1.1 Background

Even though our paper will be purely local in nature, we begin by describing its global motivation. Let F be a totally real number field, let $K/F$ be a quadratic CM extension, and let $D/K$ be a central simple algebra (CSA) of degree $2n$ . Let $*:D\to D$ be an involution of the second kind. Recall that this means that $(xy)^{*} = y^{*}x^{*}$ and that $*$ restricts to the complex conjugation on K. We further assume that $*$ is positive in the sense that $\mathrm {trd}_{D/\mathbb {Q}}(x^{*}x)> 0$ for all $x\neq 0$ . Let $\beta \in D^{\times }$ be a skew-hermitian element, meaning that $\beta ^{*} = -\beta $ . Then the algebraic group (over $\mathbb {Q}$ )

(1.1) $$ \begin{align} G := \{x\in D^{\mathrm{op}, \times} \mid x^{*}\beta x = \nu(x)\beta \text{ for some } \nu(x) \in \mathbb{G}_m\} \end{align} $$

is an inner form of a unitary similitude group in $2n$ variables and the above data can be completed into a PEL type Shimura datum $(G, X_G)$ . The resulting Shimura variety $\mathrm {Sh}_G$ can be described as a moduli space of polarized abelian varieties with D-action and level structure.

For example, we may take $(D, *)$ as the matrix algebra $M_{2n}(K)$ with transpose conjugation . Then $\beta \in GL_{2n}(K)$ can be any skew-hermitian matrix and G is the corresponding unitary similitude group $GU(K^{2n}, \beta )$ .

Coming back to the general situation, we assume that the signatures of $\beta $ are $(2n-1, 1)$ at a unique archimedean place and $(2n, 0)$ at all others. Then $\mathrm {Sh}_G$ is of dimension $2n-1$ , and our interest lies in algebraic cycles in the arithmetic middle dimension $n-1 = \lfloor (2n-1)/2\rfloor $ . By work of Li–Liu [Reference Li and Liu25, Reference Li and Liu26] for $D = M_{2n}(K)$ , it is known in many cases that the height pairings of such cycles are related to the leading terms of the Taylor expansions of certain L-functions. This relation can be understood as a generalization of the Gross–Zagier formula [Reference Gross and Zagier14, Reference Yuan, Zhang and Zhang43] to higher dimensions. It is moreover parallel to the arithmetic Gan–Gross–Prasad conjectures [Reference Gan, Gross and Prasad12], and it also represents an instance of the Beilinson–Bloch height conjectures [Reference Beĭlinson3, Reference Bloch4].

We are thus lead to the problem of constructing algebraic cycles in arithmetic middle dimension on $\mathrm {Sh}_G$ , and of relating their height pairings to the Taylor expansions of L-functions. One construction of such cycles, going back to W. Zhang, is given by imposing additional quadratic multiplication. The resulting cycles differ from the ones in [Reference Li and Liu25, Reference Li and Liu26] which are instead closely related to Kudla–Rapoport divisors. We next describe this construction in more detail:

Let $E/F$ be a totally real quadratic extension and let $E\hookrightarrow D$ be an F-linear embedding such that $x^{*} = x$ for all $x\in E$ . The centralizer $C = \mathrm {Cent}_D(E)$ then has center $EK$ and is preserved by $*$ . If we further choose these data such that $\beta $ lies in $C^{\times }$ , then we can define the algebraic group (over $\mathbb {Q}$ )

(1.2) $$ \begin{align} H := \{x\in C^{\mathrm{op}, \times} \mid x^{*}\beta x = \nu(x)\beta \text{ for some } \nu(x) \in \mathbb{G}_m\}. \end{align} $$

This is an inner form of a unitary similitude group in n variables for the quadratic extension $EK/E$ . One may always find a PEL type Shimura datum $X_H$ for H such that $(H, X_H)\to (G, X_G)$ is a morphism of Shimura data. Then we obtain a closed immersion $\mathrm {Sh}_H \hookrightarrow \mathrm {Sh}_G$ of Shimura varieties. Moreover, $\mathrm {Sh}_H$ is of the desired middle dimension $n-1$ .

The existence of the Beilinson–Bloch height pairing is still conjectural. It is expected, however, that it decomposes into a sum of local heights whose non-archimedean terms are closely related to intersection numbers on integral models. For this reason, we consider integral models $\widetilde {\mathrm {Sh}_H}\hookrightarrow \widetilde {\mathrm {Sh}_G}$ of the given Shimura varieties. Our interest then lies in the intersection numbers $I(f) = \langle \widetilde {\mathrm {Sh}_H},\, f*\widetilde {\mathrm {Sh}_H} \rangle $ for varying Hecke operators $f\in C^{\infty }_c(G(\mathbb {A}_f))$ , and their relations with leading terms of L-functions.

In this context, there is a relative trace formula (RTF) comparison approach due to Leslie–Xiao–Zhang [Reference Leslie, Xiao and Zhang24] that goes back to work of Guo [Reference Guo13] and Friedberg–Jacquet [Reference Friedberg and Jacquet11]: One RTF is formulated for the given pair of groups $(G,\, H)$ , and the other one is formulated for the pair $(G',\, H')$ defined as $(GL_{2n},\, GL_n\times GL_n)$ . The significance of the pair $(G',\, H')$ is that the L-functions of interest occur on the spectral side of its RTF. The problem thus becomes to relate the intersection numbers $I(f)$ with $(H'\times H')(\mathbb {A})$ -orbital integrals on $G'(\mathbb {A})$ .

This relation is made precise by factoring the global orbital integral into local ones and analogously decomposing the global intersection number into place-by-place contributions. The latter relies on Rapoport–Zink (RZ) uniformization. One then, finally, arrives at a purely local question – namely, that of expressing intersection numbers on moduli spaces of p-divisible groups (RZ spaces) in terms of local orbital integrals.

These ideas have recently lead to the discovery of new arithmetic fundamental lemma (AFL) identities: the linear AFL of the first author [Reference Li27], the variants from his joint work with B. Howard [Reference Howard and Li19] and with the second author [Reference Li and Mihatsch30], and the AFL for unitary groups of Leslie–Xiao–Zhang [Reference Leslie, Xiao and Zhang24]. The terminology ‘AFL’ here refers to the fact that all these are identities of intersection numbers on RZ spaces with good reduction and central derivatives of local orbital integrals for spherical Hecke functions. Put differently, they concern the places of good reduction of $\mathrm {Sh}_H$ and $\mathrm {Sh}_G$ .

We mention that the above ideas were first proposed by W. Zhang in the context of the unitary arithmetic Gan–Gross–Prasad conjecture [Reference Zhang45]. He deduced an AFL for unitary groups that he later proved with contributions of the second author and Z. Zhang [Reference Zhang48, Reference Mihatsch31, Reference Mihatsch and Zhang32, Reference Zhang49]. We refer to his ICM report [Reference Zhang47] for a survey.

Arithmetic transfer (AT) identities extend the realm of AFL identities in the sense that they express intersection numbers on RZ spaces with bad reduction in terms of orbital integrals. Their role in the global setting is similar to that of AFL identities, but for the places of bad reduction of the Shimura varieties in question. This idea was first studied systematically by Rapoport–Smithling–Zhang [Reference Rapoport, Smithling and Zhang34, Reference Rapoport, Smithling and Zhang35, Reference Rapoport, Smithling and Zhang36] in the context of the unitary arithmetic Gan–Gross–Prasad conjecture. Z. Zhang [Reference Zhang49] generalized and solved one of their cases by proving AT identities in arbitrary dimension for maximal parahoric level for unramified quadratic extensions. His result has found global applications in the work of Disegni–Zhang [Reference Disegni and Zhang8] who prove a p-adic variant of the arithmetic Gan–Gross–Prasad conjecture. Another application of AT in the global setting can be found in the work of C. Qiu [Reference Qiu33], who proved AT identities for all places and level structures to show an analog of the Gross–Zagier formula over function fields.

1.2 Arithmetic transfer for central simple algebras

Consider again the above intersection numbers $I(f) = \langle \widetilde {\mathrm {Sh}_H},\, f*\widetilde {\mathrm {Sh}_H}\rangle $ . The AT identities in the present paper provide an expression for the contributions to $I(f)$ from places of F that are split in K and inert in E. The completions of G and H at such places are essentially inner forms of general linear groups. Correspondingly, we consider an intersection problem of EL type RZ spaces for central simple algebras (CSA). It is worth singling out two special cases:

If the CSAs in question split, then the resulting moduli spaces are Lubin–Tate spaces, which leads to the intersection problem of the linear AFL mentioned above. If, however, the Hasse invariants of the CSAs in question are $1/2n$ and $1/n$ , then the intersection problem is formulated for Drinfeld’s half spaces and the cycles arise from Drinfeld’s ‘basic construction’ [Reference Drinfeld10, §Reference Beĭlinson3]. This situation comes up when the two Shimura varieties have p-adic uniformization [Reference Rapoport and Zink38, §6.40].

We will now give a precise formulation of our AT conjecture and our results.

1.3 The fundamental lemma conjecture

Let F be a p-adic local field,Footnote ¹ let $E/F$ be an unramified quadratic field extension and let $\eta :F^{\times } \to \{\pm 1\}$ be the corresponding quadratic character. Let $G' = GL_{2n}(F)$ with subgroup $H' = GL_n(F\times F)$ . For a test function $f'\in C^{\infty }_c(G')$ , an element $\gamma \in G'$ that is regular semi-simple with respect to the $(H'\times H')$ -action, and for a complex parameter $s\in \mathbb {C}$ , we consider Guo’s [Reference Guo13] orbital integral

(1.3) $$ \begin{align} \operatorname{Orb}(\gamma, f', s) = \Omega(\gamma, s)\int_{\frac{H'\times H'}{(H'\times H')_{\gamma}}} f'(h_1^{-1} \gamma h_2) |h_1h_2|^s \eta(h_2)\ dh_1 dh_2. \end{align} $$

Here, the so-called transfer factor $\Omega (\gamma , s) \in \pm q^{\mathbb {Z} s}$ is defined in a way that ensures that $\operatorname {Orb}(\gamma , f', s)$ only depends on the double coset $H'\gamma H'$ ; detailed definitions will be given in §3. Our interest lies in the central value and the first derivative which we denote by

$$ \begin{align*} \operatorname{Orb}(\gamma, f') := \operatorname{Orb}(\gamma, f', 0),\qquad \operatorname{\partial Orb} (\gamma, f') = \left.\frac{d}{ds}\right\vert_{s = 0} \operatorname{Orb}(\gamma, f', s). \end{align*} $$

Let $D/F$ be a CSA of degree $2n$ and let $E\subset D$ be a fixed embedding. The centralizer $C = \mathrm {Cent}_D(E)$ is again a CSA but over E. The Hasse invariants of D and C are related by $\mathrm {inv}_E(C) = 2\,\mathrm {inv}_F(D)$ (combine [Reference Draxl9, Corollary 9.1] with [Reference Serre39, Proposition XIII.7]). Define $(G, H) = (D^{\mathrm {op},\times }, C^{\mathrm {op},\times })$ . The reason for passing to the opposite algebra here is that $D^{\mathrm {op}}$ is isomorphic to the ring of left endomorphisms of D as left D-module. Given a test function $f\in C^{\infty }_c(G)$ and an element $g\in G$ that is regular semi-simple with respect to the $(H\times H)$ -action, we consider the orbital integral

(1.4) $$ \begin{align} \operatorname{Orb}(g, f) := \int_{\frac{H\times H}{(H\times H)_g}} f(h_1^{-1} g h_2) dh_1 dh_2. \end{align} $$

It evidently only depends on the double coset $HgH$ . Let $O_D\subseteq D$ be a maximal order such that $O_C := C\cap O_D$ is a maximal order of C and put $f_D = 1_{O_D^{\times }}$ . One can show that all such orders $O_D$ form a single H-conjugation orbit (Lemma 3.6), so the orbital integrals $\operatorname {Orb}(g, f_D)$ do not depend on the choice of $O_D$ . In dependence on the order $\ell $ of D in the Brauer group of F, we will define a specific parahoric subgroup $K'\subseteq GL_{2n}(O_F)$ . It corresponds to a parabolic of $(2n/\ell \times 2n/\ell )$ -block upper triangular matrices and is chosen compatibly with the subgroup $H'\subseteq G'$ . Let $f_D^{\prime \circ } = \mathrm {vol}(K'\cap H')^{-2} \cdot 1_{K'}$ and define $f^{\prime }_D$ as a certain $H'$ -translate of $f_D^{\prime \circ }$ ; see Definition 3.9 for details. The orbital integrals $O(\gamma , f^{\prime }_D, s)$ have a simple functional equation with respect to $s \longleftrightarrow -s$ (Proposition 3.19).

Two regular semi-simple elements $\gamma \in G'$ and $g\in G$ are said to match if the invariants (in the sense of geometric invariant theory) of the orbits $H'\gamma H'$ and $HgH$ agree. This notion defines an injection $[G_{\mathrm {rs}}]\hookrightarrow [G^{\prime }_{\mathrm {rs}}]$ of the regular semi-simple orbits of G into those of $G'$ , which allows to compare orbital integrals on the two groups. Assuming all Haar measures are chosen compatibly, we conjecture that $f^{\prime }_D$ is a smooth transfer of $f_D$ in the following sense:

Conjecture 1.1 (Fundamental Lemma, Conjecture 3.10).

Let $\gamma \in G'$ be an element that is regular semi-simple with respect to the $(H'\times H')$ -action. Then

(1.5) $$ \begin{align} \operatorname{Orb}(\gamma, f^{\prime}_D) = \begin{cases} \operatorname{Orb}(g, f_D) & \text{if there exists an element } g\in G \text{ that matches } \gamma\\ 0 & \text{if there is no such } g.\end{cases} \end{align} $$

The theorem about existence of smooth transfer of C. Zhang and H. Xue [Reference Zhang44, Reference Xue41] already states that there exists some test function $f'\in C^{\infty }_c(G')$ such that $\operatorname {Orb}(\gamma , f')$ is given by the right-hand side of (1.5) for all $\gamma $ . The new aspect of Conjecture 1.1 is that it provides $f^{\prime }_D$ as a specific candidate. The choice of $f^{\prime }_D$ is motivated by Z. Zhang’s transfer result [Reference Zhang49]. We explain this in more detail in Remark 3.14.

Conjecture 1.1 specializes to the Guo–Jacquet fundamental lemma (FL) from [Reference Guo13] if $D = M_{2n}(F)$ . By work of N. Hultberg and the second author [Reference Hultberg and Mihatsch20, Theorem A], Conjecture 1.1 is also known if the Hasse invariant of D is $1/2$ : The proof in this case is by reduction to the base change FL and to the Guo–Jacquet FL. Our main result on Conjecture 1.1 in the present paper is as follows.

Note that if D is a division algebra of degree $2n$ , then $\operatorname {Orb}(g, f_D)$ is either $0$ or an integer divisor of n (Proposition 3.13). In the case of Theorem 1.2, for example, $\operatorname {Orb}(\gamma , f^{\prime }_D)$ is either $0$ or $1$ (Theorem 8.2).

1.4 The arithmetic transfer conjecture

Denote by $\breve F$ the completion of a maximal unramified extension of F and fix an embedding $E\subset \breve F$ . Choose an isomorphism $\breve F\otimes _F C^{\mathrm {op}} \cong M_n(\breve F \times \breve F)$ that restricts to $(\mathrm {id}, \tau ) \otimes 1_n$ on the center E of C, where $\tau :E\to E$ is the Galois conjugation. Consider the conjugacy class of the ‘Drinfeld type’ minuscule cocharacter

$$ \begin{align*}\mu _H:\mathbb{G}_m \longrightarrow H_{\breve F} \cong (GL_n\times GL_n)_{\breve F},\quad t \longmapsto ((t, \ldots, t, 1),\, (t, \ldots, t)).\end{align*} $$

Let $\mu _G: \mathbb {G}_m \to G_{\breve F}$ denote its composition with $H \to G$ . Then every element $b\in B(H, \mu _H)$ defines a morphism

(1.6) $$ \begin{align} (H,\, b,\, \mu _H) \longrightarrow (G,\, b,\, \mu _G) \end{align} $$

of local Shimura data in the sense of [Reference Rapoport and Viehmann37]. We denote by $H_b \to G_b$ the automorphism groups of the H-isocrystal (resp. G-isocrystal) defined by b. In analogy with our definition of $f_D$ as the characteristic function of $O_D^{\times }$ , we consider integral models $\mathcal {M}_C$ and $\mathcal {M}_D$ for the local Shimura varieties for (1.6) at levels $O_C^{\times }$ and $O_D^{\times }$ . Their definition is as follows.

The elements of $B(H, \mu _H)$ are in bijection with isogeny classes of pairs $(\mathbb {Y}, \iota )$ , where $\mathbb {Y}$ is a strict $O_F$ -module of height $2n^2$ and dimension n over the residue field $\mathbb {F}$ of $\breve F$ , and where $\iota :O_C\to \operatorname {End}(\mathbb {Y})$ is an $O_C$ -action. The set $B(G, \mu _G)$ is similarly in bijection with pairs $(\mathbb {X}, \kappa )$ , where $\mathbb {X}/\mathbb {F}$ is a strict $O_F$ -module of height $4n^2$ and dimension $2n$ , and where $\kappa :O_D\to \operatorname {End}(\mathbb {X})$ is an $O_D$ -action. Under these bijections, the map $B(H, \mu _H)\to B(G, \mu _G)$ corresponds to the Serre tensor construction

(1.7) $$ \begin{align} (\mathbb{Y}, \iota) \longmapsto (O_D\otimes_{O_C} \mathbb{Y},\,\, \kappa(x) := x\otimes \mathrm{id}_{\mathbb{Y}}). \end{align} $$

Let $\mathcal {M}_C$ and $\mathcal {M}_D$ be the RZ spaces for $(\mathbb {Y}, \iota )$ and $(\mathbb {X}, \kappa ) = O_D\otimes _{O_C} (\mathbb {Y}, \iota )$ . These are certain EL type moduli spaces of p-divisible groups with $O_C$ -action resp. $O_D$ -action. They are formal schemes over $\operatorname {Spf} O_{\breve F}$ that are regular with semi-stable reduction and such that $\dim \mathcal {M}_D = 2\dim \mathcal {M}_C = 2n$ . Furthermore, the groups $H_b$ and $G_b$ act from the right on $\mathcal {M}_C$ resp. $\mathcal {M}_D$ and there is a closed immersion

that is equivariant with respect to $H_b\to G_b$ . This closed immersion can be defined by a Serre tensor construction as in (1.7).

Definition 1.3. Let $g\in G_b$ be a regular semi-simple element and let $\Gamma \subseteq H_b \cap g^{-1}H_b g$ be a free, discrete subgroup of covolume $1$ . Define the intersection number

$$ \begin{align*}\operatorname{Int}(g) = \langle \Gamma\backslash \mathcal{M}_C,\,\, \Gamma\backslash g\cdot \mathcal{M}_C\rangle_{\Gamma\backslash \mathcal{M}_D} \in \mathbb{Z}.\end{align*} $$

Taking the quotient by $\Gamma $ in this definition is the natural analog of taking the quotient by the stabilizer in the orbital integrals (1.3) and (1.4). The restriction to regular semi-simple g ensures that $\Gamma \backslash (\mathcal {M}_C\cap g\cdot \mathcal {M}_C)$ is a proper scheme over $\operatorname {Spec} O_{\breve F}$ with empty generic fiber. The definition is moreover independent of $\Gamma $ .

The quantity $\operatorname {Int}(g)$ only depends on the $(H_b\times H_b)$ -orbit of g, so we can use orbit matching to view it as a function on a subset of the $(H'\times H')$ -orbits in $G'$ . In this context, there is the following uniqueness and vanishing result.

The second statement already hints that $\operatorname {Int}(g)$ might be related to the first derivative $\operatorname {\partial Orb} (\gamma , f^{\prime }_D)$ for matching g and $\gamma $ . This is made precise by the AT Conjecture in its explicit form:

Conjecture 1.5 (ATC – explicit form).

There exists a correction function $f^{\prime }_{\mathrm {corr}}\in C^{\infty }_c(G')$ such that for every regular semi-simple $\gamma \in G^{\prime }_{\mathrm {rs}}$ ,

(1.8)

Conjecture 1.5 builds on our FL Conjecture. There is also a weaker form that postulates the existence of a test function $f'\in C^{\infty }_c(G')$ whose orbital integral derivatives $\operatorname {\partial Orb} (\gamma , f')$ agree with the intersection numbers $\operatorname {Int}(g)$ on the nose. We formulate and compare such variants in §4.5.

The AT Conjecture is currently known in some low-dimensional cases: Consider first the case of $D = M_{2n}(F)$ . Then $f^{\prime }_D = 1_{GL_{2n}(O_F)}$ , and it is conjectured that one may take $f^{\prime }_{\mathrm {corr}} = 0$ . This is precisely the linear AFL conjecture from [Reference Li and Mihatsch30]. By the main results of [Reference Li28] and [Reference Li and Mihatsch30], it is known to hold whenever the connected part of $\mathbb {Y}$ has height $\leq 4$ .

Assume next that the Hasse invariant of D is $1/2$ . Then it is again expected that one may take $f^{\prime }_{\mathrm {corr}} = 0$ . The main result of [Reference Hultberg and Mihatsch20] states that Conjecture 1.5 for such D follows from the linear AFL.

The main result of the present paper, to be formulated in the next section, states that Conjecture 1.5 holds for division algebras of degree $4$ . Here, $f^{\prime }_{\mathrm {corr}}$ can be chosen from the Iwahori Hecke algebra. We speculate that this is a general phenomenon, meaning that $f^{\prime }_{\mathrm {corr}}$ can always be chosen as a linear combination of indicator functions of standard parahoric subgroups. In this context, we mention related work of He–Shi–Yang [Reference He, Shi and Yang16] and He–Li–Shi–Yang [Reference He, Li, Shi and Yang17]: There, the authors prove intersection number identities for Kudla–Rapoport divisors in the presence of bad reduction. Their result involves a unique characterization of certain occurring correction terms. It would be interesting to know if similar ideas apply in the context of Conjecture 1.5.

1.5 Invariant $1/4$ and $3/4$

Assume from now on that $n = 2$ and that D denotes a central division algebra (CDA) of Hasse invariant $\lambda \in \{1/4,\, 3/4\}$ . Then C is a quaternion division algebra over E. The test function $f_D^{\prime }\in C^{\infty }_c(G')$ is an $H'$ -translate of a scalar multiple $f^{\prime }_{\mathrm {Iw}}$ of the characteristic function of an Iwahori in $G' = GL_4(F)$ . We will also consider a test function $f^{\prime }_{\mathrm {Par}}$ that is the characteristic function of a $(2\times 2)$ -block parahoric in $GL_4(F)$ .

The set $B(H, \mu _H)$ is a singleton in this situation, and the moduli space $\mathcal {M}_C$ is Drinfeld’s half plane [Reference Drinfeld10]. If $\lambda = 1/4$ , then $\mathcal {M}_D$ is the four-dimensional Drinfeld half space. If $\lambda = 3/4$ , however, then no explicit description of $\mathcal {M}_D$ is known. The two groups $H_b$ and $G_b$ are given by

$$ \begin{align*}H_b \cong GL_2(E),\qquad G_b \cong \begin{cases} GL_4(F) & \text{if } \lambda = 1/4\\ GL_2(B) & \text{if } \lambda = 3/4. \end{cases}\end{align*} $$

Here, $B/F$ denotes a quaternion division algebra.

Theorem 1.6 (Theorem 9.1).

The AT conjecture holds for D. More precisely, let $f^{\prime }_{\mathrm {corr}}$ be given by

(1.9) $$ \begin{align} f^{\prime}_{\mathrm{corr}} = \begin{cases} -4q\log(q)\cdot f^{\prime}_{\mathrm{Par}} & \text{if } \lambda = 1/4\\ 0 & \text{if } \lambda = 3/4.\end{cases} \end{align} $$

Then, for every regular semi-simple $\gamma \in G^{\prime }_{\mathrm {rs}}$ ,

(1.10) $$ \begin{align} \operatorname{\partial Orb}(\gamma, f^{\prime}_D) + \operatorname{Orb}(\gamma, f^{\prime}_{\mathrm{corr}}) = \begin{cases} 2\,\operatorname{Int}(g)\log(q) & \text{if there exists a } g\in G_b \text{ that matches } \gamma\\ 0 & \text{otherwise.} \end{cases} \end{align} $$

Why and how it is that the two cases differ by a parahoric type orbital integral is a mystery to us. The difference only emerges during the proof of Theorem 1.6 and is encoded in the $0$ -dimensional embedded components of the intersection locus.

1.6 Key aspects

Our proof of Theorem 1.6 is by determining explicitly and comparing both sides in (1.10). Key aspects are as follows:

(1) Concerning the orbital integral side, we combine three techniques to determine all occurring $\operatorname {Orb}(\gamma , f^{\prime }_{\mathrm {Par}})$ , $\operatorname {Orb}(\gamma , f^{\prime }_{\mathrm {Iw}})$ and $\operatorname {\partial Orb}(\gamma , f^{\prime }_{\mathrm {Iw}})$ . First, we work out these orbital integrals for all hyperbolic orbits. In this situation, the computation can be reduced to a Levi subgroup and hence to $GL_2$ . Second, we establish a germ expansion principle (Proposition 7.4) that allows to write each orbital integral as a linear combination of a principal germ and a unipotent germ. The principal germ can be described explicitly in all situations. Third and finally, we use the results for hyperbolic orbits and the linear relations among the various germs to also determine the remaining orbital integrals. A summary of the final results can be found in §5. In particular, Proposition 5.4 gives a formula for the derivatives $\operatorname {\partial Orb}(\gamma , f^{\prime }_D)$ when D is a division algebra of degree $4$ . The basis for all mentioned results is a combinatorial expression for $\operatorname {Orb}(\gamma , f^{\prime }_D)$ in terms of lattices; see (3.25).

(2) Concerning the intersection-theoretic side, we first prove a general formula for intersection numbers of surfaces in a $4$ -dimensional space:

Proposition 1.7 (see Corollary 10.3).

Let X be a regular $4$ -dimensional formal scheme that is locally formally of finite type over $\operatorname {Spf} O_{\breve F}$ . Let $Y_1,\,Y_2\subseteq X$ be two regular $2$ -dimensional closed formal subschemes. Assume that $Z = Y_1\cap Y_2$ is a proper scheme over $\operatorname {Spec} O_{\breve F}$ with empty generic fiber and of dimension $\leq 1$ . Let $Z^{\mathrm {pure}},\, Z^{\mathrm {art}}\subseteq Z$ be its purely $1$ -dimensional locus and the artinian subscheme of $0$ -dimensional embedded components. Then

(1.11) $$ \begin{align} \langle Y_1, Y_2\rangle_X = \mathrm{len} (\mathcal{O}_{Z^{\mathrm{art}}}) - \deg (\det \mathcal{C}_1 \vert_{Z^{\mathrm{pure}}}) - \langle Z^{\mathrm{pure}}, Z^{\mathrm{pure}}\rangle_{Y_2}. \end{align} $$

Here, $\mathcal {C}_1$ is the conormal bundle of $Y_1\subseteq X$ , and the intersection number on the very right is that of divisors on $Y_2$ .

(3) In order to compute the occurring intersection numbers $\operatorname {Int}(g)$ , we determine the three quantities on the right-hand side of (1.11). The most important input here is Drinfeld’s theorem [Reference Drinfeld10], which provides an explicit description for $\mathcal {M}_C$ and, if $\lambda = 1/4$ , for $\mathcal {M}_D$ . This in particular enables us to compute the degree of the conormal bundle:

Proposition 1.8 (see Propositions 12.6 and 13.1).

Let $P\subseteq \mathbb {F}\otimes _{O_{\breve F}} \mathcal {M}_C$ be an irreducible component of the special fiber of $\mathcal {M}_C$ . Let $\mathcal {C}$ be the conormal bundle of $\mathcal {M}_C\subseteq \mathcal {M}_D$ . Then, for both Hasse invariants $\lambda \in \{1/4,\, 3/4\}$ ,

$$ \begin{align*}\deg(\mathcal{C}\vert_P) = q^2 - 1.\end{align*} $$

Set $\mathcal {I}(g) = \mathcal {M}_C \cap g \cdot \mathcal {M}_C$ . It is left to describe $\mathcal {I}(g)^{\mathrm {pure}}$ and $\mathcal {I}(g)^{\mathrm {art}}$ , where the notation is meant in the sense of Proposition 1.7.

(4) The description of $\mathcal {I}(g)^{\mathrm {pure}}$ is in terms of the Bruhat–Tits stratification of the special fiber of $\mathcal {M}_C$ : Each of its irreducible components is isomorphic to $\mathbb {P}^1$ . Restricting to a fixed connected component $\mathcal {M}_C^0$ of $\mathcal {M}_C$ , the dual graph of its special fiber is the Bruhat–Tits tree $\mathcal {B}$ for $PGL_{2,E}$ . This is a $(q^2+1)$ -regular tree whose vertices are in bijection with homothety classes of $O_E$ -lattices $\Lambda \subseteq E^2$ . Thus, we may write

(1.12) $$ \begin{align} \mathcal{M}_C^0\cap \mathcal{I}(g)^{\mathrm{pure}} = \sum_{\Lambda\in \mathrm{Vert}(\mathcal{B})} m(g, \Lambda) [P_{\Lambda}]. \end{align} $$

One of our main auxiliary results (Theorem 11.10) describes the coefficient function $\Lambda \mapsto m(g, \Lambda )$ in terms of g.

(5) The artinian locus $\mathcal {I}(g)^{\mathrm {art}}$ is where the two cases $\lambda = 1/4$ and $\lambda = 3/4$ are substantially different. If $\lambda = 1/4$ , then $\mathcal {M}^0_C\cap \mathcal {I}(g)^{\mathrm {art}}$ consists of at most a single point of length $1$ . If $\lambda = 3/4$ , however, then every stratum $P_{\Lambda }$ that occurs in (1.12) contributes artinian embedded components of total length q. The individual lengths and positions of these components depend on g and $\Lambda $ ; see Table 2 for all possibilities. The total length of $\Gamma \backslash \mathcal {I}(g)^{\mathrm {art}}$ will, however, always match the orbital integral $4q\operatorname {Orb}(\gamma , f^{\prime }_{\mathrm {Par}})$ that makes up the difference of the two analytic sides for $\lambda = 1/4$ and $\lambda = 3/4$ in (1.10).

(6) As mentioned before, there is no explicit description of $\mathcal {M}_D$ when $\lambda = 3/4$ which makes the computation of $\mathcal {I}(g)$ trickier than in the case $\lambda = 1/4$ . We use a mix of Dieudonné theory, Cartier theory and display theory, as well as the previous results for the case of invariant $1/4$ , to achieve a precise description; see the results in §13.3 and §13.5 as well as Proposition 13.22.

1.7 Open directions

There is, of course, the question of how to prove Conjectures 3.10 and 1.5 resp. the linear AFL [Reference Li and Mihatsch30] in general. We mention here three further problems of interest.

(1) The AT conjecture of the present article concerns inner forms of $GL_{2n}$ and their moduli spaces. An equally interesting question would be to study moduli spaces for $GL_{2n}$ (Lubin–Tate spaces), but with parahoric level structure. The global motivation from §1.1 applies verbatim to that situation.

(2) Another question would be for an extension of our results to the biquadratic situation in the following sense: Instead of a single quadratic extension $E/F$ , one considers two such extensions $E_1, E_2/F$ , fixes embeddings $E_1, E_2\to D$ , and defines intersection numbers from the corresponding cycles on $\mathcal {M}_D$ . For the linear AFL, such a biquadratic generalization has been formulated by B. Howard and the first author [Reference Howard and Li19].

(3) Finally, there is the problem of relating the local quantities of the present article with global intersection numbers and L-functions. We hope to return to this question in the future.

1.8 Layout of the paper

We now give an overview of the contents of this paper. The paper consists of three parts.

In Part 1, we first give the group-theoretic setup (invariant theory, matching). We next define the orbital integrals of interest and formulate our FL conjecture. Then we introduce the moduli spaces of strict $O_F$ -modules in question and the intersection numbers $\mathrm {Int}(g)$ . We state and compare three variants of the AT conjecture.

In Part 2, we consider the case $G' = GL_4(F)$ and determine the quantities $\operatorname {Orb}(\gamma , f^{\prime }_{\mathrm {Par}})$ , $\operatorname {Orb}(\gamma , f^{\prime }_{\mathrm {Iw}})$ and $\operatorname {\partial Orb} (\gamma , f^{\prime }_{\mathrm {Iw}})$ . A summary of our results can be found in §5, the main one being the proof of the FL conjecture for $f^{\prime }_{\mathrm {Iw}}$ . The further contents of Part 2 have been described at the beginning of §1.6.

In Part 3, we consider the case of a CDA $D/F$ of degree $4$ and compute the intersection numbers $\operatorname {Int}(g)$ . Our main result is the proof of the AT conjecture in this situation. There are four main sections: In §10, we prove the intersection number formula in Proposition 1.7. In §11, we study the functions $m(g,\Lambda )$ on $\mathcal {B}$ that will later describe the multiplicities of curves in the intersection locus; see (1.12). Then, in §12, we prove the AT conjecture for $\lambda = 1/4$ . Because of Drinfeld’s description of both $\mathcal {M}_C$ and $\mathcal {M}_D$ in this situation, this does not involve any $\pi $ -divisible groups at all. Finally, in §13, we extend these results to $\lambda = 3/4$ .

Part I

The arithmetic transfer conjecture