2.1 Invariants

Fix a field F and an integer $n\geq 1$ . We consider two quadratic étale extensions $E_1/F$ and $E_2/F$ and write $\sigma_1$ and $\sigma_2$ , respectively, for their non-trivial involutions over F. Let $B = \prod_{j \in J} B^j$ be the product of a finite family $(B^j)_{j \in J}$ of central simple F-algebras. We assume that the degree $[B^j:F]= \dim_F(B^j)^{1/2}$ of each factor is even and that the total degree $[B:F] = \sum_{j \in J}[B^j:F]$ is 2n.

We write $\beta :(E_1,E_2) \to B$ for pairs $(\beta_1:E_1 \to B$ , $\beta_2:E_2 \to B)$ of embeddings as F-algebras into B. Giving a map $\beta_i:E_i \to B$ is the same as giving a tuple of maps $\beta_i^j:E_i \to B^j$ . Without further mention, we only consider maps $\beta_i^j$ that make $B^j$ into a free $E_i$ -module. This additional specification only matters if $E_i \cong F \times F$ . In particular, pairs $\beta:(E_1, E_2) \to B$ are the same as tuples of pairs $\beta^j:(E_1, E_2) \to B^j$ .

Let $E_3 \subset E_1\otimes_F E_2$ be the fixed ring under $\sigma_1\otimes \sigma_2$ . It is a quadratic étale F-algebra and we denote by $\sigma_3:E_3 \to E_3$ its non-trivial F-involution. Given $\beta $ as above, Howard and Li [Reference Howard and LiHL20, 2] define a canonical element $ \mathbf {s}_\beta \in E_3\otimes_F B$ that centralizes $\beta_1(E_1)$ and $\beta_2(E_2)$ . Their definition, which we do not need here, extends verbatim from central simple algebras to products of such; in fact, simply $ \mathbf {s}_\beta =( \mathbf {s}_{\beta^j})_{j \in J}$ . They showed in [Reference Howard and LiHL20, Proposition 2.2.3] that the reduced characteristic polynomial

\[\mathrm{charred}_{E_3 \otimes_F B/E_3}( \mathbf {s}_\beta ; T) = \prod_{j \in J} \mathrm{charred}_{E_3\otimes_FB/E_3}( \mathbf {s}_{\beta^j}; T) \in E_3[T]\]

is a square and make the following definition.

Note that $\mathrm{Inv}(\beta ) = \prod_{j \in J}\mathrm{Inv}(\beta^j)$ and that $\mathrm{Inv}(\beta )$ only depends on the $B^ \times $ -conjugation orbit of $\beta $ . For regular semi-simple $\beta $ and simple B, a converse holds as follows.

Remark 2.4. The converse direction of Proposition 2.3 does not hold for non-simple B. For example, take $E_1 = E_2 = F \times F$ and consider two regular semi-simple pairs $\beta^0, \beta^1:(E_1, E_2) \to M_2(F)$ with $\mathrm{Inv}(\beta^0) \neq \mathrm{Inv}(\beta^1)$ . Then

\[(\beta^0, \beta^1),\ (\beta^1, \beta^0):(E_1, E_2)\longrightarrow B := M_2(F) \times M_2(F)\]

are both regular semi-simple with invariant $\mathrm{Inv}(\beta^0)\mathrm{Inv}(\beta^1)$ . They are not $B^ \times $ -conjugate, however, because $\beta^0$ and $\beta^1$ are not $GL_2(F)$ -conjugate.

An immediate question is, given $E_1$ , $E_2$ , B and a regular semi-simple invariant $\delta $ as above, whether or not there exists a pair $\beta:(E_1, E_2) \to B$ with $\mathrm{Inv}(\beta ) = \delta $ . This is partially answered by the following universal construction.

1. (1) The image $\beta_{\delta,1}(E_1) \cup \beta_{\delta ,2}(E_2)$ generates $B_\delta $ as F-algebra.

2. (2) The natural maps $\beta_{\delta ,i}(E_i)\otimes_F L_\delta \to B_\delta$ are injective and make $\beta_{\delta,i}(E_i)\otimes_FL_\delta $ into a maximal commutative F-subalgebra of $B_\delta $ .

3. (3) For every product B of central simple F-algebras of total degree $[B:F] = 2n$ , and for every pair $\beta:(E_1,E_2) \to B$ of invariant $\delta $ , there is a unique embedding of F-algebras $\iota:B_\delta \to B$ such that $\beta = \iota \circ \beta_\delta $ . Conversely, if $\beta = \iota\circ \beta_\delta $ for an embedding $\iota:B_\delta \to B$ of F-algebras, then $\mathrm{Inv}(\beta ) = \delta $ .

Thus, pairs $\beta $ of invariant $\delta $ for B are the same as F-algebra maps $B_\delta \to B$ . Implicit in Proposition 2.5 is moreover the statement that the centralizer of a regular semi-simple pair $\beta $ is an étale F-algebra of degree n. Namely, it is isomorphic to the center $L_\delta \subset B_\delta $ . By the construction of $B_\delta $ in [Reference Howard and LiHL20], it may be described (up to isomorphism) by

(2.2) \begin{equation}L_\delta \cong (E_3[T]/(\delta (T)))^{\sigma_3 = \mathrm{id}},\end{equation}

where $\sigma_3$ is extended to $E_3[T]$ as $\sigma_3(T) = 1-T$ .

2.2 Matching

From now on, we let F be a non-archimedean local field with normalized absolute value $|\cdot |$ and assume $E_1,E_2$ to be étale field extensions. Put $E_0 = F \times F$ . Then the invariant $\mathrm{Inv}(\alpha )$ of a pair $\alpha =(\alpha_0, \alpha_3):(E_0, E_3) \to M_{2n}(F)$ lies in $E_3[T]$ as well.

Let $\delta \in E_3[T]$ be regular semi-simple and of degree n. Let $\alpha_\delta :(E_0,E_3) \to A_\delta $ be the universal datum for $(E_0, E_3, \delta )$ from Proposition 2.5 and let $L_\delta = \mathrm{Cent}(A_\delta )$ . By part (2) of Proposition 2.5, the $L_\delta $ -algebra $E_0\otimes_FL_\delta \cong L_\delta \times L_\delta $ embeds into $A_\delta$ which implies $A_\delta \cong M_2(L_\delta )$ . In particular, there exists an embedding $\iota:A_\delta \to M_{2n}(F)$ . By Proposition 2.5(3), $\mathrm{Inv}(\iota \circ \alpha_\delta ) = \delta $ . It follows that for every regular semi-simple invariant $\delta $ of degree n, there exists a pair $\alpha :(E_0, E_3) \to M_{2n}(F)$ with $\mathrm{Inv}(\alpha ) = \delta $ . On the other hand, given $E_1$ , $E_2$ , B and $\delta $ , there need not exist a pair $\beta :(E_1, E_2) \to B$ with $\mathrm{Inv}(\beta ) = \delta $ . Definition 2.6 is asymmetric in this sense.

2.3 Orbital integrals for $(E_1, E_2)$

The FL compares $\eta $ -twisted orbital integrals of pairs $(E_0,E_3) \to M_{2n}(F)$ with orbital integrals of matching pairs $(E_1,E_2) \to M_{2n}(F)$ . In this section, we define the second kind. Throughout, we denote by $\Lambda_{\mathrm{std}} := O_F^{2n}$ the standard lattice and by $\mathcal {H} = \mathbb {C}[GL_{2n}(O_F) \backslash GL_{2n}(F) /GL_{2n}(O_F)]$ the spherical Hecke algebra.

Definition 2.7. Let $\beta :(E_1,E_2) \to M_{2n}(F)$ be regular semi-simple and write $H_i :=\mathrm{Cent}(E_i)^ \times $ . Choose $g_i \in GL_{2n}(F)$ such that $g_i\Lambda_{\mathrm{std}}$ is $O_{E_i}$ -stable. For $f \in \mathcal {H}$ , define

(2.3) \begin{equation}O(\beta , f) := \int_{(H_1 \cap H_2) \backslash (H_1 \times H_2)} f(g_1^{-1}h_1^{-1}h_2g_2)\, dh_1 \, dh_2.\end{equation}

Here we endow $H_i \cong GL_n(E_i)$ with the Haar measure such that $\mathrm{Vol}(GL_n(O_{E_i})) = 1$ . The stabilizer $H_1 \cap H_2$ is the group of units $L^ \times $ , where $L = \mathrm{Cent}(\beta(E_1)\cup \beta (E_2)) \cong L_\delta $ is an étale F-algebra of degree n by Proposition 2.5. We endow it with the measure such that $\mathrm{Vol}(O_L^ \times ) = 1$ . Convergence of the integral follows from Lemma 2.8. The definition is then independent of the chosen $g_i$ : Any second choice would come from $H_ig_iGL_{2n}(O_F)$ and f is bi- $GL_{2n}(O_F)$ -invariant.

Proof. Set $G = GL_{2n}(F)$ . There is an isomorphism

\[G \backslash (G/H_1 \times G/H_2) \overset{\sim}{\longrightarrow} H_1 \backslash G/H_2,\quad (g_1, g_2) \longmapsto g_1^{-1}g_2.\]

By Skolem–Noether, via $(g_1, g_2) \mapsto (g_1\beta_1g_1^{-1}, g_2\beta_2g_2^{-1})$ , the source is in bijection with the G-orbits of pairs $(E_1, E_2) \to M_{2n}(F)$ . By Proposition 2.3, the orbit of $\beta $ agrees with the locus where the invariant takes value $\mathrm{Inv}(\beta )$ . It is, in particular, closed. Since the orbit of $\beta $ maps to $H_1H_2$ in the target, it follows that $(H_1 \cap H_2) \backslash (H_1 \times H_2) \subset GL_{2n}(F)$ is a closed subset as claimed.

The orbital integral $O(\beta , f)$ can also be understood as an f-weighted lattice count. Let $\mathcal {L}(\beta_i)$ denote the $O_{E_i}$ -stable lattices in $F^{2n}$ . This set is stable under the $L^ \times $ -action because $L^ \times $ centralizes $\beta_i(E_i)$ . Let $g_i \in GL_{2n}(F)$ be fixed as in Definition 2.7 and set $K_i = \mathrm{Stab}(g_i\Lambda_\mathrm{std}) \cap H_i$ . Note that $H_i$ is isomorphic to $GL_n(E_i)$ which acts transitively on the set of $O_{E_i}$ -lattices in $E_i^n$ . Thus, there is a bijection

(2.4) \begin{equation}H_i/K_i \overset{\sim}{\longrightarrow} \mathcal {L}(\beta_i),\quad h_i \longmapsto h_ig_i\Lambda_\mathrm{std}.\end{equation}

As f lies in the spherical Hecke algebra, the restriction $f\vert_{g_1^{-1}K_1h_1^{-1}h_2K_2g_2}$ is constant for every element $(h_1,h_2) \in H_1 \times H_2$ . Moreover, the volume of $L^ \times \backslash (L^ \times \cdot (h_1K_1 \times h_2K_2))$ is equal to

(2.5) \begin{equation}\mathrm{Vol}(L^ \times \cap h_1K_1 h_1^{-1} \cap h_2K_2h_2^{-1})^{-1} =\mathrm{Vol}(\mathrm{Stab}_{L^ \times }(h_1g_1\Lambda_\mathrm{std}, h_2g_2\Lambda_\mathrm{std}))^{-1}.\end{equation}

We introduce the following notation. Given two $O_F$ -lattices $\Lambda_1,\Lambda_2 \subset F^{2n}$ , we write $f(\Lambda_1,\Lambda_2)$ for the value of f at the relative position of $\Lambda_1$ and $\Lambda_2$ . That is, $f(\Lambda_1,\Lambda_2) := f(k_1^{-1}k_2)$ where the $k_i \in GL_{2n}(F)$ are chosen with $\Lambda_i = k_i\Lambda_{\mathrm{std}}$ . In addition, choose a cocompact, free, discrete subgroup $\Gamma \subseteq L^ \times $ . Combining (2.4) and (2.5) with Definition 2.7, we obtain

(2.6) \begin{align}O(\beta ,f) & = \sum_{(\Lambda_1, \Lambda_2) \in L^ \times \backslash (\mathcal {L}(\beta_1) \times \mathcal {L}(\beta_2))}\mathrm{Vol}(\mathrm{Stab}_{L^ \times }(\Lambda_1,\Lambda_2))^{-1} f(\Lambda_1,\Lambda_2) \nonumber\\& = \mathrm{Vol}(\Gamma \backslash L^ \times )^{-1}\sum_{(\Lambda_1, \Lambda_2) \in \Gamma \backslash (\mathcal {L}(\beta_1) \times \mathcal {L}(\beta_2))} f(\Lambda_1,\Lambda_2).\end{align}

2.4 The $\eta $ -twisted orbital integrals for $(E_0, E_3)$

Let $\eta :E_3^ \times \to \{\pm 1\}$ be the character from local class field theory for the extension $E_1\otimes_FE_2/E_3$ . Our aim is to define certain $\eta $ -twisted orbital integrals for regular semi-simple pairs $\alpha :(E_0,E_3) \to M_{2n}(F)$ . We only consider spherical Hecke functions in this article. For these, the definition of such integrals is only interesting when $\eta $ restricts to the trivial character on $O_{E_3}^ \times $ ; cf. [Reference Howard and LiHL20, Proposition 3.2.7]. Thus, we assume from now on that $E_1\otimes_FE_2/E_3$ is unramified. Equivalently, we assume that one out of $E_1, E_2$ is unramified over F. This leaves us with the following two cases.

1. 1. If $E_1, E_2$ are both unramified over F, then $E_3 \cong F \times F$ and $E_1\otimes_FE_2 \cong E_1 \times E_1$ is unramified over $E_3$ .

2. 2. If precisely one out of $E_1, E_2$ is ramified over F, then $E_1\otimes_FE_2$ is a biquadratic field extension of F such that $\sigma_1 \otimes \sigma_2$ acts non-trivially on the residue field. It follows that $E_3/F$ is ramified and $E_1\otimes_FE_2/E_3$ unramified.

In particular, we see that in both cases $\eta $ can be defined by $\eta (x) = (-1)^{[O_{E_3}:xO_{E_3}]}$ , where the indexFootnote ¹ is meant as $O_F$ -lattices in $E_3$ . We define $\eta $ on $GL_n(E_3)$ by $\eta (h) := \eta (\det h)$ . Denote by $|\cdot |_F$ the normalized absolute value on F. Recall that $E_0 = F \times F$ . We also define the character

(2.7) \begin{equation}\begin{aligned}|\cdot |:GL_n(E_0) & \longrightarrow \mathbb {R}^ \times ,\\(a,b) & \longmapsto \bigg|\frac{\det(a)}{\det(b)}\bigg|.\end{aligned}\end{equation}

The definition of $O(\beta , f)$ in (2.3) required the choice of auxiliary elements $g_1, g_2 \in GL_{2n}(F)$ . The situation for $\eta $ -twisted orbital integrals is similar, but we need to be more careful to ensure independence of such an auxiliary choice. Consider a regular semi-simple pair $\alpha :(E_0, E_3) \to M_{2n}(F)$ . We write $\mathcal {L}(\alpha_i)$ for the set of $\alpha_i(O_{E_i})$ -lattices in $F^{2n}$ . Any lattice $\Lambda_0 \in \mathcal {L}(\alpha_0)$ is of the form $\Lambda_0^+ \times \Lambda_0^-$ , where $\Lambda_0^+ \subset \alpha_0(1, 0)F^{2n}$ is an $\alpha_0(O_F \times 0)$ -lattice and $\Lambda_0^- \subset \alpha_0(0,1)F^{2n}$ an $\alpha_0(0 \times O_F)$ -lattice. Since $\alpha $ is regular semi-simple, one can show (omitted) that the two natural maps obtained from $\alpha_3$ ,

\[E_3\otimes_F \big(\alpha_0(1,0)\cdot F^{2n}\big)\longrightarrow F^{2n},\quad E_3\otimes_F \big(\alpha_0(0,1)\cdot F^{2n}\big) \longrightarrow F^{2n}\]

are bijective. In particular, given $(\Lambda_0,\Lambda_3) \in \mathcal {L}(\alpha_0) \times \mathcal {L}(\alpha_3)$ , there are well-defined lattice indices

(2.8) \begin{equation}[\Lambda_3 : O_{E_3} \Lambda_0^+],\quad [\Lambda_3:O_{E_3} \Lambda_0^-] \in \mathbb {Z}.\end{equation}

Definition 2.9. Let $\alpha :(E_0,E_3) \to M_{2n}(F)$ be regular semi-simple. Given $(\Lambda_0, \Lambda_3) \in \mathcal {L}(\alpha_0) \times \mathcal {L}(\alpha_3)$ and $s \in \mathbb {C}$ , the transfer factor of $(\Lambda_0,\Lambda_3)$ is defined by

(2.9) \begin{equation}\Omega(\Lambda_0, \Lambda_3, s) =(-1)^{[\Lambda_3:O_{E_3}\Lambda_0^+]} \cdot \big(q^{[\Lambda_3:O_{E_3}\Lambda_0^-] -[\Lambda_3:O_{E_3}\Lambda_0^+]}\big)^{s/2}.\end{equation}

This function has the transformation property

(2.10) \begin{equation}\Omega(h_0\Lambda_0, h_3\Lambda_3, s) = |h_0|^s\eta (h_3) \Omega(\Lambda_0, \Lambda_3,s),\quad h_0 \in H_0,\ h_3 \in H_3.\end{equation}

Choose $g_0, g_3 \in GL_{2n}(F)$ such that $g_i\Lambda_{\mathrm{std}} \in \mathcal {L}(\alpha_i)$ . For $f \in \mathcal {H}$ , define

(2.11) \begin{equation}O(\alpha , f, s) := \Omega(g_0\Lambda_{\mathrm{std}},g_3\Lambda_{\mathrm{std}}) \cdot \int_{(H_0 \cap H_3) \backslash (H_0 \times H_3)}f(g_0^{-1}h_0^{-1}h_3g_3)|h_0|^s \eta (h_3) \, dh_0 \, dh_3.\end{equation}

The integrand here is well-defined by [Reference Howard and LiHL20, Lemma 3.2.3] which states that $\eta $ and $|\cdot |$ are trivial on $H_0 \cap H_3$ . Moreover, the normalization of Haar measures in (2.11) is as in Definition 2.7 and the same remarks on convergence apply. Identity (2.10) allows (2.11) to be rewritten as

(2.12) \begin{equation}O(\alpha , f, s) = \int_{(H_0 \cap H_3) \backslash (H_0 \times H_3)} f(g_0^{-1}h_0^{-1}h_3g_3)\Omega(h_0g_0\Lambda_\mathrm{std}, h_3g_3\Lambda_\mathrm{std}) \, dh_0 \, dh_3.\end{equation}

As f is spherical, $f(g_0^{-1}h_0^{-1}h_3g_3)$ only depends on the lattices $g_0\Lambda_\mathrm{std}$ and $\Lambda_3\Lambda_\mathrm{std}$ and not on the precise choices of $g_0, g_3$ . Using that $H_i$ acts transitively on $\mathcal {L}(\alpha_i)$ , we see that $O(\alpha , f, s)$ is actually independent of $g_0$ and $g_3$ .

There is a lattice count formula for $O(\alpha ,f, s)$ that is analogous to (2.6): let L denote the centralizer of $\alpha $ and pick a cocompact subgroup $\Gamma \subset L^ \times $ as before. Then (2.12) gives

(2.13) \begin{align}O(\alpha ,f,s)& = \sum_{(\Lambda_0, \Lambda_3) \in L^ \times \backslash (\mathcal {L}(\alpha_0) \times \mathcal {L}(\alpha_3))}\mathrm{Vol}(\mathrm{Stab}_{L^ \times }(\Lambda_0,\Lambda_3))^{-1}\,f(\Lambda_0,\Lambda_3) \Omega(\Lambda_0,\Lambda_3, s)\nonumber\\& = \mathrm{Vol}(\Gamma \backslash L^ \times )^{-1}\sum_{(\Lambda_0, \Lambda_3) \in \Gamma \backslash (\mathcal {L}(\alpha_0) \times \mathcal {L}(\alpha_3))}f(\Lambda_0,\Lambda_3)\Omega(\Lambda_0,\Lambda_3, s).\end{align}

2.5 Fundamental lemma

Recall that we assumed from § 2.4 that at least one out of $E_1,E_2$ is unramified over F.

Conjecture 2.10 (Guo--Jacquet FL). Assume that $\alpha :(E_0,E_3) \to M_{2n}(F)$ and $\beta :(E_1,E_2) \to M_{2n}(F)$ are regular semi-simple and matching. Let $f \in \mathcal {H}$ be a spherical Hecke function. Then

(2.14) \begin{equation}O(\alpha , f, 0) = O(\beta ,f).\end{equation}

This has been conjectured by Guo and Jacquet [Reference GuoGuo96] for $E_1 = E_2$ . The biquadratic case ( $E_1\neq E_2$ ) has been conjectured by Howard and Li [Reference Howard and LiHL20]. The following cases are known.

1. (1) For $E_1 = E_2$ both unramified and the unit Hecke function $f = 1_{GL_{2n}(O_F)}$ , the FL is the main result of [Reference GuoGuo96].

2. (2) For general $E_1, E_2$ and $n = 1$ , the conjecture is known for the full spherical Hecke algebra, cf. [Reference Howard and LiHL20].

3. (3) For general $E_1, E_2$ and $n = 2$ , the conjecture is known for the unit element $1_{GL_{2n}(O_F)}$ , which is due to the first author [Reference LiLi25].

2.6 Reduction formula

Assume that $n = n^0 + n^1$ and that the two regular semi-simple pairs

\[\alpha :(E_0,E_3)\longrightarrow M_{2n}(F),\quad \beta :(E_1,E_2)\longrightarrow M_{2n}(F)\]

factor through $M_{2n^0}(F) \times M_{2n^1}(F)$ . Denote their components by $\alpha^0, \alpha^1$ and $\beta^0, \beta^1$ , respectively. Our aim is to relate the orbital integrals $O(\alpha ,f,s)$ and $O(\beta ,f)$ with those for the Levi factors, $O(\alpha^j, f, s)$ and $O(\beta^j, f)$ , respectively, $j =0,1$ . This relation is due to Guo [Reference GuoGuo96] when $E_1 = E_2$ .

We work with the following generators (as $ \mathbb {C}$ -vector space) of $\mathcal {H}$ . For $m \in \mathbb {Z}_{\geq 0}$ , write

(2.15) \begin{equation}f(m) = 1_{\{g \in M_{2n}(O_F),\ v(\det(g)) = m\}}.\end{equation}

Given a tuple $m = (m_1,\ldots,m_r) \in \mathbb {Z}_{\geq 0}^r$ , put

(2.16) \begin{equation}f(m) = f(m_1)*f(m_2)*\cdots * f(m_r).\end{equation}

Concretely, f(m)(g) is the number of tuples of $O_F$ -lattices $(\Lambda_r,\ldots,\Lambda_0)$ in $F^{2n}$ such that

(2.17) \begin{equation}gO_F^{2n} = \Lambda_0 \subseteq \Lambda_1 \subseteq \cdots \subseteq \Lambda_r =O_F^{2n},\quad [\Lambda_i:\Lambda_{i-1}] = m_i\ \text{for all } \textit{i}.\end{equation}

Also define $[\pi ] = 1_{\pi GL_{2n}(O_F)}$ which is invertible with inverse $[\pi ]^{-1} =1_{\pi^{-1}GL_{2n}(O_F)}$ . By Theorem B.3,

(2.18) \begin{equation}\mathcal {H} = \sum_{k \in \mathbb {Z},\ r\geq 0,\ m =(m_1,\ldots,m_r) \in \mathbb {Z}_{\geq 0}^r} \mathbb {C}\cdot [\pi ]^k*f(m).\end{equation}

We remark that this is not a direct sum, meaning there are non-trivial relations among the generators $[\pi ]^k*f(m)$ . We also mention that

(2.19) \begin{equation}O(\alpha , [\pi ]^k*f, s) = O(\alpha , f,s),\quad O(\beta , [\pi ]^k*f) = O(\beta , f)\end{equation}

for every $f \in \mathcal {H}$ and $k \in \mathbb {Z}$ . These identities are obtained from $([\pi ]^k*f)(g) =f(\pi^{-k}g)$ and a variable substitution in the definitions of $O(\alpha , f, s)$ and $O(\beta ,f)$ . Let $\mathcal {H}^j = \mathbb {C}[GL_{2n^j}(O_F) \backslash GL_{2n^j}(F) / GL_{2n^j}(O_F)]$ , for $j = 0,1$ , denote the spherical Hecke algebra of $GL_{2n^j}(F)$ . There is a partial Satake transformation

\[\mathcal {S}: \mathcal {H} \longrightarrow \mathcal {H}^0\otimes_ \mathbb {C} \mathcal {H}^1\]

which is discussed in detail in (B.2) and Proposition B.7. It is described on the above generatorsFootnote ² by

(2.20) \begin{equation}\mathcal {S}([\pi ]^k*f(m)) = \sum_{m^0, m^1 \in \mathbb {Z}_{\geq 0}^r,\ m = m^0 + m^1} q^{n^1\cdot |m^0| + n^0\cdot |m^1|} ([\pi ]^k*f(m^0)) \otimes([\pi ]^k*f(m^1)).\end{equation}

Here, $|m| := \sum_{i = 1}^r m_i$ denotes the sum of all coefficients. Taking the product of the orbital integrals for $\alpha^0$ and $\alpha^1$ and for $\beta^0$ and $\beta^1$ , respectively, defines orbital integrals

\[O^{\mathrm{Levi}}((\alpha^0,\alpha^1), f, s),\quad O^{\mathrm{Levi}}((\beta^0,\beta^1), f),\quad f \in \mathcal {H}^0\otimes_{ \mathbb {C}}\mathcal {H}^1.\]

Theorem 2.12. Given $\alpha $ and $\beta $ as above, there are the following relations of orbital integrals on $GL_{2n}(F)$ and its Levi $GL_{2n^0}(F) \times GL_{2n^1}(F)$ :

(2.21)

Here, we have used the notation $|x|_F :=|\mathrm{Nm}_{E_3/F}(x)|^{1/2}_F$ for elements $x \in E_3$ .

2.7 Vanishing order

We now combine the reduction formula with the following vanishing statement.

Proposition 2.14 [Reference Howard and LiHL20, Proposition 3.3.3]. Assume that $\alpha :(E_0,E_3) \to M_{2n}(F)$ is regular semi-simple and matches to a pair $\beta:(E_1, E_2) \to B$ , where B is a central division algebra of degree 2n over F. Then

\[O(\alpha ,f, 0) = 0.\]

This relies on a functional equation for the orbital integral that relates $O(\alpha , f, s)$ with $O(\alpha , f, -s)$ , we refer to [Reference Howard and LiHL20].

Definition 2.15 . Let $\delta $ be a regular semi-simple invariant and let $\beta_\delta :(E_1,E_2) \to B_\delta $ be the universal datum from Proposition 2.5. Denote the center by $L_\delta \subseteq B_\delta $ , write $L_\delta = \prod_{j \in J} L^j$ for its factorization into fields and let $B^j/L^j$ be the corresponding factor of $B_\delta $ . We define the order of $\delta $ as

(2.22) \begin{equation}\mathrm{ord}(\delta ) := |\{j \in J\mid \text{$B^j$ is a division algebra}\}|.\end{equation}

Corollary 2.17. Let $\alpha :(E_0,E_3) \to M_{2n}(F)$ be a regular semi-simple pair. Then

\[\mathrm{ord}_{s = 0} O(\alpha , f, s) \geq \mathrm{ord}(\mathrm{Inv}(\alpha )).\]

Let $q_F$ be the residue cardinality of F. We next consider the normalized first derivative

(2.23) \begin{equation}\mathrm{\partial Orb}(\alpha ,f) := \frac{1}{\log(q_F)} \frac{d}{ds}\bigg\vert_{s = 0}O(\alpha , f, s).\end{equation}

Corollary 2.18. Assume that $\alpha :(E_0, E_3) \to M_{2n}(F)$ is regular semi-simple and that $f \in \mathcal {H}$ is such that

(2.24) \begin{equation}O(\alpha ,f,0) = 0,\quad \mathrm{\partial Orb}(\alpha ,f) \neq 0.\end{equation}

Then $\mathrm{ord}(\mathrm{Inv}(\alpha )) = 1$ . In particular, there are uniquely determined $n^0,n^1$ with $n = n^0 + n^1$ such that $\alpha $ matches a pair $\beta :(E_1,E_2) \to D_{1/2n^0} \times M_{2n^1}(F).$ Here, $D_\lambda$ denotes the central division algebra of Hasse invariant $\lambda$ over F.

Proof. Set $\mathrm{ord}(\alpha ) = \mathrm{ord}(\mathrm{Inv}(\alpha ))$ . Corollary 2.17 provides $\mathrm{ord}(\alpha ) \leq 1$ and we are required to exclude the case $\mathrm{ord}(\alpha ) = 0$ . This relies on the functional equation [Reference Howard and LiHL20, Proposition 3.2.8], saying

(2.25) \begin{equation}O(\alpha ,f,s) = \varepsilon (\alpha ,s)O(\alpha ,f,-s)\end{equation}

-1for a factor $\varepsilon (\alpha ,s) \in \pm|F^ \times |_F^s$ that only depends on $\alpha$ . The proof of [Reference Howard and LiHL20, Proposition 3.3.3], combined with Theorem 2.12 and Lemma 2.16, shows that the sign of $\varepsilon (\alpha ,s)$ is $(-1)^{\mathrm{ord}(\alpha )}$ . Assume it were positive, say $\varepsilon (\alpha ,s) =q_F^{rs}$ . Taking the derivative $(d/ds)\vert_{s=0}$ of (2.25), we obtain that

\[\log(q_F) \mathrm{\partial Orb}(\alpha , f) = r\log(q_F)O(\alpha , f) - \log(q_F) \mathrm{\partial Orb}(\alpha , f).\]

This is impossible under our assumptions (2.24). Thus, necessarily $\mathrm{ord}(\alpha ) = 1$ . The existence of $(n^0,n^1)$ and of the pair $\beta $ is now evident from Lemma 2.16: take $n^0 =[L^{j_0}:F]$ and $n^1 = n - n^0$ , where $j_0 \in J$ is the unique index in (2.22) such that $B^{j_0}$ is division.

3.1 Quadratic cycles

Let F, $E_1$ and $E_2$ as well as $n\geq 1$ be as before. In particular, $E_1$ and $E_2$ are quadratic étale field extensions of F of which at least one is unramified. Denote by $ \breve F/F$ the completion of a maximal unramified extension and by $ \mathbb {F}$ its residue field. Let $ \breve E$ be a composite of $ \breve F$ , $E_1$ and $E_2$ , and fix embeddings $E_1, E_2 \to \breve E$ .

The theories of these two types of objects are completely parallel, at least to the extent required by this article. We also remark that the objects in part (2) are equivalent to minuscule local $O_F$ -shtuka, cf. [Reference Hartl and SinghHS19, Theorem 8.3].

Let ${ \mathbb {X}}/ \mathbb {F}$ be a strict $\pi $ -divisible $O_F$ -module of height 2n and dimension 1 and consider a pair of embeddings

\[\beta :(E_1,E_2)\longrightarrow \mathrm{End}^0({ \mathbb {X}}).\]

Considering its identity component and maximal étale quotient, ${ \mathbb {X}}$ decomposes in a unique way as a product ${ \mathbb {X}} = { \mathbb {X}}^0 \times { \mathbb {X}}^1$ with ${ \mathbb {X}}^0$ connected and ${ \mathbb {X}}^1$ étale. The heights of ${ \mathbb {X}}^0$ and ${ \mathbb {X}}^1$ are even because of the existence of $\beta $ and we write $\mathrm{ht}({ \mathbb {X}}^0) = 2n^0$ and $\mathrm{ht}({ \mathbb {X}}^1) =2n^1$ . Then

\[\mathrm{End}({ \mathbb {X}}) \cong O_D \times M_{2n^1}(O_F),\]

where $D/F$ denotes a central division algebra of Hasse invariant $1/2n^0$ . The datum of $\beta $ is, thus, equivalent to that of two pairs

\[\beta^0:(E_1,E_2) \to \mathrm{End}^0({ \mathbb {X}}^0),\quad \beta^1:(E_1 , E_2) \to \mathrm{End}^0({ \mathbb {X}}^1).\]

Proof. Write $L_\delta = \prod L^j$ and $B_\delta =\prod B^j$ as in Definition 2.15. As we showed during the proof of Corollary 2.18, there exists an F-algebra embedding $B_\delta \to D_{1/2n^0} \times M_{2n^1}(F)$ if and only if there is a unique index $j_0$ with $B^{j_0}$ a division algebra and $[L^{j_0}:F] = n^0$ . The pairs $(2n^0, 2n^1)$ with $n^0 + n^1 = n$ on the other hand are in bijection with the possible isogeny classes of ${ \mathbb {X}}$ (Dieudonné–Manin classification). Finally, any embedding of $B_\delta $ into $D_{1/2n^0} \times M_{2n^1}(F)$ has to be a product of two embeddings

\[B^{j_0}\longrightarrow D_{1/2n^0},\quad \prod_{j\neq j_0} B^j\longrightarrow M_{2n^1}(F)\]

and, hence, all such embeddings are conjugate by Skolem and Noether. This proves the uniqueness up to isogeny.

Let $M \to \mathrm{Spf} O_{ \breve E}$ denote the RZ space of ${ \mathbb {X}}$ , that is

(3.1) \begin{equation}M(S) = \left\{(X, \rho) \left\vert\begin{aligned}&\text{$X/S$ a strict $\pi $-divisible $O_F$-module}\\&\text{$\rho:\overline{S} \times _S X \to \overline{S} \times _{\mathrm{Spec} \mathbb {F}} { \mathbb {X}}$ a quasi-isogeny}\end{aligned}\right.\right\} .\end{equation}

Here, $\overline{S} := \mathbb {F}\otimes_{O_{ \breve E}} S$ denotes the special fiber of S. Analogously define, for $i = 1,2$ ,

(3.2) \begin{equation}Z(\beta_i)(S) = \left\{(X, \rho) \left\vert\begin{aligned}&\text{$X/S$ a strict $\pi$-divisible $O_{E_i}$-module}\\&\text{$\rho:\overline{S} \times _S X \to \overline{S} \times _{\mathrm{Spec} \mathbb {F}} ({ \mathbb {X}}, \beta_i)$ an $O_{E_i}$-linear quasi-isogeny}\end{aligned}\right. \right\}.\end{equation}

The above spaces are well-known to be formally of finite type and formally smooth over $\mathrm{Spf}O_{ \breve E}$ . The relative dimension of M is $2n-1$ , that of the $Z(\beta_i)$ is $n-1$ . Moreover, the reduced subschemes $M_{\mathrm{red}}$ and $Z(\beta_i)_{\mathrm{red}}$ are 0-dimensional. The forgetful maps $Z(\beta_1), Z(\beta_2) \to M$ are closed immersions and allow the $Z(\beta_i)$ to be viewed as cycles on M.

3.2 Hecke correspondences

Throughout this article, we work with a non-standard definition of integral models of Hecke operators that is derived from our non-standard presentation (2.18) of the Hecke algebra. Therefore, we begin with some general remarks on Hecke correspondences that justify this approach and provide the comparison with the definitions in [Reference LiLi22b] and [Reference Howard and LiHL20].

Write $Z^c(M)$ for the group of cycles on M of codimension c, similarly for $M \times _{\mathrm{Spf}O_{ \breve E}} M$ . By correspondence on M we mean an element of

(3.3) \begin{equation}\mathrm{Corr}(M) := \{ z \in Z^{2n-1}(M \times _{\mathrm{Spf} O_{ \breve E}} M)\vert p_1,p_2:\mathrm{Supp}(z) \to M\text{ bothfinite}\}.\end{equation}

Correspondences form a ring through convolution and act on $Z^\bullet(M)$ ; we refer to Appendix A for these definitions and their properties.

Recall that $\mathcal {H}$ denotes the spherical Hecke algebra of $GL_{2n}(F)$ with respect to $GL_{2n}(O_F)$ . Given an indicator function $1_\mu \in \mathcal {H}$ , there is a definition of Hecke correspondence $[R(1_\mu )] \in \mathrm{Corr}(M)$ . It is uniquely characterized by the requirement that its generic fiber parametrizes pairs $((X_1,\rho_1), (X_2, \rho_2))$ such that $\rho_2^{-1}\rho_1:X_1 \to X_2$ is of type $\mu $ . The general definition of [R(f)], $f \in \mathcal {H}$ , is obtained by linear extension. Given cycles $z_1,z_2$ on M that are of complementary dimension and such that $\mathrm{Supp} [R(f)] \cap (\mathrm{Supp}z_1 \times _{\mathrm{Spf} O_{ \breve E}} \mathrm{Supp} z_2)$ is artinian, there is an intersection number $(z_1,[R(f)]*z_2)$ . In this way, the Hecke action on cycles on M and the resulting intersection numbers are defined unambiguously.

An explicit construction of [R(f)] in terms of formal scheme models $R(f) \to M \times M$ is given in [Reference LiLi22b, §6.2] or [Reference Rapoport, Smithling and ZhangRSZ21, §4.3], it relies on the notion of Drinfeld level structure. By Proposition A.8, the Hecke action $[R(f)]:Z^\bullet(M) \to Z^\bullet(M)$ is then expressed as

\[[R(f)]*z = p_{1,*}[R(f) \times _{M \times M} p_2^{-1}(z)].\]

(The bracket notation indicates passage to the cycle.) Assume, furthermore, that $Z_1, Z_2 \subset M$ are closed formal subschemes of pure complementary dimension, both defined by regular sequences, and such that $R(f) \cap (Z_1 \times _{\mathrm{Spf} O_{ \breve E}} Z_2)$ is artinian. The mentioned models R(f) are even regular, so Corollary A.10 implies

(3.4) \begin{equation}\big([Z_1], [R(f)]*[Z_2]\big) = \ell_{O_{ \breve E}} \big(\mathcal {O}_{R(f) \times _{M \times M} (Z_1 \times Z_2)}\big).\end{equation}

This provides the promised link of the intersection numbers from [Reference LiLi22b, Reference LiLi25] (given by the right-hand side) with the canonical ones on the left-hand side. We now come to our definition of Hecke correspondences, which, by the above arguments, lead to the same intersection numbers.

Proposition 3.5. For any $m = (m_1,\ldots,m_r) \in \mathbb {Z}_{\geq 0}$ , the formal scheme R(m) is Cohen–Macaulay and the two projection maps $p_1,p_2:R(m) \to M$ finite flat. In particular, Definition 3.4 provides a model for the Hecke correspondence:

\[\big[[\pi]^k \underset {p_2, M, p_1}{ \times }R(m)\big] = \big[R\big([\pi]^k*f(m)\big)\big]\quad \text{in $\mathrm{Corr}(M)$}.\]

In the p-adic setting, H has to be $O_F$ -stable, of order $q_F^m$ , and such that $O_F$ acts on $X/H$ strictly, meaning $\iota (a)= a$ in $\mathrm{End}(\mathrm{Lie}(X/H))$ . Varying S and $(X,\rho)$ , this description shows that $p_1$ is relatively representable in projective schemes.

In the function field setting, we still consider $O_F$ -stable H of order $q_F^m$ . We also need to make sure that $X/H$ is again a $\pi$ -divisible $O_F$ -module, however. This is the case if and only if each truncation $(X/H)[\pi^r]$ , $r\geq 0$ , is balanced as $ \mathbb {F}_{q_F}$ -module scheme. (Being balanced is a condition on the primitive elements in the Hopf algebra of the group scheme, cf. [Reference PoguntkePog17, Definition 5.13]. This property is equivalent to being a strict $ \mathbb {F}_q$ -module scheme, cf. [Reference Hartl and SinghHS19, Remark 5.3]. It is required for the truncations of a $\pi $ -divisible $O_F$ -module by definition, cf. [Reference Hartl and SinghHS19, Definition 7.1].) We are given that all $X[\pi^r]$ are balanced. Then all $(X/H)[\pi^r]$ are balanced if and only if H is balanced, because this property is well-behaved in exact sequences. The condition for a finite locally free group scheme to be balanced is open and closed on the base, cf. [Reference PoguntkePog17, Remark 5.17]. Finally, we impose that the $O_F$ -action on $X/H$ is strict. In this way, we have again shown that $p_1$ is relatively representable in projective schemes.

Now consider the special fiber $\mathrm{Spec} \mathbb {F} \times _M R(m) \to \mathrm{Spec} \mathbb {F}$ . It is of dimension 0 because a 1-dimensional $\pi$ -divisible $O_F$ -module over $ \mathbb {F}$ has at most finitely many subgroups of order $q_F^i$ . Thus, $p_1$ is finite and we obtain $\dim R(m) \leq \dim M$ .

The map $R(m) \to M \times _{\mathrm{Spf} O_{ \breve E}} M$ is a closed immersion and $M \times _{\mathrm{Spf}O_{ \breve E}} M$ is regular. We claim that R(m) is locally defined by at most $2n - 1$ equations. Let $X_i = p_i^*(X)$ be the pullback of the universal $\pi $ -divisible $O_F$ -module from the ith component, $i = 1,2$ . Write $R(m) = V(I)$ for an ideal sheaf I and consider the Hodge filtrations of $X_1\vert_{V(I^2)}$ and $X_2\vert_{V(I^2)}$ as well as the following map on ( $O_F$ -linear) Grothendieck–Messing crystals coming from $\varphi $ .Footnote ⁴

(3.7)

Note that $I/I^2 \subset \mathcal {O}_{V(I^2)}$ is defined by the condition that it is the minimal ideal $J \subset \mathcal {O}_{V(I^2)}$ such that the universal quasi-isogeny $\rho_2^{-1}\rho_1:X_1\vert_{V(I^2)} \to X_2\vert_{V(I^2)}$ is an isogeny mod J. (Here, $(X_1, \rho_1, X_2, \rho_2)$ denotes the universal point over $M \times _{\mathrm{Spf} O_{ \breve E}}M$ .) By Grothendieck–Messing theory, $I/I^2$ equals the ideal defined by the condition that the composite map $F_1 \to \mathrm{Lie}(X_2\vert_{V(I^2)})$ in (3.7) vanishes. The source is a vector bundle of rank $2n-1$ , the target a line bundle. Thus, R(m) is locally defined by $2n - 1$ equations as claimed. In particular, also $\dim R(m) \geq \dim M$ everywhere, so R(m) is pure of dimension 2n. It follows from the smoothness of M that R(m) is Cohen–Macaulay.

The flatness of both $p_1$ and $p_2$ is now deduced from ‘miracle flatness’: a local homomorphism $R \to S$ of noetherian local rings with R regular and S Cohen–Macaulay is flat if and only if its fiber dimension is constant, see [Sta18, 00R4].

The obtained properties now extend to general r: the composition of finite flat correspondences is finite flat, so both $p_1,p_2:R(m_1,\ldots,m_r) \to M$ are finite flat. Being finite flat over a regular local ring implies Cohen–Macaulay.

Finally, the identity $[[\pi]^k \times _{p_2,M,p_1} R(m)] =[R([\pi ]^k*f(m))]$ is clear for generic fibers. Since R(m) is flat over M, it is also flat over $O_{ \breve E}$ , so the identity extends to cycles on M.

3.3 Intersection numbers

Definition 3.6. For $m = (m_1,\ldots, m_r) \in \mathbb {Z}_{\geq 0}$ and $\beta $ as before, we define $I(\beta ,m)$ by the following Cartesian diagram.

(3.8)

Proof. This follows a posteriori from the length computations in [Reference LiLi22b, Reference Howard and LiHL20]. We give an alternative a priori argument. Let $\mathrm{Spf} A \subset I(\beta ,f)$ be an affine closed formal subscheme, given by some chain $(X_\bullet, \rho) \in R(m)(\mathrm{Spf} A)$ . Then $X := X_r$ algebraizes to a strict $\pi $ -divisible $O_F$ -module X over $\mathrm{Spec} A$ that comes with a pair of embeddings

(3.9) \begin{equation}(\rho^{-1}\beta_1\rho,\ (\rho\varphi_r\cdots \varphi_1)^{-1}\beta_2(\rho\varphi_r\cdots \varphi_1)):(O_{E_1},O_{E_2}) \longrightarrow \mathrm{End}^0(X).\end{equation}

The intersection of its image with $\mathrm{End}(X)$ generates an order in a quaternion algebra B over an étale F-algebra L of degree n because $\beta $ is regular semi-simple, see Proposition 2.5. Moreover, $B\not \cong M_2(L)$ because B embeds into $\mathrm{End}^0({ \mathbb {X}})$ . Our task is to show that $\rho$ algebraizes, i.e. that $(X_\bullet, \rho) \in R(m)(\mathrm{Spec} A)$ .

It is well-known that p-divisible groups in characteristic 0 are étale. The analogous statement holds in the function field setting. Namely let $X/S$ be a strict $\pi $ -divisible $O_F$ -module. The three properties X being $\pi^ \infty$ -torsion, $\pi \in \mathcal {O}_S^ \times $ and $\iota (\pi ) = \pi $ in $\mathrm{End}(\mathrm{Lie}(X))$ imply $\mathrm{Lie}(X) = 0$ . The only strict $\pi $ -divisible $O_F$ -module of height 2n over a geometric point with $\pi \neq 0$ is, thus, $(F/O_F)^{2n}$ (up to isomorphism). Its endomorphism ring is $M_{2n}(O_F)$ and there is no possibility for a pair of invariant $\mathrm{Inv}(\beta )$ as in (3.9). Thus, A has to be $\pi^k$ -torsion, $k\gg 0$ .

Concerning $O_F$ -fields with $\pi = 0$ , the geometric isogeny class of ${ \mathbb {X}}$ is the only one of height 2n and dimension 1 that admits a pair of embeddings of invariant $\mathrm{Inv}(\beta)$ , see Lemma 3.2. Thus, all fibers of X are of the same geometric isogeny class. In particular, the connected component of identity $X^0$ is again a strict $\pi $ -divisible $O_F$ -module, cf. [Reference MessingMes72, Proposition II.4.9] (p-adic case) and [Reference Hartl and SinghHS19, Proposition 10.16] (function field case).

Let $A \to R$ be a local homomorphism to the ring of integers in a complete, algebraically closed, non-archimedean field K. Denote its residue field by k. We use index notation $X_R$ , $X_K$ , $X_k$ , etc. to denote base changes to R, K, k, etc. The map

\[\mathrm{End}({ \mathbb {X}}_R) \longrightarrow \mathrm{End}({ \mathbb {X}}_K) = O_{D_{1/2n^0}} \times M_{2n^1}(O_F)\]

is injective, so the reduction map

\[\mathrm{End}({ \mathbb {X}}_R) \longrightarrow \mathrm{End}({ \mathbb {X}}_k)\]

is a bijection. If we manage to find an isogeny $\tau :X_R \to { \mathbb {X}}_R$ , then the composition $(\rho\tau^{-1}) \tau :X_R \to { \mathbb {X}}_R$ is a $(\mathrm{Spec} R)$ -valued point of M that agrees with the given $\mathrm{Spf} R$ -valued one. But any morphism from $\mathrm{Spec} R$ to M factors through one of the closed points of M because M is a disjoint union of formal spectra of complete local rings. Since R was arbitrary, this implies the statement of the lemma.

Since R is perfect, we may write $X_R = X_R^0 \times X_R^1$ and construct $\tau $ component-wise

\[\tau^0:X_R^0\longrightarrow { \mathbb {X}}_R^0,\quad \tau^1:X_R^1\longrightarrow { \mathbb {X}}_R^1.\]

Both $X_R^1$ and ${ \mathbb {X}}_R^1$ are isomorphic to $(F/O_F)^{2n^1}$ and we take $\tau^1$ as any isomorphism. For $\tau^0$ , we consider $X^0_R \cong \mathrm{Spf} R[\![t]\!]$ as a formal group. (See [Reference Hartl and SinghHS19, Corollary 10.12] for the equivalence of formal $O_F$ -modules and connected strict $\pi $ -divisible $O_F$ -modules.) Since special and generic fiber of $X^0_R$ are both of height $2n^0$ , and since K is algebraically closed, we may normalize the coordinate such that multiplication by $\pi \in O_F$ is expressed by

\[[\pi ](t) = t^{q^{2n^0}}.\]

The power series describing addition and the $O_F$ -action all commute with $[\pi ](t)$ and now have coefficients in the finite field $ \mathbb {F}_{q^{2n^0}}$ . In other words, $X^0_R$ comes via base change from $ \mathbb {F}$ and is, thus, isomorphic to ${ \mathbb {X}}^0_R$ .

Let $L := L(\beta ) \subset \mathrm{End}^0({ \mathbb {X}})$ denote the centralizer of $\beta_1(E_1)\cup \beta_2(E_2)$ . Its units $L^ \times $ act on all objects in (3.8) by transport of structure along $\rho$ .

Proof. Since $I(\beta ,m)( \mathbb {F}) = \pi_0(I(\beta ,m))$ , this is purely a statement about $ \mathbb {F}$ -points. Given a $\pi $ -divisible group X over $ \mathbb {F}$ , we write $X = X^0 \times X^1$ for its decomposition into connected and étale part. Then every $ \mathbb {F}$ -point $(X, \rho) \in M( \mathbb {F})$ is (in a unique way) a product $(X^0 \times X^1, \rho^0 \times \rho^1)$ , where $\rho^0:X^0 \to { \mathbb {X}}^0$ and $\rho^1:X^1 \to { \mathbb {X}}^1$ are two quasi-isogenies. Moreover, giving an isogeny $\varphi_i:X_{i-1} \to X_i$ of height $m_i$ is the same as giving a pair of isogenies

\[\varphi_i^0:X_{i-1}^0\longrightarrow X_i^0,\quad \varphi_i^1:X_{i-1}^1 \longrightarrow X_i^1\]

with $\mathrm{ht}(\varphi_i^0) + \mathrm{ht}(\varphi_i^1)= m_i$ . In this way, we obtain

(3.10) \begin{equation}\pi_0(I(\beta , m)) = \coprod_{m^0 + m^1 = m} \pi_0(I(\beta^0, m^0)) \times \pi_0(I(\beta^1, m^1))\end{equation}

where $m^0,m^1 \in \mathbb {Z}_{\geq 0}^r$ . Moreover, the centralizer is a product $L = L^0 \times L^1$ and its units act diagonally on the right of (3.10). This reduces us to the two cases of ${ \mathbb {X}}$ being connected or étale.

If ${ \mathbb {X}}$ is connected, then $\pi_0(M) \cong \mathbb {Z}$ via $(X, \rho) \mapsto \mathrm{ht}(\rho)$ . The element $\pi \in F^ \times $ acts through translation by 2n, proving the statement in this case. If ${ \mathbb {X}}$ is étale, then $\pi_0(M) \cong GL_{2n}(F)/GL_{2n}(O_F)$ and the argument is the same as for the convergence of (2.3).

In fact, [R(f)] is invariant under the diagonal action of $\mathrm{Aut}^0({ \mathbb {X}})$ on $M \times _{\mathrm{Spf}O_{ \breve E}} M$ for every f. Lemmas 3.7 and 3.8 imply that $\mathrm{Supp} [R(f)] \times _{M \times M} (Z(\beta_2) \times Z(\beta_1))$ is artinian and finite modulo $L^ \times $ whenever $\beta $ is regular semi-simple.

Definition 3.9. Let $\beta $ be regular semi-simple. Endow $L = L(\beta )^ \times $ with the measure such that $\mu (O_L^ \times ) = 1$ . Let $\Gamma \subseteq L^ \times $ be a cocompact subgroup that acts freely on $\pi_0(M)$ . For $f \in \mathcal {H}$ , define the intersection number

(3.11) \begin{align}\mathrm{Int}(\beta , f) &:= \mu (\Gamma \backslash L^ \times )^{-1} \big(\Gamma \backslash [R(f)],\\Gamma \backslash (Z(\beta_2) \times Z(\beta_1))\big)\nonumber\\[6pt]&= \sum_{x \in L^ \times \backslash \pi_0(\mathrm{Supp} [R(f)] \times (Z(\beta_2) \times Z(\beta_1)))} \mu (\mathrm{Stab}(x))^{-1}\big([R(f)]_x,\ (Z(\beta_2) \times Z(\beta_1))_x\big).\end{align}

The intersection of the first expression is taken on $\Gamma \backslash (M \times M)$ . The intersections of the second happen on the connected component x. More concretely, write $f = \sum_{i \in I} \lambda_i \cdot [\pi ]^{k_i}*f(m^{(i)})$ for some finite index set I. (Recall that the $f(m^{(i)})$ were defined in (2.16).) Then, by Proposition 3.5 and Corollary A.10,

\[\mathrm{Int}(\beta ,f) = \sum_{i \in I} \lambda_i\cdot \mathrm{Int}(\beta , m^{(i)}),\]

where the summands on the right-hand side are defined by

(3.12) \begin{align}\mathrm{Int}(\beta , m) &:= \mu (\Gamma \backslash L^ \times )^{-1} \ell_{O_{ \breve E}} (\mathcal {O}_{\Gamma \backslash I(\beta ,m)})\nonumber\\[6pt]&= \sum_{x \in L^ \times \backslash \pi_0(I(\beta,m))} \mu (\mathrm{Stab}(x))^{-1} \ell_{O_{ \breve E}}(\mathcal {O}_{I(\beta ,m),x}).\end{align}

Here, we have also used the following analog of (2.19):

(3.13) \begin{equation}\mathrm{Int}(\beta ,f) = \mathrm{Int}(\beta , [\pi ]^k*f),\quad f \in \mathcal {H},\ k \in \mathbb {Z}.\end{equation}

Namely, $[\pi ]^k*R(f)$ is defined by composing R(f) with the graph of the automorphism $\pi^{-k}:M \to M$ , $(X, \rho) \mapsto (X, \pi^{-k}\rho)$ , and this automorphism maps $Z(\beta_1)$ to itself.

3.4 The AFL

Recall that $E_0 = F \times F$ , and that $E_3 \subset E_1\otimes_F E_2$ is the fixed ring under $\sigma_1\otimes \sigma_2$ . Recall further that

(3.14) \begin{equation}\mathrm{\partial Orb}(\alpha , f) = \frac{1}{\log(q_F)}\frac{d}{ds}\bigg\vert_{s = 0}O(\alpha ,f, s)\end{equation}

denotes the normalized first derivative. By Corollary 2.18 and Lemma 3.2, the following conjecture concerns precisely those invariants $\delta $ with $\mathrm{ord}(\delta ) =1$ .

Conjecture 3.10. Let $\alpha :(E_0,E_3) \to M_{2n}(F)$ be a regular semi-simple pair of embeddings that matches to a pair $\beta :(E_1, E_2) \to \mathrm{End}^0({ \mathbb {X}})$ . Then, for any $f \in \mathcal {H}$ , there is an equality

\[\mathrm{\partial Orb}(\alpha , f) = \mathrm{Int}(\beta , f).\]

It is enough to know Conjecture 3.10 for a $ \mathbb {C}$ -basis of $\mathcal {H}$ . Our main result to this end is the following analogue of the analytic reduction formula Theorem 2.12.

Theorem 3.12. For every $m \in \mathbb {Z}_{\geq 0}^r$ , the following identity holds:

(3.15) \begin{align}\mathrm{Int}(\beta , m) &= |\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-\frac{n^0\cdot n^1}{2}}\big|\mathrm{Res}\big(\mathrm{Inv}(\beta^0),\mathrm{Inv}(\beta^1)\big)\big|_F^{-1}\nonumber\\[6pt]& \quad \cdot \sum_{m = m^0 + m^1} q^{n^1\cdot |m^0| + n^0\cdot |m^1|}\,\mathrm{Int}(\beta^0, m^0) O(\beta^1, f(m^1)).\end{align}

Here, we have used the notation $|x|_F :=|\mathrm{Nm}_{E_3/F}(x)|^{1/2}_F$ for elements $x \in E_3$ .

Comparing this with (2.21), we obtain the following corollary.

Proof. We show the more precise, second statement. Assume that ${ \mathbb {X}} = { \mathbb {X}}^0 \times { \mathbb {X}}^1$ is of height $2n = 2n^0 + 2n^1$ . Consider a pair $\beta = (\beta^0,\beta^1):(E_1, E_2) \to D_{2n^0} \times M_{2n^1}(F)$ , and some $m \in \mathbb {Z}^r_{\geq 0}$ . Let $\alpha^0:(E_0, E_3) \to M_{2n^0}(F)$ match $\beta^0$ and let $\alpha^1:(E_0, E_3) \to M_{2n^1}(F)$ match $\beta^1$ . In particular, their direct sum

\[\alpha = (\alpha^0, \alpha^1):(E_0, E_3) \to M_{2n}(F)\]

matches $\beta $ . Set $\delta^j := \mathrm{Inv}(\alpha^j)= \mathrm{Inv}(\beta^j)$ in the following. In addition, define

\[\Theta = |\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-\frac{n^0\cdot n^1}{2}} \big|\mathrm{Res}\big(\delta^0, \delta^1\big)\big|_F^{-1}\]

and note that this expression equals

\[|\mathrm{Disc}_{E_3/F}|_F^{-\frac{n^0\cdot n^1}{2}}\big|\mathrm{Res}\big(\delta^0, \delta^1\big)\big|_F^{-1}.\]

Namely, taking $E_1$ to be unramified, $\mathrm{Disc}_{E_1/F}$ is trivial and

\[\mathrm{Disc}_{E_2/F} = \mathrm{Disc}_{E_1\otimes_FE_2/E_1} = \mathrm{Disc}_{E_1\otimes_F E_3/E_1} =\mathrm{Disc}_{E_3/F},\]

where the outer identities follow from the unramifiedness of $E_1/F$ and where the middle one follows from $E_1\otimes_FE_3 \cong E_1\otimes_FE_2$ . Now we apply Theorem 2.12. Substituting the concrete expression for the partial Satake transformation $\mathcal {S}(f(m))$ from (2.20), taking the derivative at $s =0$ , and applying the vanishing of $O(\alpha^0,f(m^0))$ for every $m^0 \in \mathbb {Z}^r_{\geq 0}$ (see Proposition 2.14), this theorem states that

(3.16) \begin{equation}\mathrm{\partial Orb}(\alpha , f(m)) = \Theta \cdot \sum_{m = m^0+ m^1} q^{n^1\cdot |m^0| + n^0\cdot |m^1|}\,\mathrm{\partial Orb}(\alpha^0, f(m^0)) O(\alpha^1, f(m^1)).\end{equation}

On the other hand, Theorem 3.12 states that

(3.17) \begin{equation}\mathrm{Int}(\beta , m) = \Theta \cdot \sum_{m = m^0 + m^1} q^{n^1\cdot |m^0| + n^0\cdot |m^1|}\, \mathrm{Int}(\beta^0, m^0) O(\beta^1, f(m^1)).\end{equation}

The FL (Conjecture 2.10), which we assumed to hold, states that $O(\alpha^1, f(m^1))$ equals $O(\beta^1, f(m^1))$ for all $m^1$ . Furthermore, our assumption also is that $\mathrm{\partial Orb}(\alpha^0, f(m^0)) =\mathrm{Int}(\beta^0, m^0)$ for all occurring $m^0$ . Comparing (3.16) and (3.17) then shows that $\mathrm{\partial Orb}(\alpha , f(m)) = \mathrm{Int}(\beta, m)$ as was to be shown. The last statement of the corollary follows from the known cases of the AFL in Remark 3.11.

The proof of Theorem 3.12 is given in the next section. Here, we conclude with the promised example on connected component counts.

Example 3.14. Assume that ${ \mathbb {X}} = { \mathbb {X}}^0$ is connected and that $E = E_1 = E_2$ is unramified over F. Let $L/F$ be an unramified field extension of degree n and choose an embedding $L \to D = \mathrm{End}^0({ \mathbb {X}})$ . Write $B = \mathrm{Cent}_D(L)$ for its centralizer and let $\beta :(E, E) \to B$ be a pair of embeddings of F-algebras whose image generates B over F. This situation can only exist if n is odd because $E\otimes_FL$ then embeds into B by Proposition 2.5(2) and, hence, has to be a field. Denote by $Z = Z(O_L, \beta ) \subseteq M$ the closed formal subscheme of $(X,\rho)$ such that

\[\rho^{-1} O_L\rho,\quad \rho^{-1}\beta_1(O_E)\rho,\quad \rho^{-1}\beta_2(O_E)\rho\ \mathrm{all}\ \subseteq\mathrm{End}(X).\]

Let $\Gamma = \pi^ \mathbb {Z}$ . (Note that $\pi $ is also a uniformizer for L.) Then $\Gamma \backslash Z(O_L, \beta )$ has 2n connected components that correspond to the 2n possible characters of $O_{\beta_1(E)L}$ , acting on the Lie algebra. Imposing compatible Kottwitz conditions for the two $O_E$ -actions leaves us with only n connected components. Each is isomorphic to the intersection $\Gamma \backslash I(\beta \otimes \mathrm{id}_L, 1_{GL_2(O_L)})$ that occurs in the formulation of the linear AFL for $GL_{2,L}$ . Thus,

\[\ell_{O_{ \breve E}}(\mathcal {O}_{\Gamma \backslash Z}) = n\mathrm{Int}(\beta \otimes \mathrm{id}_L, 1_{GL_2(O_L)}).\]

Let $\alpha :(E_0,E_3) \to M_{2n}(F)$ match $\beta $ . The centralizer of $\alpha $ is isomorphic to L. In (2.13), we may sum only over the sets

\[\mathcal {L}(O_L[\alpha_0]) \subseteq \mathcal {L}(\alpha_0),\quad \mathcal {L}(O_L[\alpha_3]) \subseteq \mathcal {L}(\alpha_3)\]

of lattices $\Lambda $ that are also $O_L$ -stable to obtain a rational function in $q^s$ ,

(3.18) \begin{align}&O(O_L[\alpha ],1_{GL_{2n}(O_F)},s)\nonumber\\&\quad = \sum_{(\Lambda_0, \Lambda_3) \in L^ \times \backslash (\mathcal {L}(O_L[\alpha_0]) \times \mathcal {L}(O_L[\alpha_3]))}\mathrm{Vol}(\mathrm{Stab}_{L^ \times }(\Lambda_0,\Lambda_3))^{-1}\delta (\Lambda_0,\Lambda_3) \Omega(\Lambda_0,\Lambda_3, s),\end{align}

with $\delta (\Lambda_0,\Lambda_3) = 1$ if $\Lambda_0 = \Lambda_3$ and 0 otherwise. Then

(3.19) \begin{equation}\begin{aligned}O(O_L[\alpha ], 1_{GL_{2n}(O_F)}, s) & = O(\alpha\otimes \mathrm{id}_L, 1_{GL_2(O_L)}, s),\\\mathrm{\partial Orb}(O_L[\alpha ], 1_{GL_{2n}(O_F)}) & = n\mathrm{\partial Orb}(\alpha \otimes \mathrm{id}_L, 1_{GL_2(O_L)}),\end{aligned}\end{equation}

where the factor n stems from the difference in normalization by $q_F$ or $q_L = q_F^n$ . In this way, we have recovered the AFL conjecture over L for $n = 2$ and the unit Hecke function, but only with the right convention for connected component counting.

4.1 Fibration of lattice sets

Let $V = V^0\oplus V^1$ be a direct sum decomposition of $V = F^{2n}$ with $V^0 \cong F^{2n^0}$ and $V^1 \cong F^{2n^1}$ . Denote by $p^j:V \to V^j$ the two projection maps. The fundamental observation is that there is a bijection of $O_F$ -lattices $X \subseteq V$ and triples

\[\left\{(X^0, X^1, s) \left\vert\begin{aligned}&\text{$X^0 \subset V^0,\ X^1 \subset V^1$ both $O_F $-lattices,}\\&\text{$s:X^1 \to V^0/X^0 $any $O_F$-linear map}\end{aligned}\right. \right\}.\]

It is given by

(4.1) \begin{equation}X \longmapsto \big(X^0 := X \cap V^0,\ X^1 :=p^1(X),\ s_X :=[X^1\overset{t}{\longrightarrow}X\overset{p^0}{\longrightarrow}V^0/X^0]\big),\end{equation}

where $t:X^1 \to X$ is any choice of splitting for $p^1$ . The inverse is given by

\[(X^0, X^1, s) \longmapsto X^0 + \{(\widetilde s(x^1),x^1),\ x^1 \in X^1\},\]

where $\widetilde s:X^1 \to V^0$ is any lift of s. The bijection moreover satisfies the following two functoriality properties.

1. (1) Assume $X \subseteq Y$ . Then clearly $X^0 \subseteq Y^0$ and $X^1 \subseteq Y^1$ . We claim that also the following diagram commutes.

   (4.2)

   
   Namely $(\widetilde s_X(x^1), x^1) \in X$ lying in Y means that there is $y^0 \in Y^0$ such that $\widetilde s_X(x^1) = \widetilde s_Y(x^1) +y_0$ . Conversely, if for two lattices X and Y one has $X^j \subseteq Y^j$ and also that (4.2) commutes, then $X \subseteq Y$ .

2. (2) Let $\zeta = (\zeta^0, \zeta^1) \in \mathrm{End}(V^0) \times \mathrm{End}(V^1) \subset \mathrm{End}(V)$ be an endomorphism. Then X is $\zeta $ -stable if and only if each $X^j$ is $\zeta^j$ -stable and $\zeta^0 \circ s = s \circ \zeta^1$ .

4.2 Application to orbital integrals

Consider a regular semi-simple pair $\alpha:(E_0,E_3) \to M_{2n}(F)$ . Let B be the quaternion algebra generated by the image of $\alpha $ , denote by $L = \mathrm{Cent}(B)$ its center. Assume that $L = L^0 \times L^1$ is a product with eigenspace decomposition $V =V^0\oplus V^1$ . Given a tuple $m =(m_1,\ldots,m_r) \in \mathbb {Z}_{\geq 0}^r$ , define

(4.3) \begin{equation}\mathcal {L}(\alpha ,m) = \left\{(X_0, X_1, \ldots,X_r) \left\vert\begin{aligned}&\text{$X_0 \subset V$ an $O_{E_3} $-lattice, $X_r \subset V$ an $O_{E_0}$-lattice},\\&\text{$X_i \subset V$ an $O_F $-lattice for $i=1,\ldots,r-1$},\\&\text{$X_{i-1} \subseteq X_i$ of index $m_i$}\end{aligned}\right. \right\}.\end{equation}

Note that this definition matches (2.17). The above discussion provides a map

(4.4) \begin{equation}\begin{aligned}\mathcal {L}(\alpha ,m) & \longrightarrow\coprod_{m^0,m^1 \in \mathbb {Z}_{\geq 0}^r,\ m =m^0+m^1} \mathcal {L}(\alpha^0,m^0) \times \mathcal {L}(\alpha^1,m^1)\\(X_0,\ldots,X_r) & \longmapsto \big((X_0^0,\ldots,X_r^0),(X_0^1,\ldots,X_r^1)\big).\end{aligned}\end{equation}

We use the shorter notation $X_\bullet$ , $X_\bullet^0$ and $X_\bullet^1$ in the following. Let $\Gamma^j \subset L^{j, \times }$ be a subgroup with $\Gamma^j \times \mathcal {O}_{L^j}^ \times \cong L^{j, \times }$ and put $\Gamma = \Gamma^0 \times \Gamma^1 \subseteq L^ \times $ . Then (4.4) is $\Gamma $ -equivariant. Using (2.13) and the description of f(m) in (2.17), we obtain

(4.5) \begin{equation}O\big(\alpha ,[\pi ]^k*f(m), s\big) = \sum_{X_\bullet \in \Gamma \backslash \mathcal {L}(\alpha ,m)} \Omega(X_r, X_0, s).\end{equation}

The same results give

(4.6) \begin{align}&O\big(\alpha^0, [\pi ]^k*f(m^0), s\big)O\big(\alpha^1, [\pi ]^k*f(m^1), s\big)\nonumber\\&\quad = \sum_{(X_\bullet^0, X_\bullet^1) \in \Gamma^0 \backslash \mathcal {L}(\alpha^0,m^0) \times \Gamma^1 \backslash \mathcal {L}(\alpha^1, m^1)}\Omega(X_r^0, X_0^0, s) \Omega(X_r^1, X_0^1,s).\end{align}

Observe now that the transfer factor from (2.9) is multiplicative: if $X_\bullet \mapsto (X_\bullet^0, X_\bullet^1)$ in (4.4), then

(4.7) \begin{equation}\Omega(X_r, X_0, s) = \Omega(X_r^0, X_0^0, s)\,\Omega(X_r^1, X_0^1, s).\end{equation}

By the explicit description of the partial Satake transform (2.20), the reduction formula Theorem 2.12 (for $E_0$ and $E_3$ ) is now reduced to the following proposition.

Proposition 4.1. Each fiber of (4.4) over $\mathcal {L}(\alpha^0,m^0) \times \mathcal {L}(\alpha^1,m^1)$ has

(4.8)

many elements. In other words, given any pair $(X_\bullet^0,X_\bullet^1) \in \mathcal {L}(\alpha^0,m^0) \times \mathcal {L}(\alpha^1,m^1)$ , there are (4.8) many tuples $(\gamma_0,\ldots,\gamma_r)$ of $O_F$ -linear maps $\gamma_i:X^1_i \to V^0/X^0_i$ that fit into a commutative diagram

(4.9)

and such that $\gamma_0$ is $O_{E_3}$ -linear, and such that $\gamma_r$ is $O_{E_0}$ -linear.

Given a regular semi-simple pair $\beta :(E_1,E_2) \to M_{2n}(F)$ there is a completely analogous definition of a lattice set $\mathcal {L}(\beta , m)$ . If $\beta $ factors through $M_{2n^0}(F) \times M_{2n^1}(F)$ , all the above considerations apply to provide a fibration

(4.10) \begin{equation}\mathcal {L}(\beta , m)\longrightarrow \coprod_{m = m^0 + m^1} \mathcal {L}(\beta^0, m^0) \times \mathcal {L}(\beta^1, m^1).\end{equation}

In this way, Theorem 2.12 (for $E_1$ and $E_2$ ) reduces to the following statement.

Proposition 4.2. Each fiber of (4.10) over $\mathcal {L}(\beta^0,m^0) \times \mathcal {L}(\beta^1,m^1)$ has

(4.11) \begin{equation}|\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-\frac{n^0\cdot n^1}{2}}\big|\mathrm{Res}\big(\mathrm{Inv}(\beta^0), \mathrm{Inv}(\beta^1)\big)\big|_F^{-1}\cdot q^{n^1|m^0|+n^0\cdot |m^1|}\end{equation}

many elements. In other words, given any pair $(X_\bullet^0,X_\bullet^1) \in \mathcal {L}(\beta^0,m^0) \times \mathcal {L}(\beta^1,m^1)$ , there are (4.11) many tuples $(\gamma_0,\ldots,\gamma_r)$ of $O_F$ -linear maps $\gamma_i:X^1_i \to V^0/X^0_i$ that fit into a commutative diagram

(4.12)

and such that $\gamma_0$ is $O_{E_2}$ -linear, and such that $\gamma_r$ is $O_{E_1}$ -linear.

4.3 Fibration of RZ spaces

Our aim is to adapt the previous arguments to strict $\pi $ -divisible $O_F$ -modules. We begin with some general observations about the connected-étale sequence. Let S be a scheme such that $\pi \in \mathcal {O}_S$ is locally nilpotent. Assume that $X/S$ has fiberwise constant étale rank. Then, by [Reference MessingMes72, Proposition II.4.9] and [Reference Hartl and SinghHS19, Proposition 10.16], respectively, it is an extension

\[0 \longrightarrow X^0 \longrightarrow X \longrightarrow X^1 \longrightarrow 0\]

of its maximal étale quotient by its identity component. We view $\pi $ -divisible $O_F$ -modules as sheaves for the fpqc-topology in the following. Then the maximal étale quotient $X^1$ has the canonical presentation

(4.13) \begin{equation}0 \longrightarrow T \longrightarrow V \longrightarrow X^1 \longrightarrow 0,\end{equation}

where

\[T := TX^1 := \underset {\longleftarrow}{\lim}_{n\geq 0}X^1[\pi^n]\quad \text{and}\quad V := VX^1 :=TX^1[\pi^{-1}]\]

denote the Tate module and rational Tate module of $X^1$ , respectively. The natural map $V \to X^1$ in (4.13) is defined by

\[\pi^{-n}(\ldots,x_2,x_1,x_0) \longmapsto x_n,\quad \text{where}\ (\ldots,x_2,x_1,x_0) \in TX^1.\]

Applying $\mathrm{Hom}(-, X^0)$ and taking into account that multiplication by $\pi $ is an automorphism of V while $\mathrm{Hom}(T,X^0)$ is torsion, we obtain an exact sequence

(4.14) \begin{equation}0 \longrightarrow \mathrm{Hom}(T, X^0) \longrightarrow \mathrm{Ext}^1(X^1,X^0) \longrightarrow \mathrm{Ext}^1(V, X^0).\end{equation}

The leftmost term here is torsion, the rightmost term torsion-free, so

\[\mathrm{Hom}(T, X^0) = \mathrm{Ext}^1(X^1, X^0)_{\mathrm{torsion}}.\]

By definition, an extension

\[0 \longrightarrow X^0 \longrightarrow X \longrightarrow X^1 \longrightarrow 0\]

here is torsion if and only if there exists an integer $N\geq 1$ and the following commutative diagram.

(4.15)

Apply these considerations now to $(X,\rho)$ , the universal pair over M. Let

(4.16) \begin{equation}0\longrightarrow X^0\longrightarrow X\longrightarrow X^1 \longrightarrow 0\end{equation}

be its connected-étale sequence and let $S \to M$ be a point. Both $X^0$ and $X^1$ inherit a strict $O_F$ -action and $\rho$ provides a pair of quasi-isogenies

(4.17) \begin{equation}\rho^0:\overline S \times _S X^0 \longrightarrow \overline S \times _{\mathrm{Spec} \mathbb {F}} { \mathbb {X}}^0,\quad \rho^1:\overline S \times _S X^1 \longrightarrow\overline S \times _{\mathrm{Spec} \mathbb {F}} { \mathbb {X}}^1\end{equation}

over the special fiber $\overline S = \mathbb {F}\otimes_{O_{ \breve E}} S$ . It follows that $\overline S \times _{S} X$ is isogeneous to $\overline S \times _S (X^0 \times X^1)$ and hence X is isogeneous to $X^0 \times X^1$ . The extension class $S \times _M X$ is thus torsion by Proposition 4.3(1) and comes from a map $TX^1 \to X^0$ .

Let $M_{{ \mathbb {X}}^0}$ and $M_{{ \mathbb {X}}^1}$ denote the RZ spaces for the two indicated strict $\pi$ -divisible $O_F$ -modules. Denote by $(Y^0,\rho^0)$ and $(Y^1, \rho^1)$ the universal points of $M_{{ \mathbb {X}}^0}$ and $M_{{ \mathbb {X}}^1}$ , respectively. By abuse of notation, we continue to write $(Y^j, \rho^j)$ in place of $p_j^*(Y^j, \rho^j)$ for their pullbacks to $M_{{ \mathbb {X}}^0} \times M_{{ \mathbb {X}}^1}$ . The Hom-functor $\underline{\mathrm{Hom}}(TY^1, Y^0)$ is then itself a strict $\pi $ -divisible $O_F$ -module. (In fact, since $M_{{ \mathbb {X}}^0} \times M_{{ \mathbb {X}}^1}$ is a union of formal spectra of strict complete local rings, it is isomorphic to $(Y^0)^{2n^1}$ .) Thus, we obtain a map

(4.18) \begin{equation}\begin{aligned}M_{ \mathbb {X}} & \longrightarrow \underline{\mathrm{Hom}}(TY^1, Y^0)\\(X,\rho) & \longmapsto \big((X^0,\rho^0),\ (X^1,\rho^1),\ \text{extension class of } \textit{X}\big).\end{aligned}\end{equation}

Here, we view $\underline{\mathrm{Hom}}(TY^1, Y^0)$ as a sheaf on $M_{{ \mathbb {X}}^0} \times M_{{ \mathbb {X}}^1}$ and $(X^j,\rho^j)$ denotes the image point in $M_{{ \mathbb {X}}^j}$ . Proposition 4.3 proves the following proposition.

Proposition 4.4 The map (4.18) is an isomorphism. It moreover has the following functoriality property. Given $\zeta = (\zeta^0,\zeta^1) \in \mathrm{End}^0({ \mathbb {X}}^0) \times \mathrm{End}^0({ \mathbb {X}}^1)$ , the closed formal subscheme $\mathcal {Z}(\zeta ) \subseteq M_{ \mathbb {X}}$ where $\rho^{-1}\zeta \rho$ is a homomorphism agrees with the subfunctor of $\zeta $ -linear homomorphisms in

(4.19) \begin{equation}\big(\mathcal {Z}(\zeta^0) \times \mathcal {Z}(\zeta^1)\big) \times _{(M_{{ \mathbb {X}}^0} \times M_{{ \mathbb {X}}^1})} \underline{\mathrm{Hom}}(TY^1, Y^0).\end{equation}

Here, an S-valued point of (4.19) is called $\zeta $ -linear if

\[(\rho^0)^{-1}\zeta^0\rho^0\circ h = h\circ T((\rho^1)^{-1}\zeta^1\rho^1).\]

Consider now a regular semi-simple pair $\beta:(E_1,E_2) \to \mathrm{End}^0({ \mathbb {X}})$ , write $\beta^j:(E_1,E_2) \to \mathrm{End}^0({ \mathbb {X}}^j)$ , $j = 0,1$ for its components. Then we have the three intersections $I(\beta ,m)$ , $I(\beta^0,m^0)$ and $I(\beta^1,m^1)$ that result from (3.8) applied to ${ \mathbb {X}}$ , ${ \mathbb {X}}^0$ and ${ \mathbb {X}}^1$ , respectively. The above considerations provide a map

(4.20)

Proposition 4.5. The map (4.20) is finite locally free. Its degree over $I(\beta^0,m^0) \times I(\beta^1,m^1)$ equals

(4.21) \begin{equation}|\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-\frac{n^0\cdot n^1}{2}}\big|\mathrm{Res}\big(\mathrm{Inv}(\beta^0),\mathrm{Inv}(\beta^1)\big)\big|_F^{-1}\cdot q^{n^1\cdot |m^0|+n^0\cdot |m^1|}.\end{equation}

The proof is given in the next section. Here we note the following description of $I(\beta ,m)$ , which is provided by Proposition 4.3 and Proposition 4.4. Namely

\[I(\beta ,m) \subset \prod_{i = 0}^r \underline{\mathrm{Hom}}_{O_F}(TY_i^1, Y_i^0)\vert_{\coprod_{m = m^0 + m^1} I(\beta^0, m^0) \times I(\beta^1, m^1)}\]

is the subfunctor of those tuples $(\gamma_i:TY_i^1 \to Y_i^0)_{i = 0}^r$ such that the diagram

(4.22)

commutes, such that $\gamma_0$ is $O_{E_2}$ -linear, and such that $\gamma_r$ is $O_{E_1}$ -linear.

4.4 Degree computation

In this section we prove Propositions 4.1, 4.2 and 4.5 in parallel, the arguments being the same for all three cases. Our main interest lies in the RZ space intersection, so we use the terminology of $\pi $ -divisible $O_F$ -modules. Fix a pair $\beta :(E_1,E_2) \to \mathrm{End}^0({ \mathbb {X}})$ , a tuple $m \in \mathbb {Z}_{\geq 0}^r$ and a decomposition $m = m^0 + m^1$ . Let $Y_\bullet^1 \in I(\beta^1, m^1)(S)$ and $Y_\bullet^0 \in I(\beta^0,m^0)(S)$ be two fixed S-valued points where S is local artinian with residue field $ \mathbb {F}$ . Our aim is to compute the fiber over $(Y_\bullet^0, Y_\bullet^1)$ of

(4.23) \begin{equation}I(\beta ,m)\longrightarrow \coprod_{m = m^0 + m^1} I(\beta^0,m^0) \times I(\beta^1,m^1).\end{equation}

The group $Y_\bullet^1$ only comes in via its Tate module, so we introduce the notation $T_\bullet := TY_\bullet^1$ and put $Y_\bullet = Y_\bullet^0.$

For the proofs in the analytic settings, one considers instead a pair $\alpha :(E_0,E_3) \to M_{2n}(F)$ (respectively, $\beta :(E_1,E_2) \to M_{2n}(F)$ ) that preserves a decomposition $V = V^0\oplus V^1$ . Fix lattice chains $\Lambda_\bullet \in \mathcal {L}(\alpha^0,m^0)$ and $T_\bullet \in \mathcal {L}(\alpha^1,m^1)$ , (respectively, $\Lambda_\bullet \in \mathcal {L}(\beta^0,m^0)$ and $T_\bullet \in \mathcal {L}(\beta^1,m^1)$ ). Taking $S = \mathrm{Spec} \mathbb {F}$ , the datum $\Lambda_\bullet$ is equivalent to that of a chain $Y_\bullet := \Lambda_\bullet \otimes_{O_F} (F/O_F)$ of étale $\pi $ -divisible $O_F$ -modules of type $(\alpha^0,m^0)$ (respectively, $(\beta^0, m^0)$ ). Similarly, $T_\bullet$ is the chain of Tate modules of $T_\bullet \otimes_{O_F} (F/O_F)$ . In this way, all arguments about $\pi $ -divisible $O_F$ -modules in the following apply literally and provide the proofs of Propositions 4.1 and 4.2. We only formulate the case of $(E_1, E_2)$ , however, leaving it to the reader to substitute for $(E_0, E_3)$ .

Consider the following $\pi $ -divisible $O_F$ -modules over S,

(4.24) \begin{equation}\begin{aligned}H_i & = \mathrm{Hom}_{O_F}(T_i, Y_i),\quad i = 0,\ldots,r,\\C_i & = \mathrm{Hom}_{O_F}(T_{i-1}, Y_i),\quad i = 1,\ldots,r,\end{aligned}\end{equation}

as well as

(4.25) \begin{equation}\begin{aligned}H_0^+ & = \mathrm{Hom}_{O_{E_2}}(T_0, Y_0),\quad H_0^- = \mathrm{Hom}_{O_{E_2}\text{-conj}}(T_0, Y_0),\\H_r^+ & = \mathrm{Hom}_{O_{E_1}}(T_r, Y_r),\quad H_r^- = \mathrm{Hom}_{O_{E_1}\text{-conj}}(T_r, Y_r).\end{aligned}\end{equation}

Here, we used the notation

\[\mathrm{Hom}_{O_{E_2}\text{-conj}}(T_0, Y_0) = \{f:T_0 \to Y_0 \mid f(at) = a^{\sigma_2}f(t)\ \forall\ a \in O_{E_2}\}\]

in the first line and an analogous notation in the second. We now define a homomorphism

(4.26) \begin{equation}\Phi =\begin{pmatrix}p_2^-&&&\\-R_1&L_1&&\\&\ddots&\ddots&\\&&-R_r&L_r\\&&&p_1^-\end{pmatrix}:\bigoplus_{i = 0}^r H_i \longrightarrow H_0^-\oplus \bigoplus_{i = 1}^r C_i \oplus H_r^-\end{equation}

such that $\ker(\Phi )$ precisely describes the tuples $(\gamma_0, \ldots, \gamma_r)$ in (4.9), (4.12) and (4.22). The components $R_i$ and $L_i$ are the composition to the right and left in each square,

(4.27) \begin{equation}\begin{aligned}R_i:H_{i-1} & \longrightarrow C_i,\quad \gamma \longmapsto [Y_{i-1} \to Y_i]\circ \gamma ,\quad i = 1, \ldots, r\\L_i:H_i & \longrightarrow C_i,\quad \gamma \longmapsto \gamma \circ [T_{i-1} \to T_i],\quad i= 1,\ldots, r.\end{aligned}\end{equation}

The condition $L_i(\gamma_i) - R_i(\gamma_{i-1}) = 0$ precisely expresses commutativity of the following square.

The kernels of $p_2^-:H_0 \to H_0^-$ and $p_1^-:H_r \to H_r^-$ , on the other hand, should precisely be the $O_{E_2}$ -linear (respectively, $O_{E_1}$ -linear) elements. Pick elements $\zeta_i$ such that $O_{E_i} = O_F[\zeta_i]$ . Define

(4.28) \begin{equation}\begin{aligned}p_2^-:H_0 & \longrightarrow H^-_0,\quad f \longmapsto [t \mapsto f(\zeta_2t) - \zeta_2f(t)],\\p_1^-:H_r& \longrightarrow H^-_r,\quad f \longmapsto [t \mapsto f(\zeta_1t) - \zeta_1f(t)].\end{aligned}\end{equation}

It is easily checked that $p_2^-$ and $p_1^-$ have image in $H^-_0$ and $H^-_r$ , respectively, as claimed. For example,

\begin{align*}p_1^-(f)(\zeta_1t) & = f(\zeta_1^2t) - \zeta_1f(\zeta_1t)\\& = (\zeta_1 + \zeta_1^{\sigma_1}) f(\zeta_1t) - \zeta_1\zeta_1^{\sigma_1} f(t) - \zeta_1 f(\zeta_1t)\\& = \zeta_1^{\sigma_1} [f(\zeta_1t) - \zeta_1f(t)]\\& = \zeta_1^{\sigma_1}p_1^-(f)(t).\end{align*}

This concludes the definition of (4.26).

Lemma 4.6. Let $E/F$ be an étale quadratic extension and T an $O_E$ -lattice. Let Y be a $\pi$ -divisible group with $O_E$ -action, write $O_E =O_F[\zeta ]$ . Then $\mathrm{Hom}_{O_E}(T, Y)$ is again a $\pi $ -divisible $O_F$ -module and equals the kernel of

\[p:\mathrm{Hom}_{O_F}(T, Y) \longrightarrow \mathrm{Hom}_{O_E\text{-conj}}(T, Y),\ f \longmapsto [t \mapsto f(\zeta t) - \zeta f(t)].\]

In particular, the following sequence is exact,

(4.29) \begin{equation}0 \longrightarrow \mathrm{Hom}_{O_E}(T, Y)\overset{\iota }{\longrightarrow} \mathrm{Hom}_{O_F}(T, Y) \overset{p}{\longrightarrow} \mathrm{Hom}_{O_E\text{-conj}}(T, Y) \longrightarrow 0.\end{equation}

Proof. Choosing a basis of T, it suffices to consider $T = O_E$ . Then both $\mathrm{Hom}_{O_E}(O_E, Y)$ and $\mathrm{Hom}_{O_E\text{-conj}}(O_E, Y)$ are isomorphic to Y by the map $f \mapsto f(1)$ . The choice of $\zeta $ also allows us to identify

\[\mathrm{Hom}_{O_F}(O_E, Y) \cong Y^2,\ f \longmapsto (f(1), f(\zeta )).\]

In these coordinates, p is given by $p(y_1,y_2) = y_2 - \zeta y_1$ . Its kernel is isomorphic to Y via $(y_1, \zeta y_1) \mapsto y_1$ and, hence, a $\pi $ -divisible $O_F$ -module as claimed. Then the cokernel of $\iota $ is a $\pi$ -divisible $O_F$ -module and has the same height as $\mathrm{Hom}_{O_E\text{-conj}}(T, Y)$ , so maps isomorphically onto it.

In particular, $\ker(\Phi )$ precisely describes the fiber of (4.23). The maps $R_i$ and $L_i$ are isogenies, so we have quasi-isogenies $L_i^{-1}R_i:H_{i-1}\dashrightarrow H_i$ that allow to define a quasi-homomorphism

\[\widetilde{p}_1^- := p_1^- L_r^{-1}R_rL_{r-1}^{-1}R_{r-1}\cdots L_1^{-1}R_1: H_0 \dashrightarrow H_r^-.\]

Lemma 4.7 The homomorphism $\Phi $ is an isogeny if and only if the quasi-homomorphism

\[\begin{pmatrix}p_2^-\\\widetilde{p}_1^-\end{pmatrix}: H_0\dashrightarrow H_0^-\oplus H_r^-\]

is a quasi-isogeny. If both are quasi-isogenies, then their degrees are related by

(4.30) \begin{equation}\deg(\Phi ) = \deg\begin{pmatrix}p_2^-\\\widetilde {p}_1^-\end{pmatrix} \cdot \prod_{i=1}^r\deg(L_i).\end{equation}

Proof. We may multiply $\Phi $ from the left by a quasi-isogeny of degree 1 in the following way:

\[\begin{pmatrix}I_{\frac{n}{2}}&&&&\\&\quad \!\!\! I_{n}&&&\\&&\quad \!\!\! \ddots&&\\&&&\quad \!\!\! -p_1^-L_r^{-1}&\quad \!\!\! I_{\frac{n}{2}}\\&&&\quad \!\!\! I_{n}&\quad \!\!\! \end{pmatrix}\begin{pmatrix}p_2^-&&&\quad \!\!\! \\-R_1&\quad \!\!\! L_1&&\quad \!\!\! \\&\quad \!\!\! \ddots&\quad \!\!\! \ddots&\quad \!\!\! \\&&\quad \!\!\! -R_r&\quad \!\!\! L_r\\&&&\quad \!\!\! p_1^-\end{pmatrix}=\begin{pmatrix}p_2^-&&&&\quad \!\!\! \\-R_1&\quad \!\!\! L_1&&&\quad \!\!\! \\&\quad \!\!\! \ddots&\quad \!\!\! \ddots&&\\&&\quad \!\!\! R_{r-1}&\quad \!\!\! L_{r-1}&\\&&&\quad \!\!\! p_1^-L_r^{-1}R_r&\\&&&\quad \!\!\! -R_r&\quad \!\!\! L_r\end{pmatrix}.\]

Iterating this operation leaves us with

\[\begin{pmatrix}p_1^-&&&\quad \\\widetilde{p}_2^-&&&\quad \\-R_1&L_1&&\quad \\&\quad \ddots&\ddots&\quad \\&&\quad -R_r&\quad L_r\end{pmatrix}.\]

The degree of the diagonal here is the right-hand side of (4.30) and the claim follows.

We now shift our attention to the framing object. The link with Lemma 4.7 will be given at the end of the section. Write

\[N = N({ \mathbb {X}}),\quad N^0 = N({ \mathbb {X}}^0),\quad N^1 = N({ \mathbb {X}}^1)\]

for the isocrystals of the respective $O_F$ -modules. These are $ \breve F$ -vector spaces of dimensions 2n, $2n^0$ and $2n^1$ , respectively. In addition, put

\[P = N(\mathrm{Hom}_{O_F}(T{ \mathbb {X}}^1, { \mathbb {X}}^0)) = \mathrm{Hom}(N^1,N^0)\]

which has $ \breve F$ -dimension $4n^0n^1$ . Define the following subspaces of P, each of half that dimension,

(4.31) \begin{equation}\begin{aligned}P^+_1 &= \mathrm{Hom}_{E_1}(N^1,N^0),\quad P^-_1 = \mathrm{Hom}_{E_1\text{-conj}}(N^1, N^0),\\P^+_2 &= \mathrm{Hom}_{E_2}(N^1,N^0),\quad P^-_2 = \mathrm{Hom}_{E_2\text{-conj}}(N^1, N^0).\end{aligned}\end{equation}

We also consider the projection maps to $P^{\pm}_i$ , for $i = 1,2$ ,

(4.32) \begin{equation}\begin{aligned}q_i^+:P & \longrightarrow P^+_i,\quad f \longmapsto [t \mapsto f(\zeta_it) -\zeta_i^{\sigma_i}f(t)],\\q_i^-:P &\longrightarrow P^-_i,\quad f \longmapsto [t \mapsto f(\zeta_it) - \zeta_i^{\sigma_i}f(t)].\end{aligned}\end{equation}

The given points $(Y_\bullet^j, \rho_\bullet^j,\varphi_\bullet^j) \in I(\beta^j,m^j)(S)$ define lattices in the above $ \breve F$ -vector spaces. Let $M(-)$ denote the integral Dieudonné module functor over the special point of S. Set

(4.33) \begin{equation}\begin{aligned}M^0_1 &= N(\rho_r^0)(M(Y_r^0)),\quad M^0_2 = N(\rho_0^0)(M(Y_0^0)),\\M^1_1 &= N(\rho_r^1)(M(Y_r^1)),\quad M^1_2 = N(\rho_0^1)(M(Y_0^1)).\end{aligned}\end{equation}

Then $M^0_1 \subset N^0$ and $M^1_1 \subset N^1$ are $O_{E_1}$ -stable, whereas $M^0_2$ and $M^1_2$ are $O_{E_2}$ -stable. We note for later use that there is an inclusion $M^0_2 \subseteq M^0_1$ of index $|m^0|$ because of the chain of isogenies

\[Y_0^0 \overset{\varphi^0_1}{\longrightarrow} Y_1^0 \overset{\varphi^0_2}{\longrightarrow} \cdots \overset{\varphi^0_r}{\longrightarrow} Y_r^0\]

that satisfies $\rho_0^0 = \rho_r^0\circ \varphi^0_r \circ \cdots \circ \varphi^0_1$ . In the same way, $M^1_2 \subseteq M^1_1$ with index $|m^1|$ .

Passing to Hom-spaces, these lattices provide Dieudonné lattices in P and the $P_i^{\pm}$ ,

(4.34) \begin{equation}\begin{aligned}\Lambda_i & = \mathrm{Hom}_{O_F}(M_i^1, M_i^0),\\\Lambda_i^+ & = \mathrm{Hom}_{O_{E_i}}(M_i^1, M_i^0),\\\Lambda_i^- & = \mathrm{Hom}_{O_{E_i}\text{-conj}}(M_i^1, M_i^0).\end{aligned}\end{equation}

Just like in (4.29), the projections from (4.32) restrict to surjections $q_i^{\pm}:\Lambda_i\twoheadrightarrow\Lambda_i^{\pm}$ . Let us adopt the following notion. Given two $O_{ \breve F}$ -lattices or $O_F$ -lattices $\Lambda $ and $\Lambda '$ , we write $f:\Lambda \dashrightarrow \Lambda '$ for homomorphisms $f:\Lambda [\pi^{-1}] \to \Lambda'[\pi^{-1}]$ together with the datum of the two lattices. We call such morphisms quasi-homomorphisms. We call a quasi-homomorphism f a quasi-isogeny if the map $f:\Lambda[\pi^{-1}] \to \Lambda '[\pi^{-1}]$ is bijective. In this case, we define the degree of f by $\deg(f) = q^{[\Lambda ':f(\Lambda )]}$ . For example, $\Lambda_1$ and $\Lambda_2$ are both lattices in the $ \breve F$ -vector space P. The inclusion maps define a quasi-isogeny $\Lambda_2\dashrightarrow \Lambda_1$ with

(4.35) \begin{align}\deg(\Lambda_2\dashrightarrow \Lambda_1) & = q^{[\Lambda_1:\Lambda_2]}\nonumber\\& = q^{[M^0_1:M^0_2] \dim_{ \breve F}(N^1) - [M^1_1:M^1_2] \dim_{ \breve F}(N^0)}\nonumber\\&= q^{2n^1|m^0| - 2n^0|m^1|}.\end{align}

Proposition 4.8 Assume that $\beta $ is regular semi-simple. Then the quasi-homomorphism

\[(q_1^-, q_2^-):\Lambda_2 \dashrightarrow \Lambda_1^- \times \Lambda_2^-\]

is a quasi-isogeny of degree

(4.36) \begin{equation}|\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-\frac{n^0\cdot n^1}{2}}\big|\mathrm{Res}\big(\mathrm{Inv}(\beta^0),\mathrm{Inv}(\beta^1)\big)\big|_F^{-1}\cdot q^{n^1\cdot |m^0| - n^0 \cdot |m^1|}.\end{equation}

Here, we have used the notation $|x|_F :=|\mathrm{Nm}_{E_3/F}(x)|^{1/2}_F$ for elements $x \in E_3$ .

We first prove two lemmas.

Lemma 4.9 If one of the six maps appearing below is a quasi-isogeny, then all of them are. In this case, the degrees of

(4.37) \begin{equation}\begin{array}{c}(q_1^-, q_2^-): \Lambda_1 \dashrightarrow\Lambda_1^- \times \Lambda_2^-,\\[6pt]q_2^-\vert_{\Lambda_1^+}:\Lambda_1^+\dashrightarrow \Lambda_2^-\quad \mathrm{and}\quad q_2^+\vert_{\Lambda_1^-}:\Lambda_1^-\dashrightarrow\Lambda_2^+\end{array}\end{equation}

all agree. Similarly, the degrees of

(4.38) \begin{equation}\begin{array}{c}(q_1^-, q_2^-):\Lambda_2 \dashrightarrow\Lambda_1^- \times \Lambda_2^-,\\[6pt]q_1^-\vert_{\Lambda_2^+}:\Lambda_2^+\dashrightarrow \Lambda_1^-\quad \mathrm{and}\quad q_1^+\vert_{\Lambda_2^-}:\Lambda_2^-\dashrightarrow\Lambda_1^+\end{array}\end{equation}

all agree.

Proof. We prove the claim for the first triple of maps. Consider the exact sequence from (4.29) to obtain the following commutative diagram.

(Note that $q_1^-:\Lambda_1 \to \Lambda_1^-$ is surjective even if $E_1/F$ is ramified; the argument is as for the map p in (4.29).) The claim follows from this diagram for the two maps $(q_1^-,q_2^-)$ and $q_2^-\vert_{\Lambda_1^+}$ in (4.37). Consider now any quasi-isogeny $z \in \mathrm{End}^0_F({ \mathbb {X}}^0)$ that Galois commutes with both $E_1$ and $E_2$ in the sense that $z\zeta_i = \zeta_i^{\sigma_i}z$ for both $i= 1,2$ . (Such elements always exist; a specific choice is given by the element $ \mathbf {z}$ from [Reference Howard and LiHL20, (2.4.2)]. It Galois commutes with both $E_i$ by [Reference Howard and LiHL20, Proposition 2.4.2] and is invertible because $\beta^0$ is regular semi-simple. The latter implies $ \mathbf {t}$ in [Reference Howard and LiHL20, Proposition 2.4.2] to be invertible by definition, then [Reference Howard and LiHL20, Proposition 2.4.1] implies $ \mathbf {z}$ invertible.) The element z intertwines $q_2^-\vert_{\Lambda_1^+}$ and $q_2^+\vert_{\Lambda_1^-}$ in the sense that the following diagram commutes.

Since z is a quasi-isogeny and since the horizontal maps have the same degree, this proves the claimed shared properties of the first triple. The case of the second triple is the same by symmetry. Moreover, $\Lambda_1$ and $\Lambda_2$ are lattices in the same isocrystal, so the first three maps are quasi-isogenies if and only if the second triple is.

Lemma 4.10. The endomorphism $q_1^+q_2^-\vert_{\Lambda_1^+}:\Lambda_1^+ \to \Lambda_1^+$ is a quasi-isogeny of degree

\[|\mathrm{Disc}_{E_1/F}\cdot \mathrm{Disc}_{E_2/F}|_F^{-n^0\cdot n^1}\big|\mathrm{Res}\big(\mathrm{Inv}(\beta^0), \mathrm{Inv}(\beta^1)\big)\big|_F^{-2}.\]

Proof. For any $f \in \Lambda_1^+$ , we have $\beta_1^0(x)\circ f = f\circ \beta_1^1(x)$ for every $x \in E_1$ . From the definition in (4.32) we obtain that

(4.39)

with the two elements

(4.40) \begin{equation}\begin{aligned}w(\beta^0) & = \big(\beta_2^0(\zeta_2) \circ \beta_1^0(\zeta_1) + \beta_1^0(\zeta^{\sigma_1}_1)\circ \beta_2^0(\zeta^{\sigma_2}_2)\big),\\w(\beta^1) & = \big(\beta_2^1(\zeta_2) \circ \beta_1^1(\zeta_1) + \beta_1^1(\zeta^{\sigma_1}_1)\circ \beta_2^1(\zeta^{\sigma_2}_2)\big).\end{aligned}\end{equation}

We have also used the simple observation

(4.41) \begin{equation}f\circ \zeta_i - \zeta_i \circ f = \zeta_i^{\sigma_i}\circ f - f \circ \zeta_i^{\sigma_i}\end{equation}

for all maps f, which results from $f \circ \mathrm{tr}(\zeta_i) = \mathrm{tr}(\zeta_i) \circ f.$ The elements $w(\beta^0)$ and $w(\beta^1)$ are precisely the elements $ \mathbf w$ for the two pairs $\beta^0$ and $\beta^1$ from [Reference Howard and LiHL20, (2.4.1)]. They have the property that each of them is both $O_{E_1}$ - and $O_{E_2}$ -linear, i.e. $w(\beta^i)$ centralizes $\beta^i$ , see [Reference Howard and LiHL20, Proposition 2.4.2]. Note that $\mathrm{End}_{E_1}^0({ \mathbb {X}}^1) \cong M_{n^1}(E_1)$ , also write $D = \mathrm{End}_{E_1}^0({ \mathbb {X}}^0)$ . Thus, we may view

\[w(\beta^0) \in D,\quad w(\beta^1) \in M_{n^1}(E_1)\]

and $q_1^+ q_2^-\vert_{\Lambda_1^+} =w(\beta^1)\otimes 1 - 1 \otimes w(\beta^0) \in M_{n^1}(E_1) \otimes_{E_1} D$ . We recall a basic fact about the resultant. Assume we are given $A \in M_n(k)$ and $B \in M_m(k)$ where k is any field. Then

\[\det\big[A\otimes 1 - 1 \otimes B: M_{n \times m}(k)\longrightarrow M_{n \times m}(k),\quad f \longmapsto Af - fB\big] = \mathrm{Res}(P_A, P_B),\]

where $P_A$ and $P_B$ denote the characteristic polynomials of A and B. (This identity is clear if A and B are diagonalizable. The general case follows from a Zariski density argument and by passing to the algebraic closure of k.) Put differently, the determinant of $A\otimes 1 - 1 \otimes B \in M_n(k)\otimes_kM_m(k)$ , viewed as $(nm \times nm)$ -matrix is $\mathrm{Res}(P_A, P_B)$ . Since determinants may be computed after scalar extension, these considerations also apply to central simple algebras and we obtain

(4.42) \begin{align}\deg(q_1^+ q_2^-\vert_{\Lambda_1^+}) &= \big\vert\mathrm{Nrd}_{M_{n^1}(D)/E_1}\big(w(\beta^1)\otimes 1 - 1 \otimes w(\beta^0)\big)\big\vert^{-1}_{E_1}\nonumber\\&= \big\vert\mathrm{Res}(\mathrm{charred}(w(\beta^0)), \mathrm{charred}(w(\beta^1)))\big\vert^{-1}_{E_1},\nonumber\\&= \big\vert\mathrm{Res}(\mathrm{charred}(w(\beta^0)), \mathrm{charred}(w(\beta^1)))^2\big\vert^{-1}_{F}.\end{align}

It is shown in [Reference Howard and LiHL20, Proposition 2.4.1] that the two characteristic polynomials here are related to the invariants of $\beta^0$ and $\beta^1$ by a linear transformation. More precisely, one has $w(\beta^i) =d \mathbf {s}_{\beta^i} + c$ in $E_3 \otimes_FM_{2n^1}(F)$ and in $E_3\otimes_F D_{1/2n^0}$ , respectively, where c, $d \in E_3$ , are the following two elements,

\[c = -(\zeta_1\otimes \zeta_2^{\sigma_2} +\zeta_1^{\sigma_1}\otimes \zeta_2),\quad d =(\zeta_1-\zeta_1^{\sigma_1})\otimes (\zeta_2 -\zeta_2^{\sigma_2}).\]

(These a priori lie in $E_1\otimes_FE_2$ but are clearly $\sigma_1\otimes \sigma_2$ -invariant.) It follows that if $\mu_1,\ldots,\mu_{n^i} \in \bar F$ are the eigenvalues of $\mathrm{Inv}(\beta^i)$ with respect to a homomorphism $\rho:E_3 \to \bar F$ , then $d\mu_1+c,\ldots,d\mu_{n^i}+c$ are the eigenvalues of $\mathrm{charred}(w(\beta^i))$ with respect to $\rho$ . The resultant term in (4.42) is thus equal to

(4.43) \begin{equation}\big\vert\big((\zeta_1 -\zeta_1^{\sigma_1})(\zeta_2-\zeta_2^{\sigma_2})\big)^{2n^0\cdot n^1}\mathrm{Res}(\mathrm{Inv}(\beta^0),\mathrm{Inv}(\beta^1))^2\big\vert_F^{-1}.\end{equation}

Since $|\zeta_i - \zeta_i^{\sigma_i}|_F =|\mathrm{Disc}_{E_i/F}|_F^{1/2}$ , the lemma is proved.