcal{F}_{Z',U'}^{\sharp}[d^Z]$ to $Y_{Z,U'} := U' \times_{Z^{\text{tor}}} Y_{Z}^{\text{tor}}$, which is a shifted perverse sheaf because $U' \to Z'$ is smooth of constant relative dimension. Since $Z_{[\sigma]} \cong C \to Z$ is smooth of constant relative dimension, the further pullback $\mathcal{G}_{Z',U'''}^{\sharp}$ of $\mathcal{G}_{Z',U'}^{\sharp}$ to $Y_{U'''}$, which coincides with the pullback of $\mathcal{G}_{Z'}^{\sharp}$ under $Y_{U'''} \to Y_{Z'}$, is also a shifted perverse sheaf. Using the isomorphisms $\iota_{U'''}$ and $\iota_{\bar{U}'''}$ above up to shifting, the pullback of $\mathcal{F}_{Z}^{\sharp}[d^Z]$ under $Y_{U'''} \to Y_{\bar{Z}}^{\sharp}$ is isomorphic to $\mathcal{G}_{Z',U'''}^{\sharp}$. Since $U''' \to U'$ is smooth and has connected geometric fibers, by [5, Prop. 4.2.5 and Cor. 4.2.6.2], this uniquely descends to an isomorphism between $\mathcal{F}_{Z',U'}^{\sharp}[d^Z]$ and $\mathcal{G}_{Z',U'}^{\sharp}$ over the image of $Y_{U'''}$ in $Y_{Z'}$. Hence, the pullback $\mathcal{G}_{Z',\bar{U}'}^{\sharp}$ of $\mathcal{G}_{Z'}^{\sharp}$ to $Y_{Z,\bar{U}'} := \bar{U}' \times_{Z^{\text{tor}}} Y_{Z}^{\text{tor}}$, which necessarily has no support in $Y_{Z,\bar{U}'} - Y_{Z,U'}$, defines an extension of the partially defined $\mathcal{G}$ over the image of $Y_{\bar{U}'''} \to Y_{Z,\bar{U}'}$. By varying $x$ (and $y$), we obtain étale neighborhoods which induce an open covering of the preimage of $Z'$ in $Z^{\text{tor}}$, and hence by descent an extension of $\mathcal{G}$ over a larger open subscheme containing this preimage, with the desired $\iota_{Z'}$ and its extensions over some collection $\bar{U}_{Z'}$ as in Definition 4.1.3. Thus, by inductively repeating this process until we exhaust all strata $Z'$ of $\bar{Z}$, the lemma follows. $\square$

Now we are ready to state and prove our generalizations of Morel’s results:

**Theorem 4.5.26.** Suppose $\mathcal{T} = \text{Spec}(k)$, where $k$ is a finite field of characteristic $p$. Let $\mathcal{Y}$ be a well-positioned subset of $(X_{\mathcal{H}})_{\mathcal{T}}$, with associated $\mathcal{Y}^{\sharp} = \{Y_{\bar{Z}}^{\sharp}\}_{\bar{Z}}$ as in Definition 2.2.1, and with partial minimal and toroidal compactifications $\mathcal{Y}^{\text{min}}$ and $\mathcal{Y}_{\Sigma}^{\text{tor}}$ as in Definition 2.3.1 and Theorem 2.3.2.

Suppose that $V = V_{\xi}$ is an algebraic representation of $G \otimes_{\mathbb{Z}} \mathbb{Q}$ on a finite-dimensional vector space over $\mathbb{Q}_{\ell}$ that is pure of weight $-a \in \mathbb{Z}$ (as in Definition 4.5.2), and that $\mathcal{F}$ is a well-positioned pure perverse sheaf of weight $b \in \mathbb{Z}$ over $Y_{\Sigma}^{\text{tor}}$, equipped with a collection $\mathcal{F}^{\sharp} = \{(\mathcal{F}_{\bar{Z},\iota_Z}^{\sharp}, \iota_{\bar{Z}})\}_{\bar{Z}}$ as in Definition 4.1.3 such that $\mathcal{F}_{\bar{Z}}^{\sharp}[-d^Z]$ is a pure perverse sheaf of weight $b - d^Z$ over $Y_{\bar{Z}}^{\sharp}$, where $d^Z := \dim((X_{\mathcal{H}})_{\mathcal{T}}) - \dim(Z_{\mathcal{T}})$, for each $\bar{Z}$.

Let $\bar{Z}$ be a stratum of $X_{\mathcal{H}}^{\text{min}}$. Let $i_{\mathcal{Y}}^{\text{min}}$, $i_{\mathcal{Y}}^{\text{min}}$, etc be defined as in the paragraphs preceding Theorems 4.3.10 and 4.3.16. Let $\mathcal{F}_{\mathcal{Y}}$, $\mathcal{V}_{\mathcal{Y}}$, and $\mathcal{V}_{\mathcal{Y},\bar{Z}}^{\sharp,\le a}$ denote the pullbacks of $\mathcal{F}_{\mathcal{Y}}$, $\mathcal{V}_{\mathcal{Y}}$, and $\mathcal{V}_{\mathcal{Y},\bar{Z}}^{\sharp,\le a}$ (see Definition 4.5.22) to $\mathcal{Y}$, $\mathcal{Y}$, and $\mathcal{Y}_{\bar{Z}}$, respectively. Then,
under Assumptions 4.3.1 and 4.5.1, we have an equality

$$ (4.5.27) \quad [i_{\mathcal{Y}}^{\min,*} j_{\mathcal{Y},*}^{\min}(\mathcal{V}_{\mathcal{Y}}^{L} \mathcal{F}_{\mathcal{Y}})] = \sum_{\mathcal{C} \triangleright \mathcal{Z}} (-1)^{r(\mathcal{C})-1} [\mathcal{V}_{\mathcal{C},\mathcal{Y}_{\mathcal{Z}}}^{\mathcal{C},\le a} \otimes_{\mathcal{Z}}^{L} \mathcal{F}_{\mathcal{Z}}^{\mathcal{C}}] $$

in the Grothendieck group of $D_c^b(\mathcal{Y}_Z, \bar{\mathcal{Q}}_\ell)$. (See Definitions 4.5.7, 4.5.12, and 4.5.22.)

*Proof.* We shall closely follow the arguments in [56, especially the proofs of Thm. 4.2.1 and Prop. 4.2.3]. Let $w_{\le a+b}$ and $w_{>a+b}$ denote the weight truncation functors defined in [56, Sec. 3.1, p. 36]. By Lemma 4.5.6, $\mathcal{V}_{\mathcal{Y}}^L \otimes \mathcal{F}_{\mathcal{Y}}$ is a pure perverse sheaf of weight $a+b$. Therefore, by [56, Thm. 3.1.4], there is a canonical isomorphism

$$ (4.5.28) \quad j_{\mathcal{Y},*}^{\min}(\mathcal{V}_{\mathcal{Y}}^{L} \mathcal{F}_{\mathcal{Y}}) \xrightarrow{\sim} w_{\le a+b} R j_{\mathcal{Y},*}^{\min}(\mathcal{V}_{\mathcal{Y}}^{L} \mathcal{F}_{\mathcal{Y}}) $$

in $D_c^b(\mathcal{Y}^{\min}, \bar{\mathcal{Q}}_\ell)$. For each $\mathcal{C} = \{\mathcal{Z}^{(c)}\}_c$ with indices ordered as $c_0 > c_1 > \cdots > c_r(\mathcal{C})$ as in Lemma 4.5.8, and for each $0 \le i \le r(\mathcal{C})$, let $i_{\mathcal{C}}^{(i)} : \mathcal{Y}_{\mathcal{Z}^{(c_i)}} \to \mathcal{Y}^{\min}$ denote the canonical morphism, so that $i_{\mathcal{C}}^{(0)} = j_{\mathcal{Y}}^{\min}$ and $i_{\mathcal{C}}^{(r(\mathcal{C}))} = i_{\mathcal{Y}}^{\min}$. By [56, Prop. 3.1.3 and 3.3.4; cf. Thm. 3.3.5], we have an equality

$$ (4.5.29) \quad [i_{\mathcal{Y}}^{\min,*} w_{\le a+b} R j_{\mathcal{Y},*}^{\min}(\mathcal{V}_{\mathcal{Y}}^{L} \mathcal{F}_{\mathcal{Y}})] = \sum_{\mathcal{C} \triangleright \mathcal{Z}} (-1)^{r(\mathcal{C})-1} [w_{\le a+b} i_{\mathcal{C}}^{(r(\mathcal{C})),*} R i_{\mathcal{C},*}^{(r(\mathcal{C})-1)} \\ w_{>a+b} i_{\mathcal{C}}^{(r(\mathcal{C})-1),*} \cdots R i_{\mathcal{C},*}^{(2)} w_{>a+b} i_{\mathcal{C}}^{(1),*} R i_{\mathcal{C},*}^{(1)} w_{>a+b} i_{\mathcal{C}}^{(1),*} R j_{\mathcal{Y},*}^{\min}(\mathcal{V}_{\mathcal{Y}}^{L} \mathcal{F}_{\mathcal{Y}})] $$

in the Grothendieck group of $D_c^b(\mathcal{Y}_Z, \bar{\mathcal{Q}}_\ell)$.

Let $\mathcal{Q}$ be as in Lemma 4.5.8, which is $\bigcap_{0 \le i \le r(\mathcal{C})} \mathcal{P}^{(i)}$ as in the paragraph following Lemma 4.5.14. For each $0 \le i \le r(\mathcal{C})$, let $\mathcal{U}^{(i)}$, $\mathcal{M}^{(i)}$, $\mathcal{G}_l^{(i)}$, and $\mathcal{G}_h^{(i)}$ be defined as in Definition 3.3.8 for $\mathcal{P}^{(i)}$, with $\Gamma_{\mathcal{P}^{(i)}} \subset \mathcal{G}_l^{(i)}(\mathcal{Q})$ defined by the corresponding $\mathcal{Z}^{(c_i)}$ as in Definition 4.3.4, and let $\mathcal{u}^{(i)} := \text{Lie} \, \mathcal{U}^{(i)}(\mathcal{Q}_\ell)$. Let $\mathcal{Q}^{(i)} := \bigcap_{0 \le j \le i} \mathcal{P}^{(j)}$, let $\mathcal{U}_{\mathcal{Q}^{(i)}}$ denote the unipotent radical of $\mathcal{Q}^{(i)}$, and let $\mathcal{u}_{\mathcal{Q}^{(i)}} := \text{Lie} \, \mathcal{U}_{\mathcal{Q}^{(i)}}(\mathcal{Q}_\ell)$. Let $\mathcal{Q}_l^{(i)} := \ker((\mathcal{Q}^{(i)}/\mathcal{U}^{(i)}) \to \mathcal{G}_l^{(i)})$, which is a parabolic subgroup of $\mathcal{G}_l^{(i)}$, with unipotent radical $\mathcal{U}_{\mathcal{Q}_l^{(i)}} := \mathcal{U}_{\mathcal{Q}^{(i)}/\mathcal{U}^{(i)}}$ and Levi quotient $\mathcal{M}_{\mathcal{Q}_l^{(i)}} := \mathcal{Q}_l^{(i)}/\mathcal{U}_{\mathcal{Q}_l^{(i)}}$ (cf. Lemma 4.5.8). Let $\Gamma_{\mathcal{Q}^{(i)}}$ be the image of $\Gamma_{\mathcal{P}^{(i)}} \cap \mathcal{Q}_l^{(i)}(\mathcal{Q})$ under the canonical homomorphism $\mathcal{Q}_l^{(i)}(\mathcal{Q}) \to \mathcal{M}_{\mathcal{Q}_l^{(i)}}(\mathcal{Q})$ (cf. Definition 4.5.12). When $i \ge 1$, also consider the canonical homomorphism $\mathcal{Q}^{(i)}/\mathcal{U}^{(i-1)} \to \mathcal{G}_h^{(i-1)}$, whose kernel is $\mathcal{Q}_l^{(i-1)}$, and whose image is a parabolic subgroup $\mathcal{P}_h^{(i)}$ of $\mathcal{G}_h^{(i-1)}$, with unipotent radical $\mathcal{U}_h^{(i)} \cong \mathcal{U}^{(i)}/(\mathcal{U}^{(i)} \cap \mathcal{U}^{(i-1)}) \cong \mathcal{U}_{\mathcal{Q}^{(i)}/\mathcal{U}_{\mathcal{Q}^{(i-1)}}}$ and Levi quotient $\mathcal{M}_h^{(i)} \cong \mathcal{G}_h^{(i)} \times \mathcal{G}_h^{(i)}$, where $\mathcal{G}_h^{(i)} := \mathcal{M}_{\mathcal{Q}_l^{(i)}/\mathcal{M}_{\mathcal{Q}_l^{(i-1)}}}$. (Note that we have $\mathcal{U}^{(0)} = 1$, $\mathcal{G}_h^{(0)} = \mathcal{G} \otimes_{\mathcal{Z}} \mathcal{Q}$, $\mathcal{Q}_l^{(0)} = 1$, $\mathcal{U}_{\mathcal{Q}_l^{(0)}} = 1$, and $\mathcal{M}_{\mathcal{Q}_l^{(0)}} = 1$.) Let $\Gamma_{\mathcal{P}_h^{(i)}}$ be the image of $\Gamma_{\mathcal{Q}^{(i)}}$ under the canonical morphism $\mathcal{M}_{\mathcal{Q}_l^{(i)}} \to \mathcal{G}_h^{(i)}$, and let $\mathcal{u}_h^{(i)} := \text{Lie} \, \mathcal{U}_h^{(i)}(\mathcal{Q}_\ell)$. By definition, we have $\Gamma_{\mathcal{P}_h^{(i)}} \cong \Gamma_{\mathcal{Q}^{(i)}}/\Gamma_{\mathcal{Q}^{(i-1)}}$ and $\mathcal{u}_h^{(i)} \cong \mathcal{u}_{\mathcal{Q}^{(i)}/\mathcal{u}_{\mathcal{Q}^{(i-1)}}}$. We also have $\mathcal{M}_{\mathcal{Q}_l^{(i)}} \cong \prod_{1 \le j \le i} \mathcal{G}_h^{(j)}$ and $\mathcal{M}_{\mathcal{Q}_l} \cong \prod_{1 \le i \le r(\mathcal{C})} \mathcal{G}_h^{(i)}$.
Suppose that, for some $1 \le i < r(\mathcal{C})$, we have a canonical isomorphism

$$ (4.5.30) \qquad \begin{aligned} & i_{\mathcal{C}}^{(i),*} R i_{\mathcal{C},*}^{(i-1)} w_{>a+b} \cdots R i_{\mathcal{C},*}^{(1)} w_{>a+b} i_{\mathcal{C}}^{(1),*} R i_{\mathcal{C},*}^{(0)} (\mathcal{V}_{\mathcal{Y}}^{L} \otimes \mathcal{F}_{\mathcal{Y}}) \\ & \cong \mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a, \text{pre}} \otimes \mathcal{F}_{\mathcal{Z}^{(c_i)}}^{\ge i} \end{aligned} $$

in $D_c^b(\mathcal{Y}_{Z^{(c_i)}}, \bar{\mathcal{Q}}_\ell)$, where $\mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a, \text{pre}}$ is associated with

$$ (4.5.31) \qquad \mathrm{RInv}(\Gamma_{Q^{(i)}}, w_{<d_{Q^{(i-1)}}-a}^{Q,(i-1)} \cdots w_{<d_{Q^{(1)}}-a}^{Q,(1)} (\mathrm{RInv}(\mathbf{u}_{Q^{(i)}}, V|_{Q^{(i)}}))) $$

in $D^b(G_h^{(i)} \otimes_{\mathbb{Z}} \bar{\mathcal{Q}}, \bar{\mathcal{Q}}_\ell)$ (cf. Definition 4.3.8 and (4.5.20)). When $i = 1$, this is just Theorem 4.3.16 (with no truncation needed in (4.5.31)).

Since $\mathcal{F}_{\mathcal{Z}^{(c_i)}}^{\ge i}[-d^{Z^{(c_i)}}]$ is a pure perverse sheaf of weight $b - d^{Z^{(c_i)}}$ over $\mathcal{Y}_{Z^{(c_i)}}$, and since $d_{Q^{(i)}} = d_{Z^{(c_i)}}$, by the analogue of Lemma 4.5.6 for $Z^{(c_1)}$, and by [56, proof of Lem. 4.1.2, based on Prop. 2.1.4] (cf. Lemma 4.5.4), we have a canonical isomorphism

$$ (4.5.32) \qquad w_{>a+b} (\mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a, \text{pre}} \otimes_{\mathcal{Z}^{(c_i)}}^{L} \mathcal{F}_{\mathcal{Z}^{(c_i)}}^{\ge i}) \cong \mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a} \otimes_{\mathcal{Z}^{(c_i)}}^{L} \mathcal{F}_{\mathcal{Z}^{(c_i)}}^{\ge i} $$

in $D_c^b(\mathcal{Y}_{Z^{(c_i)}}, \bar{\mathcal{Q}}_\ell)$, where $\mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a}$ is associated with

$$ (4.5.33) \qquad \mathrm{RInv}(\Gamma_{Q^{(i)}}, w_{<d_{Q^{(i)}}-a}^{Q,(i)} \cdots w_{<d_{Q^{(1)}}-a}^{Q,(1)} (\mathrm{RInv}(\mathbf{u}_{Q^{(i)}}, V|_{Q^{(i)}}))) $$

in $D^b(G_h^{(i)} \otimes_{\mathbb{Z}} \bar{\mathcal{Q}}, \bar{\mathcal{Q}}_\ell)$.

Since $\Gamma_{P_h^{(i+1)}} \cong \Gamma_{Q^{(i+1)}}/\Gamma_{Q^{(i)}}$, since $\Gamma_{Q^{(i)}} \subset M_{Q^{(i)}}(\mathbb{Q})$, since $M_{Q^{(i)}}$ commutes with $G_h^{(i)}$ and hence also with $U_h^{(i+1)}$, and since the cocharacters $s^{Q,(1)}, \dots, s^{Q,(i)}$ act trivially on $\mathbf{u}_h^{(i+1)} \cong \mathbf{u}_{Q^{(i+1)}}/\mathbf{u}_{Q^{(i)}}$, we have a canonical isomorphism

$$ (4.5.34) \qquad \begin{aligned} & \mathrm{RInv}(\Gamma_{P_h^{(i+1)}}, w_{<d_{Q^{(i)}}-a}^{Q,(i)} \mathrm{RInv}(\mathbf{u}_h^{(i+1)}), \\ & \qquad (\mathrm{RInv}(\Gamma_{Q^{(i)}}, w_{<d_{Q^{(i-1)}}-a}^{Q,(i-1)} \cdots w_{<d_{Q^{(1)}}-a}^{Q,(1)} (\mathrm{RInv}(\mathbf{u}_{Q^{(i)}}, V|_{Q^{(i)}})))|_{P_h^{(i+1)}}) \\ & \cong \mathrm{RInv}(\Gamma_{Q^{(i+1)}}, w_{<d_{Q^{(i)}}-a}^{Q,(i)} \cdots w_{<d_{Q^{(1)}}-a}^{Q,(1)} (\mathrm{RInv}(\mathbf{u}_{Q^{(i+1)}}, V|_{Q^{(i+1)}}))). \end{aligned} $$

By Lemmas 2.3.16 and 4.5.24, and by applying Theorem 4.3.16 to the pullback to $\mathcal{Y}_{Z^{(c_{i+1})}}$ of $R(\mathcal{Y}_{Z^{(c_i)}} \hookrightarrow \mathcal{Y}_{Z^{(c_i)}}^{\min})_* (\mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_i)}}}^{\ge a} \otimes_{\mathcal{Z}^{(c_i)}}^{L} \mathcal{F}_{\mathcal{Z}^{(c_i)}}^{\ge i})$, we obtain (4.5.30) with $i$ replaced with $i+1$. In the final step, when $i+1 = r(\mathcal{C})$, so that $Z^{(c_r(\mathcal{C}))} = \mathbb{Z}$, the same argument for showing (4.5.32) gives a canonical isomorphism

$$ (4.5.35) \qquad w_{\le a+b} (\mathcal{V}_{\mathcal{C}, \mathcal{Y}_{Z^{(c_r(\mathcal{C}))}}}^{\ge a, \text{pre}} \otimes_{\mathcal{Z}^{(c_r(\mathcal{C}))}}^{L} \mathcal{F}_{\mathcal{Z}^{(c_r(\mathcal{C}))}}^{\ge i}) \cong \mathcal{V}_{\mathcal{C}, \mathcal{Y}_{\mathcal{Z}^{(c_r(\mathcal{C}))}}}^{\ge a} \otimes_{\mathcal{Z}^{(c_r(\mathcal{C}))}}^{L} \mathcal{F}_{\mathcal{Z}^{(c_r(\mathcal{C}))}}^{\ge i} $$

in $D_c^b(\mathcal{Y}_{\mathcal{Z}}, \bar{\mathcal{Q}}_\ell)$, and hence we obtain a canonical isomorphism

$$ (4.5.36) \qquad \begin{aligned} & w_{\le a+b} i_{\mathcal{C}}^{(r(\mathcal{C})),*} R i_{\mathcal{C},*}^{(r(\mathcal{C})-1)} w_{>a+b} \cdots R i_{\mathcal{C},*}^{(1)} w_{>a+b} i_{\mathcal{C}}^{(1),*} R i_{\mathcal{C},*}^{(0)} (\mathcal{V}_{\mathcal{Y}}^{L} \otimes \mathcal{F}_{\mathcal{Y}}) \\ & \cong \mathcal{V}_{\mathcal{C}, \mathcal{Y}_{\mathcal{Z}}}^{\ge a, \text{pre}} \otimes_{\mathcal{Z}}^{L} \mathcal{F}_{\mathcal{Z}}^{\ge i}. \end{aligned} $$

in $D_c^b(\mathcal{Y}_{\mathcal{Z}}, \bar{\mathcal{Q}}_\ell)$. Now, just combine (4.5.28), (4.5.29), and (4.5.36). $\square$
**Theorem 4.5.37.** Suppose that $V = V_\xi$ and $-a \in \mathbb{Z}$ are as in Theorem 4.5.26. Let $d := \dim((X_H)_\eta)$. Consider the nearby cycle functors $R\Psi_{X_H}$ etc defined by some choices of geometric points $\bar{\eta}$ and $\bar{s}$ above the generic and special points $\eta$ and $s$ of $S$, respectively, as in [43, Sec. 5.1], where $S = \text{Spec}(O_K)$, and where $K$ is the $v$-adic completion of $F_0$ at some place $v|p$, as in Section 4.4. Then, under Assumptions 4.3.1 and 4.5.1, we have an equality

$$ (4.5.38) \qquad [i_{\bar{s}}^{\min,*} j_{\bar{s},!*} R\Psi_{X_H}(V[d])] = \sum_{\mathcal{C} \triangleright Z} (-1)^{r(\mathcal{C})-1} [V_{\mathcal{C},Z_{\bar{s}}}^{\mathfrak{h},\le a} \otimes^{L} R\Psi_Z(\bar{Q}_{\ell})[d]] $$

in the Grothendieck group of $D_c^b(Z_{\bar{s}} \times \bar{\eta}, \bar{Q}_{\ell})$.

*Proof.* We shall closely follow [76], with additional inputs from [59] and [43, Sec. 5], and from this article. Since the argument is global, let us consider $M_H$, $M_H^{\min}$, etc over $\text{Spec}(F_0)$ as in Assumption 2.1.1. Note that these are the generic fibers for the analogues of $X_H$, $X_H^{\min}$, etc, with the base scheme $S$ replaced with some analogue over $\text{Spec}(O_{F_0,(q)})$ for any good prime $q \ne p$ for $(\mathcal{O}, \star, L, \langle \cdot, \cdot \rangle, h_0)$ and $\mathcal{H}$ (in the sense that $\mathcal{H}$ is maximal hyperspecial at $q$).

Consider the categories $\mathcal{M}(M_H)$, $\mathcal{M}(M_H^{\min})$, etc of horizontal perverse sheaves which admit weight filtrations, as in [59, Prop. 6.1]. For $V = V_\xi$ as in Theorem 4.5.26, the associated sheaf $\mathcal{V}[d]$ over $M_H$ (which can be defined using any analogues of $X_H$ as above over $\text{Spec}(O_{F_0,(q)})$ for some good $q \ne p$) is an object in $\mathcal{M}(X_H)$, and it follows from [59, Cor. 8.1.4] and the same argument of the proof of Theorem 4.5.26 (with $\mathcal{F}$ and $\mathcal{F}_Z^{\mathfrak{h}}$ being $\bar{Q}_{\ell}[d]$ for each $Z$) that, after pullback from $\text{Spec}(F_0)$ to $\eta = \text{Spec}(K)$, we have an equality

$$ (4.5.39) \qquad [i_{\eta}^{\min,*} j_{\eta,!*}^{\min}(\mathcal{V}[d])] = \sum_{\mathcal{C} \triangleright Z} (-1)^{r(\mathcal{C})-1} [V_{\mathcal{C},Z_{\eta}}^{\mathfrak{h},\le a}[d]] $$

in the Grothendieck group of $D_c^b(Z_\eta, \bar{Q}_\ell)$, and hence also an equality

$$ (4.5.40) \qquad [j_{\eta,!*}^{\min}(\mathcal{V}[d])] = \sum_{Z'} \sum_{\mathcal{C}' \triangleright Z'} (-1)^{r(\mathcal{C}')-1} [i_{\eta,!*}^{\prime,\min} V_{\mathcal{C}',Z_{\eta}}^{\mathfrak{h},\le a}[d]] $$

in the Grothendieck group of $D_c^b((X_H^{\min})_\eta, \bar{Q}_\ell)$, where the objects associated with $Z'$ are denoted with an additional prime.

By [43, Thm. 5.26], we have a canonical isomorphism

$$ (4.5.41) \qquad R\Psi_{X_H^{\min}} j_{\eta,!*}^{\min}(\mathcal{V}[d]) \simeq j_{\bar{s},!*}^{\min} R\Psi_{X_H}(\mathcal{V}[d]) $$

in $D_c^b((X_H^{\min})_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$. For each $Z'$, by decomposing the canonical morphism $i'^{,\min}: Z' \to X_H^{\min}$ as a composition of the open immersion $Z' \hookrightarrow Z'^{,\min}$ and the closed immersion $Z'^{,\min} \to X_H^{\min}$ using Lemma 2.3.16, by applying [43, Thm. 5.23] to $Z' \hookrightarrow Z'^{,\min}$, and by the proper base change theorem (cf. [2, XII, 5.1] and [15, XIII, (2.1.7.1)]), we have a canonical isomorphism

$$ (4.5.42) \qquad i_{\bar{s},!*}^{\prime,\min} R\Psi_{Z'}(V_{\mathcal{C}',Z_{\eta}}^{\mathfrak{h},\le a}) \simeq R\Psi_{X_H^{\min}}(i_{\eta,!*}^{\prime,\min} V_{\mathcal{C}',Z_{\eta}}^{\mathfrak{h},\le a}) $$

in $D_c^b((X_H^{\min})_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$. On the other hand, since $V_{\mathcal{C}',Z_{\eta}}^{\mathfrak{h},\le a}$ is quasi-isomorphic to a direct sum of shifted lisse sheaves, we have a canonical isomorphism

$$ (4.5.43) \qquad V_{\mathcal{C}',Z_{\bar{s}}}^{\mathfrak{h},\le a} \otimes^{L} R\Psi_{Z'}(\bar{Q}_{\ell}) \simeq R\Psi_{Z'}(V_{\mathcal{C}',Z_{\eta}}^{\mathfrak{h},\le a}) $$
in $D_c^b(Z'_s \times \bar{\eta}, \bar{Q}_\ell)$ (as we have seen in Section 4.4). Thus, we obtain the equality

$$ (4.5.44) \quad [j_{\bar{s},!*}^{\min} R\Psi_{X_H}(V[d])] = \sum_{Z'} \sum_{\mathcal{C}' \succ Z'} (-1)^{r(\mathcal{C}')-1} [i_{\bar{s},!}^{\prime,\min}(V_{\mathcal{C}',Z'}^{\natural,\le a} \otimes_L R\Psi_{Z'}(\bar{Q}_\ell)[d])] $$

in the Grothendieck group of $D_c^b((X_H^{\min})_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$ by combining (4.5.39), (4.5.41), (4.5.42), and (4.5.43). By applying $i_{\bar{s}}^{\min,*}$ to (4.5.44), we also obtain the equality (4.5.38) in the Grothendieck group of $D_c^b(Z_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$, as desired. $\square$

**Remark 4.5.45.** In order to establish (4.5.39) and (4.5.40), instead of resorting to results in [59], we may resort to Laumon’s Cebotarev density theorem for perverse sheaves as in [44, Thm. 1.1.2], by applying Theorem 4.5.26 to $Y = (X_H)_s$ and $\mathcal{F} = \Lambda(X_{H,\Sigma}^{\text{tor}})_s[d]$ (for some smooth $\Sigma$), after replacing $X_H$ etc with their analogues over $\text{Spec}(\mathcal{O}_{F_0,(q)})$, for all good primes $q \nmid p^\ell$, as in the proof of [77, Thm. 1.4.8].

**Corollary 4.5.46.** With the same setting as in Theorem 4.5.37, the perverse sheaf $j_{\bar{s},!*}^{\min}(R\Psi_{X_H}(V[d]))$ has no support contained in the boundary of $(X_H^{\min})_{\bar{s}}$.

*Proof.* As in the case of (4.5.43), we have a canonical isomorphism

$$ (4.5.47) \quad \mathcal{V} \otimes_L R\Psi_{X_H}(\bar{Q}_\ell[d]) \xrightarrow{\sim} R\Psi_{X_H}(V[d]) $$

in $D_c^b((X_H^{\min})_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$. Since Assumption 4.5.1 implies Assumption 3.7.7, by Lemma 3.7.9, Proposition 3.7.13, and Lemma 4.2.23, we have the following facts: The pure isotypic subquotients of $R\Psi_{X_H}(\bar{Q}_\ell[d])$ extend to well-positioned perverse sheaves $\mathcal{F}_\alpha$ over $(X_{H,\Sigma}^{\text{tor}})_{\bar{s}}$, labeled by some index $\alpha$. Each such $\mathcal{F}_\alpha$ is equipped with a collection $\mathcal{F}_\alpha^\natural = \{(\mathcal{F}_{\alpha,Z}^\natural, \iota_{\alpha,Z})\}_Z$, where $\mathcal{F}_{\alpha,Z}^\natural[-d^Z]$ is a pure isotypic subquotient of $R\Psi_Z(\bar{Q}_\ell[d_Z])$, with $d_Z := \text{dim}(Z_\eta)$ and $d^Z := d - d_Z$ as usual, for each $Z$. Since $\mathcal{F}_\alpha$ and $\mathcal{F}_{\alpha,Z}^\natural$ have isomorphic pullbacks to schemes $U$ smooth over $Z$ of relative dimension $d^Z$, they have the same weight. Moreover, pure isotypic subquotients of $R\Psi_Z(\bar{Q}_\ell[d_Z])$ are exactly such $\mathcal{F}_{\alpha,Z}^\natural[-d^Z]$, over all $\alpha$. By applying Theorem 4.5.26 to such $\mathcal{F}_\alpha$ and $\mathcal{F}_{\alpha,Z}^\natural$, by summing up identities like (4.5.27) over all $Z$ and $\alpha$ (which have the same form regardless of the possibly varying weights of $\mathcal{F}_\alpha$), and by (4.5.44) and (4.5.47), we obtain an identity

$$ (4.5.48) \quad [j_{\bar{s},!*}^{\min}(R\Psi_{X_H}(V[d]))] = \sum_{\alpha} [j_{\bar{s},!*}^{\min}(\mathcal{V} \otimes_L (\mathcal{F}_{\alpha}|_{(X_H)_{\bar{s}}}))] $$

in the Grothendieck group of $D_c^b((X_H^{\min})_{\bar{s}} \times \bar{\eta}, \bar{Q}_\ell)$. For each $\alpha$, since $\mathcal{V}$ is quasi-isomorphic to a direct sum of shifted lisse sheaves, and since $\mathcal{F}_\alpha$ is the middle perversity extension of a shifted (pure isotypic) lisse étale sheaf over an irreducible smooth scheme, $j_{!*}^{\min}(\mathcal{V} \otimes_L (\mathcal{F}_{\alpha}|_{(X_H)_{\bar{s}}}))$ has no support contained in the boundary of $(X_H^{\min})_{\bar{s}}$. Thus, the corollary follows from (4.5.48), as desired. $\square$

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UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MN 55455, USA

E-mail address: kwlan@math.umn.edu

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