where $\varphi$ is in a certain set of parameters, $I_{\varphi}$ is an algebraic group over $\mathbb{Q}$ (the centralizer of $\varphi$), $X^p(\varphi)$ involves the finite adeles outside of $p$ and $X_p(\varphi)$ is a purely $p$-adic object. The conjecture also includes a description of the actions of the Frobenius morphism and of the Hecke operators.

It would take too long to explain what the parameters $\varphi$ are, but we can say that they give rise to triples $(\gamma_0, \gamma, \delta)$, where:

* $\gamma_0$ is a semisimple element of $G(\mathbb{Q})$, given up to $G(\overline{\mathbb{Q}})$-conjugacy;

* $\gamma = (\gamma_\ell)_{\ell \neq p}$ is an element of $G(\mathbb{A}_f^p)$, given up to $G(\mathbb{A}_f^p)$-conjugacy, and such that $\gamma_\ell$ and $\gamma_0$ are conjugated under $G(\overline{\mathbb{Q}}_\ell)$ for every $\ell \neq p$;

* $\delta$ is an element of $G(L)$, where $L$ is an unramified extension of degree $r = [\mathbb{F}_q : \mathcal{O}/\wp]$ of $F_\wp$, such that, if we denote by $\sigma \in \text{Gal}(L/\mathbb{Q}_p)$ the lift of the (arithmetic) Frobenius, then $N(\delta) = \delta\sigma(\delta) \dots \sigma^{r-1}(\delta)$ is $G(\overline{\mathbb{Q}}_p)$-conjugate to $\gamma_0$.

There are some more conditions on the triple $(\gamma_0, \gamma, \delta)$, see for example Section 2 of [62]. One of these conditions is that a certain invariant $\alpha(\gamma_0; \gamma, \delta)$ defined by Kottwitz should vanish, and it implies that there exists an inner form $I$ of $G_{\gamma_0}$, the centralizer of $\gamma_0$ in $G$, such that $I_{\mathbb{Q}_v} \simeq G_{\mathbb{Q}_v, \gamma_v}$ for $v \neq p, \infty$, $I_{\mathbb{Q}_p}$ is isomorphic to the twisted centralizer of $\delta$ (see page 169 of [62]) and $I_{\mathbb{R}}$ is anisotropic modulo the center of $G$. We then take $I_\varphi = I$,

$$X^p(\varphi) = \{g \in G(\mathbb{A}_f^p)/K^p \mid g^{-1}\gamma g \in K^p\}$$

and

$$X_p(\varphi) = \{g \in G(L)/G(\mathcal{O}_L) \mid g^{-1}\delta\sigma(g) \in G(\mathcal{O}_L)\mu_h(\varpi_L)G(\mathcal{O}_L)\},$$

where we extended $G$ to a reductive group scheme over $\mathbb{Z}_p$, $\varpi_L$ is a uniformizer of $L$, and $\mu_h$ is the morphism $h_{\mathbb{C}} \circ r : \text{GL}_{1,\mathbb{C}} \to \text{G}_{\mathbb{C}}$, seen as a conjugacy class of morphisms $\text{GL}_{1,\overline{\mathbb{Q}}_p} \to \text{G}_{\overline{\mathbb{Q}}_p}$ that is stable by $\text{Gal}(\overline{\mathbb{Q}}_p/L)$; as $G$ is quasi-split over $L$ for $r$ big enough, we may assume up to taking $r$ big enough that $\mu_h$ is defined over $L$ (see Lemma 1.1.3 of [59]).

If the Shimura datum $(G, h)$ is PEL of type $A$ or $C$, so that $M_K$ has a modular description as in Definition 2.10, then the triples $(\gamma_0, \gamma, \delta)$ should parametrize the $\mathbb{Q}$-isogeny classes of triples $(A, \lambda, \iota)$ as in the moduli problem. This parametrization rests on Honda-Tate theory, which classifies abelian varieties $A$ over finite fields using their Frobenius, seen as a central element of $\text{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. See Part III of [62] for the case of Siegel modular varieties, and [64] for the PEL cases of type $A$ and $C$.

The Langlands-Rapoport conjecture is not known for general Shimura varieties, because we do not even have integral models for general Shimura varieties. It is also not known for Shimura varieties of abelian type, except in some particular cases such as the case of quaternionic Shimura varieties (see the paper [102] of Reimann), but there are some weaker versions that are proved and suffice for the application to the Kottwitz conjecture. Here is my understanding of the current situation:

* If $(G, h)$ is of PEL type and $G_{\text{der}}$ is simply connected (i.e. $G$ is of type $A$ or $C$), then Kottwitz formulated a (strictly) weaker form of the Langlands-Rapoport conjecture in [62] and proved this reformulation in [64]. This version is enough to provide a counting-point formula that can be compared to the Arthur-Selberg trace formula (see below).
* Dong Uk Lee (see [77]) generalized Kottwitz's results for Shimura varieties of Hodge type. The methods, and even the formulation itself, do not seem to be generalizable to Shimura varieties of abelian type.

* Before that, Kisin has proved his own weaker version of the Langlands-Rapoport conjecture in [55] for Shimura varieties of abelian type. This version of the conjecture is not obviously weaker or stronger than Kottwitz's version. One of its advantages is that it makes it easier to go from Shimura varieties of Hodge type to Shimura varieties of abelian type, and one drawback is that it does not imply a counting-point formula that we can use with the Arthur-Selberg trace formula.

* In [56], Kisin, Shin and Zhu, working with Shimura varieties of abelian type, formulated and proved a version of the Langlands-Rapoport conjecture that is weaker than the original conjecture but stronger than Kottwitz's and Kisin's previous versions, and that is enough for the application to the Kottwitz conjecture. (In fact they did more than that, as they also "stabilized" the expression that they got; see Remark 3.17(1).)

The upshot is that, if $g \in G(\mathbb{A}_f)$ has a trivial component at the prime number $p$ under $\wp$ and $f^\infty = \mathbb{1}_{KgK} \in \mathcal{H}_K$, we get a formula for the trace of $\text{Frob}_{\wp}^r \cdot f^\infty$ on the $\ell$-adic cohomology of $M_K(G, h)$ involving terms such as orbital integrals for $f^\infty$ (i.e. integrals of $f$ over $G(\mathbb{A}_f)$-conjugacy orbits of elements of $G(\mathbb{Q})$) and twisted orbital integrals of a function (depending on $r$) at $p$; a priori we only get this for $r$ big enough, but then the identity can be extended to all non-negative $r$. <fnref n="35" /> This is the kind of input that we can plug into the geometric side of the Arthur-Selberg trace formula, see 3.4.4. The spectral side of the trace formula will then give us an expression that can be massaged into what we want, that is, in cases when the simplest form of the Kottwitz conjecture applies, the trace of $\text{Frob}_{\wp}^r \cdot f^\infty$ on the virtual representation

$$ \sum_{i \ge 0} (-1)^i \sum_{\pi \in \Pi_G} a(\pi_f) \pi_f^K \otimes (r_\mu \circ \sigma_\pi) $$
of $\mathcal{H}_K \times \text{Gal}(\bar{F}/F)$. So we win.

**REMARK 3.17.** (1) The sentence “This is the kind of input that we can plug into the geometric side of the Arthur-Selberg trace formula” in the previous paragraph is sweeping a lot of difficulties under the rug. If we are looking at cases where $G$ has no endoscopy (such as the simple Shimura varieties of Kottwitz), then the formula given by the Langlands-Rapoport conjecture is not too far from the geometric side of the trace formula (see Section 4 of [63]). In general, we must first perform a complicated process known as “stabilization”, which uses difficult results such as the fundamental lemma, cf. 3.3.2. See Section 4 of Kottwitz’s paper [62] for an explanation of stabilization in the simpler case when $G_{\text{der}}$ is simply connected, and the book [56] of Kisin-Shin-Zhu for the case of general Shimura data.

(2) Note that we never used Matsushima's formula in our outline of the proof of the Kottwitz conjecture. In fact, though it serves as a guide, Matsushima's formula is not logically necessary to the proof.

<footnote n="35">³⁵This does require some work. We need to see that both sides are finite sums of terms of the form $a \cdot b^r$ with $a, b \in \mathbb{C}$, which is not obvious for the twisted orbital integrals.</footnote>
3.4.4. The Arthur-Selberg trace formula for a compact quotient. We consider the situation of 3.1.1, so G is a connected reductive group over $\mathbb{Q}$, $S_G$ is the maximal $\mathbb{Q}$-split torus in the center of G and $A_G = S_G(\mathbb{R})^0$. We write $L_G^2$ for $L^2(G(\mathbb{Q})\backslash G(\mathbb{A})/A_G, \mathbb{C})$, with the action of $G(\mathbb{A})$ given by right translations.

We also assume that $G_{\text{der}}$ is of $\mathbb{Q}$-rank 0, so that the quotient $G(\mathbb{Q})\backslash G(\mathbb{A})/A_G$ is compact. Then the representation $L_G^2$ is semi-simple, and we have

$$ (3.3) \qquad L_G^2 \simeq \bigoplus_{\pi \in \Pi_G} \pi^{m(\pi)}. $$

Fix a Haar measure on $G(\mathbb{A})$. If $f$ is a smooth function with compact support on $G(\mathbb{A})$, then we write $R(f)$ for the action of $f$ on $L_G^2$ by right convolution. Thanks to our hypothesis on $G$, the operator $R(f)$ is of trace class, and the goal of the Arthur-Selberg trace formula is to give two expressions of its trace.

**THEOREM 3.18** (See Section 1 of [7] or [15]). We have

$$ \begin{aligned} \mathrm{Tr}(R(f)) &= \sum_{\pi \in \Pi_G} m(\pi) \mathrm{Tr}(\pi(f)) \\ &= \sum_{\gamma \in G(\mathbb{Q})/\sim} \mathrm{vol}(G_{\gamma}(\mathbb{Q})\backslash G_{\gamma}(\mathbb{A})/A_G) \int_{G_{\gamma}(\mathbb{A})\backslash G(\mathbb{A})} f(x^{-1}\gamma x)dx, \end{aligned} $$

where, in the first formula, $\pi(f)$ is the operator $\int_G f(x)\pi(x)dx$ acting on the space of $\pi$ and, in the second formula, the sum is over all elements of $G(\mathbb{Q})$ modulo conjugation and, if $\gamma \in G(\mathbb{Q})$, we denote by $G_{\gamma}$ the centralizer of $\gamma$ in $G$.

Note that we need to choose Haar measures on the groups $G_{\gamma}(\mathbb{A})$ to make sense of the second formula for $R(f)$ (we also use the counting measure on $G_{\gamma}(\mathbb{Q})$), but that the result does not depend on that choice.

The first formula for $R(f)$ is called the *spectral side*. It follows from the isomorphism of (3.3).

The second formula for $R(f)$ is called the *geometric side*. We can deduce it by noting that $R(f)$ is an integral operator with kernel

$$ K(x, y) = \sum_{\gamma \in G(\mathbb{Q})} f(x^{-1}\gamma y), $$

so the trace of $R(f)$ must be equal to $\int_G K(x, x)dx$. See Section 1 of [7] for more details.

**3.5. The general Kottwitz conjecture.** Let us finally explain briefly how the Kottwitz conjecture generalizes to arbitrary Shimura varieties. There are two sources of difficulty:

* The fact that general Shimura varieties are not compact, which introduces complications in the geometry: we will have to deal with compactifications and choose which cohomology theory to use.

* Endoscopy, which will make the Arthur-Selberg trace formula harder to analyze.

**REMARK 3.19.** Although we are concentrating on the case of constant coefficients for simplicity, we could also consider cohomology with coefficients in a local system defined by a representation of the group G, as in Section 5 of Pink's paper [97]. In fact, it is necessary to work in this more general situation to make some
inductive arguments work, but it does not complicate the calculation of cohomology
in an essential way.

3.5.1. *Compactifications.* There are many ways to compactify locally symmet-
ric spaces, see for example the book [20] of Borel and Ji for a detailed presentation.
However, in these notes we are interested in compactifications of Shimura varieties
that have the structure of an algebraic variety, and then there are two choices:

(1) The Baily-Borel compactification, also called minimal Satake compactification, Satake-Baily-Borel compactification or just minimal compactification. Its main advantage is that it is canonical, so for example Hecke correspondences will extend to the Baily-Borel compactifications of Shimura varieties, and that its boundary is stratified by lower-dimensional Shimura varieties.<fnref n="36" /> Its main drawback is that it is very singular in general; for example, the boundary of the Baily-Borel compactification of the Siegel modular variety $M_{d,n}$ has codimension $d$.

(2) The toroidal compactifications, which map to the Baily-Borel compactification. As the plural indicates, they are not canonical and depend on extra data, so we cannot extend the action of the Hecke algebra to the co-homology of any fixed toroidal compactification. On the other hand, we can always find toroidal compactifications which are smooth projective and whose boundary is a divisor with normal crossings, and the boundary of a toroidal compactification is stratified by so-called *mixed Shimura varieties*, which are a generalization of Shimura varieties to certain non-reductive groups. Even if we were only interested in the Baily-Borel compactification, the toroidal compactifications would be a very useful tool to study it; they are also interesting in their own right.

Sections I.4, I.5 and III.4 of the book [20] contain a review of Satake and Baily-Borel compactifications of locally symmetric spaces as topological spaces and complex analytic spaces. The fact that the Baily-Borel compactification is a projective algebraic variety over $\mathbb{C}$ was proved by Satake for Siegel modular varieties, and by Baily and Borel in general, see the paper [12]. As for toroidal compactifications of complex Shimura varieties, they are studied in the book [11] of Ash, Mumford, Rapoport and Tai. In his thesis [96], Pink constructed canonical models of the toroidal compactifications over the reflex field, and deduced that the Baily-Borel compactification also descends to a projective algebraic variety over that field. As in the case of Shimura varieties, the story of integral models is not yet complete,<fnref n="37" /> and as for canonical models, the crucial part is to construct models of the toroidal compactifications (we then get a model of the Baily-Borel compactification as a byproduct). The case of modular curves is treated in the paper [33] of Deligne-Rapoport, and that of Siegel modular varieties in the book [41] of Chai and Faltings. The next case to be handled systematically was that of PEL type Shimura varieties, in Lan's thesis [67]. Then Madapusi Pera treated the case of Hodge type Shimura varieties in his paper [80]. The case of abelian type Shimura varieties seems to be within reach, but as far as I know it has not been written yet.

<footnote n="36">³⁶Almost: the boundary strata are actually quotients of Shimura varieties by finite groups.</footnote>

<footnote n="37">³⁷And as in the case of Shimura varieties, we only discuss places of good reduction here.</footnote>
3.5.2. Intersection cohomology and Zucker’s conjecture. It is perfectly possible to try calculating compact support cohomology of noncompact Shimura varieties with the methods outlined in 3.4, this is for example what Laumon does in [73], [75]. The issue is that the Arthur-Selberg trave formula will be harder to use, because there is no simple spectral description of the cohomology such as the one given by Matsushima’s formula in the compact case. Indeed, the generalization of Matsushima’s formula, which is a theorem of Borel and Casselman (see [19]) gives a spectral description of the $L^2$ cohomology of the Shimura variety, so it would be simpler to study that. The next problem is that $L^2$ cohomology has a very non-algebraic definition (it is the cohomology of the complex of $L^2$ differential forms, for a certain metric on the locally symmetric space); it has an action of the Hecke algebra, but it is not clear at all a priori how to make the absolute Galois group of the reflex field act on it.

Fortunately, this problem is solved by Zucker’s conjecture, which says that there should be a Hecke-equivariant isomorphism between $L^2$ cohomology of a Shimura variety and intersection cohomology of its Baily-Borel compactification. Intersection cohomology is a cohomology theory adapted to singular spaces. It was first introduced by Goresky and MacPherson (see [45]) as a modification of Betti cohomology for stratified spaces, and then further studied by them in [46] using a sheaf-theoretic interpretation due to Deligne. Then, in the book [14], Beilinson, Bernstein, Deligne and Gabber put intersection cohomology into the wider context of perverse sheaves: if $X$ is a possibly singular space, its intersection cohomology is the cohomology of a particular complex of sheaves on it, called the intersection complex, which is a simple object in the abelian category of perverse sheaves (for the middle perversity). We have been deliberately vague about the precise nature of $X$, as one of the advantages of the sheaf-theoretic point of view is that it can be adapted to many different situations; for example, $X$ could be an algebraic variety over a field and the intersection complex could be an étale complex with $Q_l$-coefficients; or $X$ could be an algebraic variety over $\mathbb{C}$ and the intersection complex could be a complex of $\mathbb{Q}$-vector spaces for the analytic topology, or a mixed Hodge module.<fnref n="38" /> In any case, the intersection cohomology of $X$ satisfies some of the properties that we would expect from the cohomology of a nonsingular variety:

* Poincaré duality if $X$ is proper;

* the hard Lefschetz theorem if $X$ is projective;

* purity if $X$ is proper (in the appropriate sense, e.g. in the sense of Deligne (see [36])) if $X$ is a variety over a finite field or as a Hodge structure if $X$ is a variety over $\mathbb{C}$ and we are using mixed Hodge modules).

It also satisfies comparison theorems between the different cohomology theories. (See for example Chapter 6 of [14] for the comparison between the étale and Betti theories.)

3.5.3. The Borel-Casselman theorem. We come back to Shimura varieties. Let $(G, h)$ be a Shimura datum, $K \subset G(\mathbb{A}_f)$ be a small enough level, $X$ be the Shimura variety $M_K(G, h)$, $X^*$ be the Baily-Borel compactification of $X$ and $IH^*(X^*)$ be its intersection cohomology. We have seen that Hecke correspondences extend to

<footnote n="38">³⁸We are avoiding the cases of non-algebraic $X$ or integral coefficients, as there is no unique “middle perversity” there. But the theory still makes sense, it is just slightly more complicated.</footnote>
Baily-Borel compactifications. Although intersection cohomology is not as func-
torial as usual cohomology, we can still make the Hecke algebra at level $K$ act
on $IH^*(X^*)$ (because Hecke operators are finite morphisms between Baily-Borel
compactifications and become finite étale when restricted to Shimura varieties).
Zucker’s conjecture (see [125, (6.20)]) says that, if we use the Betti version of
$IH^*(X^*)$ with complex coefficients, then there is a Hecke-equivariant isomorphism
between $IH^*(X^*)$ and $L^2$ cohomology of $X$. Also, Zucker’s conjecture is actu-
ally a theorem, as it has been proved (at least) four times: by Looijenga ([78]),
Looijenga-Rapoport ([79]), Saper-Stern ([104]) and Saper ([103]). If we combine
it with the generalization of Matsushima’s formula due to Borel and Casselman
([19]), we finally get the following theorem (note the similarity with Theorem 3.1):

THEOREM 3.20. Let $(G, h)$ be a Shimura datum. Then we have a $G(\mathbb{A}_f)$-equivariant isomorphism of graded $\mathbb{C}$-vector spaces

$$
\lim_{K} IH^*(M_K(G, h)^*(\mathbb{C})) \simeq \bigoplus_{\pi \in \Pi_G} \pi_f \otimes H^*(\mathfrak{g}, K_\infty; \pi_\infty)^{m(\pi)},
$$

where $G(\mathbb{A}_f)$ acts on the factors $\pi_f$ on the right hand side, $M_K(G,h)^*$ is the Baily-Borel compactification of $M_K(G,h)$ and we are using the Betti version of intersection cohomology (with constant coefficients for simplicity).

3.5.4. The Kottwitz conjecture. Let $F$ be the reflex field of $(G, h)$. If we use the étale $\ell$-adic version of intersection cohomology, then we get cohomology groups $\lim_{\underset{K}{\rightarrow}} IH^*(M_K(G, h)^*_{\overline{F}}, \overline{Q}_\ell)$ equipped with commuting actions of $G(\mathbb{A}_f)$ and $\text{Gal}(\overline{F}/F)$, which become isomorphic as $G(\mathbb{A}_f)$-modules to the cohomology groups of Theorem 3.20 if we choose any isomorphism $\overline{Q}_\ell \simeq \mathbb{C}$. We introduce the following virtual representation of $G(\mathbb{A}_f) \times \text{Gal}(\overline{Q}_\ell/\overline{Q}_\ell)$:

$$
IH = \sum_{i \ge 0} (-1)^i [\lim_{K} IH^i(M_K(G, h)^*_{\overline{F}}, \overline{Q}_\ell)],
$$

where the brackets denote the class of a representation in the group of virtual
representations. The sum is actually finite, because the intersection cohomology
group vanishes for $i \ge 2 \dim(M^K(G, h))$, and this dimension is independent of the
level $K$. We have a decomposition

$$
IH = \sum_{\pi_f} [\pi_f] \otimes \sigma(\pi_f),
$$

where $\pi_f$ goes through the set of equivalence classes of irreducible representations
of $G(\mathbb{A}_f)$ and each $\sigma(\pi_f)$ is a virtual representation of $\text{Gal}(\overline{F}/F)$. By Theorem 3.20,
we know that the only $\pi_f$ that can occur are the finite parts of discrete automorphic
representations of $G$, and we have a formula for the dimension of $\sigma(\pi_f)$. The
Kottwitz conjecture, stated on page 201 of [62], gives a precise description of $\sigma(\pi_f)$.

We will not state a precise form of the conjecture, as it would require introduc-
ing too much notation, but we will try to give its flavor. The most naive guess is
that we should have something like

$$
\sigma(\pi_f) = \pm \sum_{\pi' \in \Pi_G, \pi'_f \simeq \pi_f} m(\pi')[(r_\mu \circ \sigma_{\pi'})],
$$

where $r_\mu$ and $\sigma_\pi$ are as in Conjecture 3.8, but this is not true because of en-
doscopy. What we actually expect is as follows. As mentioned in Remark 3.6,
discrete automorphic representations of $G$ should be classified by Arthur parameters $\psi: \mathcal{L}_Q \times SL_2(\mathbb{C}) \to \mathbb{L}^L G$. The parameters corresponding to the representations appearing in the cohomology of Shimura varieties are expected to factor through the motivic Galois group of $\mathbb{Q}$ and to be defined over a finite extension of $\mathbb{Q}$ in $\mathbb{C}$, and there is an explicit recipe giving $\sigma_\pi$ from the parameter corresponding to $\pi$; we denote by $V_\psi$ the space of the representation $r_\mu$ with the action of $\text{Gal}(\overline{F}/F)$ coming from $\sigma_\pi$. 39 Let $S_\psi$ be the centralizer of $\psi$ in $\hat{G}$; this group, or rather its quotient $S_\psi/S_\psi^0Z(\hat{G})^{\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$, should control the structure of the Arthur packet of representations associated to $\psi$ and the multiplicities of these representations. We can decompose $V_\psi$ as a direct sum

$$V_{\psi} = \bigoplus_{\nu} V_{\psi, \nu},$$

where $\nu$ runs over characters of $S_\psi$ and $V_{\psi, \nu}$ is the part where $S_\psi$ acts by $\nu^{-1}$. Then the Kottwitz conjecture says that

$$\sigma(\pi_f) = \sum_{\psi} \sum_{\nu} n(\pi_f, \nu)[V_{\psi, \nu}],$$

where $\psi$ runs over Arthur parameters whose packets contains a representation with finite part $\pi_f$, and $n(\pi_f, \nu)$ is the product of a sign and of a multiplicity (which is the multiplicity of $\nu$ in some finite-dimensional representation of $S_\psi$ depending on $\pi_f$).

3.5.5. *Proving the Kottwitz conjecture.* The general strategy of 3.4 still applies with some adaptations:

(1) The specialization theorem still holds for intersection cohomology, as long as we have toroidal compactifications that are proper and smooth over the base ring (which is the case for Hodge type Shimura varieties at places of good reduction). This follows from the specialization theorem for the cohomology of the toroidal compactifications and from the decomposition theorem (see Corollaires 5.4.2 and 5.4.6 of [14]), which says that intersection cohomology of the Baily-Borel compactification is a direct factor of this cohomology.

(2) Deligne's conjecture makes sense (and is known) for general complexes and cohomological correspondences, not just for constant coefficients. It reduces the calculation of the trace of a Hecke operator twisted by a power of the Frobenius on intersection cohomology to counting fixed points (in the Baily-Borel compactification) and calculating the stalks of the intersection complex at these fixed points.

(3) The Langlands-Rapoport conjecture makes no distinction between compact and noncompact Shimura varieties. It also theoretically gives a description of points of the Baily-Borel compactification over a finite field, as this compactification is stratified by Shimura varieties (or explicit finite quotients thereof).

<footnote>39We are cheating here: if we want an action of $\text{Gal}(\overline{F}/F)$ and not of the motivic Galois group of $F$, we have to take as coefficients an $\ell$-adic completion of the field of definition of $\psi$.</footnote>
(3') The intersection complex can be quite difficult to calculate, but in the case of Shimura varieties there is an explicit description given by the theory of weighted cohomology of Goresky-Harder-MacPherson (see [44]); in particular, it is a complex of local systems on each stratum of the Baily-Borel compactification. Once the Langlands-Rapoport conjecture (or an appropriate weakening) is known, this reduces the calculation of the cohomology to a complicated combinatorial problem.

(4) As for the Arthur-Selberg trace formula, we now need to use the stable trace formula for a noncompact quotient, fortunately in a somewhat simple form (see Section 29 of Arthur's overview [7] for an introduction to the stable trace formula, and Arthur's paper [6] for the simplifications that occur when calculating the cohomology of Shimura varieties). The stabilization process that we must apply to the point-counting formula coming from (3) and (3') is also more complicated, because we need to stabilize the terms coming from the boundary.<fnref n="40" /> For an example of this process, see Yihang Zhu's thesis [124].

In the end, we expect to get the following formula for $h \in \mathcal{H}_K$ a Hecke operator trivial at $p$, Frob$_\phi$, the geometric Frobenius at a place $\phi$ of $F$ above $p$ and $j \in \mathbb{N}$:

$$ (3.4) \quad \mathrm{Tr}(h \times \mathrm{Frob}_{\phi}, IH^*(M_K(G, h) \underline{\otimes}_{\mathbb{F}_p} \overline{\otimes}_{\mathcal{Q}_\ell})) = \sum_H \iota(G, H) \mathrm{ST}_H((h^p)^H \varphi_{p,j} \varphi_\infty), $$

where $H$ runs over elliptic endoscopic groups of $G$ (to be more precise, it should run over elliptic endoscopic triples), $\iota(G, H)$ is a positive rational number, $\mathrm{ST}_H$ is the stable trace formula for $H$, $(h^p)^H$ is the so-called endoscopic transfer to $H(\mathbb{A}_f^p)$ of the prime-to-$p$ part of $h$, $\varphi_{p,j}$ is a function in the spherical Hecke algebra of $G(\mathbb{Q}_p)$ that only depends on $j$ (not on $h$), and $\varphi_\infty$ is a function on $G(\mathbb{R})$ that depends only on the coefficients of the cohomology (here we are taking constant coefficients, but we could take coefficients in any algebraic representation of $G$).

(5) The last step is to transform the right-hand side of the previous formula into an expression of the form $\sum_{\pi_f} \mathrm{Tr}(h, \pi_f) \mathrm{Tr}(\mathrm{Frob}_{\phi}^j, \sigma(\pi_f))$. Kottwitz explains a way to do this in Section 10 of [62], but his method requires Arthur's conjectural classification of discrete automorphic representations, which is only available for classical and unitary groups (see [8], [89] and [52]) so far, while the groups of Shimura varieties of PEL type (i.e. the best understood Shimura varieties) tend to be general symplectic, orthogonal or unitary groups (i.e. GSp, GO and GU); there are some results for general symplectic groups, see Bin Xu's thesis [121] and his paper [122], though as far as I understand they are not sufficient to completely prove the Kottwitz conjecture. It also requires the agreement of the various constructions of the classification of representations of real groups, that was proved for classical and unitary groups in papers

<footnote n="40">⁴⁰Some of these complications already arise in the compact case with endoscopy. Indeed, even when the group $G$ has no nontrivial parabolic subgroups, its endoscopic groups almost certainly do because they are quasi-split, and this produces “boundary terms” in their stable trace formulas on the right-hand side of identity (3.4). These terms must cancel each other out, as there are no corresponding terms on the left-hand side of (3.4).</footnote>
of Arancibia-Moeglin-Renard ([5]), Arancibia ([3]), Arancibia-Mezo ([4]) and Adams-Arancibia-Mezo ([2]).

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