The trace formula is best understood as a machine that equates two ways of taking the trace of an operator on automorphic forms:

$$\boxed{\text{spectral data of automorphic representations} \;=\; \text{geometric data of conjugacy classes}.}$$

For cohomology of a locally symmetric space $X=\Gamma\backslash G/K$, the bridge to the trace formula is Matsushima's formula: cohomology is built out of those automorphic representations whose archimedean component has nonzero $(\mathfrak g,K)$-cohomology. An Euler–Poincaré test function then turns the automorphic trace formula into a formula for the Euler characteristic. To obtain an individual Betti number $\dim H^q$, one uses a finer projector or the $q$-form heat kernel.

There are three closely related but distinct formulas:

setting	formula	what it counts
$SL_2(\mathbf Z)$, holomorphic forms	Eichler–Selberg trace formula	$\operatorname{tr}(T_n\mid S_k)$, hence $\dim S_k$ by $n=1$
$\Gamma\backslash\mathbf H$, Maass forms	Selberg trace formula	Laplace spectrum versus closed geodesics
general reductive $G$	Arthur–Selberg trace formula	automorphic representations versus rational conjugacy classes and Levi subgroups

The compact quotient case contains the essential mechanism and admits a fairly complete proof in a few pages; the full noncompact Arthur trace formula is a book-length theorem involving truncation, Eisenstein series, weighted orbital integrals and normalized intertwining operators. Arthur's exposition explicitly develops the compact formula first and then constructs the general formula from truncation. (Clay Mathematics Institute)

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1. The classical Selberg trace formula

Let

$$G=PSL_2(\mathbf R),\qquad K=PSO(2),\qquad G/K\simeq\mathbf H,$$

and let $\Gamma\subset G$ be torsion-free and cocompact. Then

$$X=\Gamma\backslash\mathbf H$$

is a compact hyperbolic surface. Write the eigenvalues of its positive Laplacian as

$$0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots , \qquad \lambda_j=\frac14+r_j^2.$$

For an appropriate even test function $h(r)$, let $g$ be its Fourier transform. Selberg's trace formula is

$$\boxed{ \sum_j h(r_j) = \frac{\operatorname{area}(X)}{4\pi} \int_{-\infty}^{\infty} r h(r)\tanh(\pi r)\,dr + \sum_{[P]} \sum_{m\geq1} \frac{\ell(P)} {2\sinh(m\ell(P)/2)} g(m\ell(P)). }$$

Here $P$ ranges over primitive closed geodesics and $\ell(P)$ is their length. Thus

$$\text{eigenvalues of }\Delta \quad\longleftrightarrow\quad \text{closed geodesics}.$$

A standard modern statement of this formula appears, for example, in the cited hyperbolic-surface literature. (Springer)

For the modular surface $SL_2(\mathbf Z)\backslash\mathbf H$, the quotient is noncompact and has elliptic fixed points. Consequently there are additional elliptic and parabolic terms and, most importantly, a term involving the logarithmic derivative of the scattering matrix. That term represents the continuous spectrum generated by Eisenstein series. (doczz.net)

This distinction is already a prototype for the general theory:

$$\boxed{\text{compact quotient: discrete spectrum only}}$$

versus

$$\boxed{\text{noncompact arithmetic quotient: discrete + continuous Eisenstein spectrum}.}$$

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2. The abstract compact trace formula

Now forget $SL_2$ temporarily.

Let $G$ be a unimodular Lie group and let

$$\Gamma\subset G$$

be a discrete cocompact subgroup. Consider the right regular representation

$$(R(g)\phi)(x)=\phi(xg)$$

on

$$L^2(\Gamma\backslash G).$$

For

$$f\in C_c^\infty(G)$$

define

$$R(f)=\int_G f(g)R(g)\,dg.$$

The fundamental trace formula is

$$\boxed{ \sum_{\pi\in\widehat G} m_\Gamma(\pi)\operatorname{tr}\pi(f) = \sum_{[\gamma]_\Gamma} \operatorname{vol}(\Gamma_\gamma\backslash G_\gamma) \,O_\gamma(f), } \tag{TF}$$

where

$$O_\gamma(f) = \int_{G_\gamma\backslash G} f(x^{-1}\gamma x)\,dx,$$

$G_\gamma$ is the centralizer of $\gamma$ in $G$, and $m_\Gamma(\pi)$ is the multiplicity of the irreducible unitary representation $\pi$ in $L^2(\Gamma\backslash G)$. This is precisely the compact Selberg trace formula in its representation-theoretic form. (Clay Mathematics Institute)

Proof

This proof is worth knowing because nearly everything in the general trace formula is a sophisticated version of it.

For $\phi\in L^2(\Gamma\backslash G)$,

$$\begin{aligned} (R(f)\phi)(x) &= \int_G f(g)\phi(xg)\,dg\\ &= \int_G f(x^{-1}y)\phi(y)\,dy. \end{aligned}$$

Passing from $G$ to $\Gamma\backslash G$,

$$(R(f)\phi)(x) = \int_{\Gamma\backslash G} K_f(x,y)\phi(y)\,dy$$

with kernel

$$\boxed{ K_f(x,y) = \sum_{\gamma\in\Gamma} f(x^{-1}\gamma y). }$$

Because $f$ has compact support and $\Gamma$ is discrete, the sum is locally finite.

Since $\Gamma\backslash G$ is compact and $f$ is smooth, $K_f$ is a smooth kernel on a compact manifold. Hence $R(f)$ is trace class and

$$\operatorname{Tr}R(f) = \int_{\Gamma\backslash G}K_f(x,x)\,dx.$$

Therefore

$$\operatorname{Tr}R(f) = \int_{\Gamma\backslash G} \sum_{\gamma\in\Gamma} f(x^{-1}\gamma x)\,dx.$$

Group the elements of $\Gamma$ by conjugacy classes. If $\Gamma_\gamma$ is the centralizer of $\gamma$,

$$\Gamma\cdot\gamma = \{\delta^{-1}\gamma\delta: \delta\in\Gamma_\gamma\backslash\Gamma\}.$$

Thus

$$\begin{aligned} \operatorname{Tr}R(f) &= \sum_{[\gamma]} \int_{\Gamma\backslash G} \sum_{\delta\in\Gamma_\gamma\backslash\Gamma} f(x^{-1}\delta^{-1}\gamma\delta x)\,dx\\ &= \sum_{[\gamma]} \int_{\Gamma_\gamma\backslash G} f(x^{-1}\gamma x)\,dx. \end{aligned}$$

Now decompose

$$\Gamma_\gamma\backslash G \longrightarrow G_\gamma\backslash G$$

with fiber

$$\Gamma_\gamma\backslash G_\gamma.$$

Fubini gives

$$\int_{\Gamma_\gamma\backslash G} f(x^{-1}\gamma x)\,dx = \operatorname{vol}(\Gamma_\gamma\backslash G_\gamma) \int_{G_\gamma\backslash G} f(x^{-1}\gamma x)\,dx.$$

Hence

$$\operatorname{Tr}R(f) = \sum_{[\gamma]} \operatorname{vol}(\Gamma_\gamma\backslash G_\gamma) O_\gamma(f). \tag{geometric side}$$

On the other hand, because the quotient is compact,

$$L^2(\Gamma\backslash G) \simeq \widehat{\bigoplus}_{\pi\in\widehat G} m_\Gamma(\pi)\,\pi$$

with finite multiplicities. Consequently

$$R(f) \simeq \bigoplus_\pi m_\Gamma(\pi)\pi(f),$$

and hence

$$\operatorname{Tr}R(f) = \sum_\pi m_\Gamma(\pi)\operatorname{tr}\pi(f). \tag{spectral side}$$

Equating the two expressions proves (TF).

That kernel computation is essentially Arthur's compact-quotient derivation. (Clay Mathematics Institute)

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3. Why modular forms occur in this formula

A holomorphic modular form of weight $k$ can be regarded representation-theoretically as a function on

$$SL_2(\mathbf R)$$

transforming under

$$K=SO(2)$$

according to the character

$$e^{i\theta}\mapsto e^{ik\theta}.$$

Holomorphic cusp forms correspond to holomorphic discrete-series representations

$$D_k^+$$

of $SL_2(\mathbf R)$.

Thus the dimension of $S_k(\Gamma)$ is essentially the multiplicity of the representation $D_k^+$ in the automorphic spectrum.

The idea is therefore:

$$\text{choose }f_\infty$$

such that

$$\operatorname{tr}\pi_\infty(f_\infty) = \begin{cases} 1,&\pi_\infty=D_k^+,\\ 0,&\text{other relevant }\pi_\infty. \end{cases}$$

Then the spectral side literally counts weight-$k$ cusp forms.

This is the representation-theoretic origin of the Eichler–Selberg trace formula.

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4. The Eichler–Selberg trace formula

For the full modular group let

$$S_k=S_k(SL_2(\mathbf Z))$$

with $k\geq4$ even, and let $T(n)$ be the usual Hecke operator.

Define

$$p_k(t,n) = \operatorname{Coeff}_{X^{k-2}} \frac1{1-tX+nX^2}.$$

Equivalently,

$$p_k(t,n) = \sum_{0\leq r\leq k/2-1} \binom{k-2-r}{r} (-n)^rt^{\,k-2-2r}.$$

Let $H(D)$ denote the Kronecker–Hurwitz class number. The Eichler–Selberg trace formula in Zagier's normalization is

$$\boxed{ \operatorname{tr}(T(n)\mid S_k) = -\frac12 \sum_{\substack{t\in\mathbf Z\\t^2\leq4n}} p_k(t,n)H(4n-t^2) - \frac12 \sum_{d\mid n} \min(d,n/d)^{\,k-1}. } \tag{ES}$$

Zagier states exactly this form and explains its relation with the trace on the full space $M_k$. (MPIM Bonn Personal Pages)

Notice the analogy with the abstract trace formula. The discriminant

$$t^2-4n$$

classifies conjugacy types of matrices of determinant $n$; the class numbers $H(4n-t^2)$ count the relevant elliptic conjugacy classes.

So the class-number sum is not accidental: it is the geometric side of a trace formula.

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5. Obtaining $\dim S_k$ from the trace formula

This is a particularly clean example of the principle

$$\boxed{\text{dimension}=\text{trace of identity}.}$$

Since

$$T(1)=1,$$

put $n=1$ in (ES):

$$\dim S_k = -\frac12 \sum_{t^2\leq4} p_k(t,1)H(4-t^2) -\frac12. \tag{1}$$

The possible $t$'s are

$$-2,-1,0,1,2.$$

Use the class-number conventions

$$H(0)=-\frac1{12},\qquad H(3)=\frac13,\qquad H(4)=\frac12.$$

First,

$$\frac1{1-2X+X^2} = \frac1{(1-X)^2},$$

so

$$p_k(2,1)=k-1.$$

Since $k-2$ is even,

$$p_k(-2,1)=k-1$$

as well.

Next,

$$\frac1{1+X^2} = 1-X^2+X^4-\cdots ,$$

so

$$p_k(0,1) = (-1)^{(k-2)/2}.$$

Finally

$$\frac1{1-X+X^2}$$

has coefficients of period six. For even $k$,

$$a_k:=p_k(1,1)=p_k(-1,1) = \begin{cases} -1,&k\equiv0\pmod6,\\ 1,&k\equiv2\pmod6,\\ 0,&k\equiv4\pmod6. \end{cases}$$

Substitution into (1) gives

$$\begin{aligned} \dim S_k &= -\frac12 \left[ 2(k-1)\left(-\frac1{12}\right) + 2a_k\left(\frac13\right) + (-1)^{(k-2)/2}\frac12 \right] -\frac12\\[2mm] &= \boxed{ \frac{k-1}{12} -\frac{a_k}{3} -\frac14(-1)^{(k-2)/2} -\frac12. } \tag{2} \end{aligned}$$

Simplifying by congruence classes modulo $12$,

$$\boxed{ \dim S_k(SL_2(\mathbf Z)) = \begin{cases} \left\lfloor k/12\right\rfloor-1, & k\equiv2\pmod{12},\\[1mm] \left\lfloor k/12\right\rfloor, &\text{otherwise}, \end{cases}} \qquad k\ge4\text{ even}.$$

For example,

$$\dim S_{12}=1,\qquad \dim S_{14}=0,\qquad \dim S_{24}=2.$$

So even the familiar dimension formula for modular forms can be viewed as an $n=1$ specialization of a trace formula rather than as a Riemann–Roch computation.

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6. From modular forms to cohomology: Eichler–Shimura

Let

$$V_{k-2} = \operatorname{Sym}^{k-2}\mathbf C^2.$$

The Eichler–Shimura isomorphism identifies modular forms with degree-one cohomology:

$$\boxed{ H^1\!\left(\Gamma,V_{k-2}\right) \simeq S_k(\Gamma) \oplus \overline{S_k(\Gamma)} \oplus E_k(\Gamma), } \tag{ESh}$$

where $E_k$ denotes the Eisenstein contribution. Equivalently,

$$H^1\simeq M_k\oplus\overline{S_k}.$$

The parabolic/interior part is

$$\boxed{ H^1_{\rm par}(\Gamma,V_{k-2}) \simeq S_k(\Gamma)\oplus\overline{S_k(\Gamma)}. }$$

This is a concrete rank-one model of automorphic cohomology. (Ashwin Iyengar)

For $\Gamma=SL_2(\mathbf Z)$ and $k\ge4$ even, the Eisenstein space is one-dimensional, so

$$\dim H^1_{\rm par} = 2\dim S_k$$

and

$$\boxed{ \dim H^1 = 2\dim S_k+1. }$$

Combining this with the trace-formula calculation above gives

$$\boxed{ \dim H^1 = 2\left( \left\lfloor\frac{k}{12}\right\rfloor - \mathbf 1_{k\equiv2\pmod{12}} \right)+1 }$$

for $k\ge4$ even.

That is already an explicit computation of the cohomology of a locally symmetric orbifold using automorphic traces.

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7. General locally symmetric spaces

Now let $G$ be a connected semisimple real group with finite center, let

$$K\subset G$$

be maximal compact, and put

$$\widetilde X=G/K.$$

For a torsion-free lattice $\Gamma$,

$$X_\Gamma=\Gamma\backslash G/K$$

is a locally symmetric manifold.

Let

$$\mathfrak g=\mathfrak k\oplus\mathfrak p$$

be the Cartan decomposition. Let $V_\xi$ be a finite-dimensional representation giving a local system $\mathcal E_\xi$ on $X_\Gamma$.

The fundamental result is Matsushima's formula.

For a compact quotient,

$$\boxed{ H^q(X_\Gamma,\mathcal E_\xi) \simeq \bigoplus_{\pi\in\widehat G} m_\Gamma(\pi)\, H^q(\mathfrak g,K;\pi^\infty\otimes V_\xi). } \tag{M}$$

In adelic notation, for level $K_f$,

$$\boxed{ H^q(S_{K_f},\mathcal E_\xi) \simeq \bigoplus_{\pi} m(\pi)\, H^q(\mathfrak g,K_\infty; \pi_\infty\otimes V_\xi) \otimes \pi_f^{K_f}. } \tag{M-ad}$$

For noncompact spaces the same formula describes $L^2$-cohomology using the discrete automorphic spectrum; this is often referred to as the Borel–Casselman/Matsushima formula. (Princeton Math)

Taking dimensions,

$$\boxed{ b_q(S_{K_f},\mathcal E_\xi) = \sum_\pi m(\pi)\, \dim H^q(\mathfrak g,K_\infty; \pi_\infty\otimes V_\xi) \, \dim\pi_f^{K_f}. } \tag{3}$$

This equation is the basic answer to the question “how do automorphic forms compute cohomology?”

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8. Proof of Matsushima's formula

The argument is conceptually simple once the correct language is used.

A differential $q$-form on

$$X_\Gamma=\Gamma\backslash G/K$$

can be lifted to $G$. At the identity coset, the tangent space of $G/K$ is

$$\mathfrak p.$$

Therefore differential forms with coefficients in $V_\xi$ identify with

$$\Omega^q(X_\Gamma,\mathcal E_\xi) \simeq \operatorname{Hom}_K \left( \wedge^q\mathfrak p, C^\infty(\Gamma\backslash G)\otimes V_\xi \right). \tag{4}$$

But the right-hand side is precisely the degree-$q$ term of the relative Lie algebra cochain complex

$$C^q(\mathfrak g,K;W) = \operatorname{Hom}_K(\wedge^q\mathfrak p,W)$$

with

$$W=C^\infty(\Gamma\backslash G)\otimes V_\xi.$$

Under (4), the exterior derivative becomes the relative Lie algebra differential. Consequently

$$H^q(X_\Gamma,\mathcal E_\xi) \simeq H^q\!\left( \mathfrak g,K; C^\infty(\Gamma\backslash G)\otimes V_\xi \right). \tag{5}$$

If the quotient is compact,

$$L^2(\Gamma\backslash G) = \widehat{\bigoplus}_{\pi} m_\Gamma(\pi)\pi.$$

Hodge theory says that cohomology is represented by harmonic forms. The Hodge Laplacian is an elliptic operator and, by the Matsushima–Kuga formula, can be expressed in terms of the Casimir operator of $G$, up to the scalar contributed by the coefficient system.

Hence only representations with the appropriate infinitesimal character can contribute to the kernel. Since the harmonic space is finite-dimensional, decomposition into $G$-isotypical components may be performed before taking harmonic vectors. Thus

$$H^q \simeq \bigoplus_\pi m_\Gamma(\pi) H^q(\mathfrak g,K;\pi^\infty\otimes V_\xi),$$

which is (M).

Conceptually,

$$\boxed{ \text{Hodge theory} + \text{spectral decomposition of }L^2(\Gamma\backslash G) = \text{Matsushima formula}. }$$

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9. The cohomological trace formula

Define

$$d_q(\pi,\xi) = \dim H^q(\mathfrak g,K; \pi\otimes V_\xi).$$

Matsushima says

$$b_q = \sum_\pi m(\pi)d_q(\pi,\xi).$$

Thus if one could construct $f_q$ satisfying

$$\operatorname{tr}\pi(f_q) = d_q(\pi,\xi),$$

then the spectral side of the trace formula would be exactly $b_q$.

This is the central strategy.

The easiest universal function to construct does not isolate $q$; it isolates the alternating sum.

Define

$$\chi(\mathfrak g,K;\pi\otimes V_\xi) = \sum_q (-1)^q \dim H^q(\mathfrak g,K;\pi\otimes V_\xi).$$

Matsushima immediately yields

$$\boxed{ \chi(X_\Gamma,\mathcal E_\xi) = \sum_\pi m_\Gamma(\pi) \chi(\mathfrak g,K;\pi\otimes V_\xi). } \tag{6}$$

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10. Why the Euler characteristic is especially easy

The relative cochain spaces are

$$C^q = \operatorname{Hom}_K (\wedge^q\mathfrak p,\pi\otimes V_\xi).$$

For every finite complex,

$$\sum_q(-1)^q\dim H^q = \sum_q(-1)^q\dim C^q.$$

Therefore

$$\chi(\mathfrak g,K;\pi\otimes V_\xi) = \sum_q (-1)^q \dim \operatorname{Hom}_K (\wedge^q\mathfrak p,\pi\otimes V_\xi). \tag{7}$$

Using $K$-characters,

$$\dim\operatorname{Hom}_K(W,V) = \int_K \chi_V(k)\overline{\chi_W(k)}\,dk.$$

But

$$\sum_q(-1)^q \chi_{\wedge^q\mathfrak p}(k) = \det(1-\operatorname{Ad}(k)|_{\mathfrak p}).$$

Hence formally

$$\chi(\mathfrak g,K;\pi\otimes V_\xi) = \int_K \Theta_\pi(k)\chi_\xi(k) \det(1-\operatorname{Ad}(k)|_{\mathfrak p})\,dk. \tag{8}$$

This makes the existence of a special trace-function very plausible.

Clozel and Delorme constructed an Euler–Poincaré function

$$f_{\rm EP,\xi}\in C_c^\infty(G)$$

such that

$$\boxed{ \operatorname{tr}\pi(f_{\rm EP,\xi}) = \chi(\mathfrak g,K; \pi\otimes V_\xi). } \tag{EP}$$

Such functions are cuspidal: their orbital integrals vanish on nonelliptic classes. (ScienceDirect)

This local theorem is one of the most important points in the cohomological use of the trace formula.

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11. Applying the trace formula

Insert

$$f=f_{\rm EP,\xi}$$

into the compact trace formula.

The spectral side becomes

$$\begin{aligned} \sum_\pi m_\Gamma(\pi)\operatorname{tr}\pi(f_{\rm EP,\xi}) &= \sum_\pi m_\Gamma(\pi) \chi(\mathfrak g,K;\pi\otimes V_\xi)\\ &= \chi(X_\Gamma,\mathcal E_\xi). \end{aligned}$$

Therefore

$$\boxed{ \chi(X_\Gamma,\mathcal E_\xi) = \sum_{[\gamma]} \operatorname{vol} (\Gamma_\gamma\backslash G_\gamma) O_\gamma(f_{\rm EP,\xi}). } \tag{Coh-TF}$$

This is the basic cohomological trace formula.

It is a Lefschetz-type formula:

$$\boxed{ \text{alternating cohomology dimension} = \text{sum of geometric orbital contributions}. }$$

At finite adelic level $K_f$, put

$$e_{K_f} = \frac{\mathbf 1_{K_f}}{\operatorname{vol}(K_f)}.$$

Then

$$\operatorname{tr}\pi_f(e_{K_f}) = \dim\pi_f^{K_f}.$$

Consequently

$$f=f_{\rm EP,\xi}\otimes e_{K_f}$$

has spectral side

$$\sum_\pi m(\pi) \chi(\mathfrak g,K;\pi_\infty\otimes V_\xi) \dim\pi_f^{K_f},$$

which by Matsushima is

$$\boxed{ \chi(S_{K_f},\mathcal E_\xi). }$$

Thus

$$\boxed{ \chi(S_{K_f},\mathcal E_\xi) = J_{\rm geom} (f_{\rm EP,\xi}\otimes e_{K_f}). } \tag{9}$$

That is perhaps the cleanest general formulation of “use the trace formula to compute cohomology.”

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12. Why only elliptic elements survive

A crucial property is

$$O_\gamma(f_{\rm EP,\xi})=0$$

if $\gamma$ is nonelliptic.

Therefore (Coh-TF) reduces to elliptic conjugacy classes.

If $\Gamma$ is torsion-free and cocompact, a nontrivial elliptic element cannot occur: an elliptic element lies in a compact subgroup, and discreteness forces the cyclic subgroup it generates to be finite. Torsion-freeness then forces

$$\gamma=1.$$

Hence

$$\chi(X_\Gamma,\mathcal E_\xi) = \operatorname{vol}(\Gamma\backslash G) f_{\rm EP,\xi}(1). \tag{10}$$

With Euler–Poincaré normalization of Haar measure,

$$f_{\rm EP,\xi}(1)=\dim V_\xi,$$

so

$$\boxed{ \chi(X_\Gamma,\mathcal E_\xi) = \dim V_\xi\, \operatorname{vol}_{EP}(\Gamma\backslash G). } \tag{11}$$

This is the representation-theoretic form of Gauss–Bonnet. Harder's arithmetic Gauss–Bonnet theorem extends this philosophy to arithmetic quotients and Euler–Poincaré measures. (Numdam)

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13. Example: a compact hyperbolic surface

For a compact hyperbolic surface,

$$\operatorname{area}(X)=4\pi(g-1).$$

Gauss–Bonnet gives

$$\chi(X) = -\frac{\operatorname{area}(X)}{2\pi} = 2-2g.$$

Since

$$b_0=b_2=1,$$

we obtain

$$1-b_1+1 = -\frac{\operatorname{area}(X)}{2\pi},$$

hence

$$\boxed{ b_1 = 2+\frac{\operatorname{area}(X)}{2\pi} = 2g. }$$

This is a very simple instance where the trace-formula/Euler-characteristic computation plus elementary topological information recovers an individual cohomology dimension.

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14. How to compute an individual $\dim H^q$

The Euler–Poincaré function gives

$$\sum_q(-1)^q b_q,$$

not $b_q$ separately.

There are two important ways around this.

The cleanest universally valid method for compact quotients is the heat kernel.

Let

$$\Delta_q$$

be the Hodge Laplacian on $q$-forms with coefficients in $\mathcal E_\xi$. Its spectrum is discrete:

$$0=\lambda_{q,1}=\cdots=\lambda_{q,b_q} < \lambda_{q,b_q+1}\leq\cdots.$$

Therefore

$$\operatorname{Tr}(e^{-t\Delta_q}) = \sum_j e^{-t\lambda_{q,j}},$$

and

$$\boxed{ b_q = \lim_{t\to\infty} \operatorname{Tr}(e^{-t\Delta_q}). } \tag{12}$$

Lift the heat kernel to $G/K$. If $k_{t,q}(g)$ denotes the corresponding $G$-equivariant matrix-valued kernel, its automorphic periodization is

$$K_{t,q}(x,y) = \sum_{\gamma\in\Gamma} k_{t,q}(x^{-1}\gamma y).$$

Exactly the same argument as in the compact trace formula gives

$$\operatorname{Tr}(e^{-t\Delta_q}) = \sum_{[\gamma]} \operatorname{vol} (\Gamma_\gamma\backslash G_\gamma) O_\gamma(k_{t,q}). \tag{13}$$

Therefore

$$\boxed{ b_q(X_\Gamma,\mathcal E_\xi) = \lim_{t\to\infty} \sum_{[\gamma]} \operatorname{vol} (\Gamma_\gamma\backslash G_\gamma) O_\gamma(k_{t,q}). } \tag{14}$$

This is an exact geometric trace formula for the individual Betti number.

On the spectral/representation side,

$$\operatorname{Tr}(e^{-t\Delta_q}) = \sum_\pi m_\Gamma(\pi) \operatorname{Tr} (e^{-t\Delta_{\pi,q}}),$$

and

$$\lim_{t\to\infty} \operatorname{Tr} (e^{-t\Delta_{\pi,q}}) = \dim H^q(\mathfrak g,K;\pi\otimes V_\xi).$$

Thus (12) becomes precisely

$$b_q = \sum_\pi m_\Gamma(\pi) \dim H^q(\mathfrak g,K;\pi\otimes V_\xi),$$

recovering Matsushima's dimension formula.

So there is a beautiful triangle:

$$\boxed{ \begin{array}{ccc} \text{heat kernel} &\longrightarrow& \text{Betti number}\\ \downarrow && \uparrow\\ \text{orbital integrals} &\longleftrightarrow& \text{automorphic representations}. \end{array}}$$

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15. Pseudo-coefficients and representation-by-representation counting

Sometimes one knows from representation theory that the only representations contributing to $H^q$ are particular cohomological representations

$$A_{\mathfrak q}(\lambda).$$

One can then try to choose a pseudo-coefficient $f_\pi$ satisfying

$$\operatorname{tr}\pi(f_\pi)=1$$

and

$$\operatorname{tr}\pi'(f_\pi)=0$$

for other representations in a suitable class.

Putting

$$f=f_\pi\otimes e_{K_f}$$

into the trace formula gives

$$\sum_{\Pi:\Pi_\infty\simeq\pi} m(\Pi)\dim\Pi_f^{K_f}$$

on the spectral side.

If

$$\dim H^q(\mathfrak g,K;\pi\otimes V_\xi)=d$$

and no other degree contributes, then the corresponding contribution to cohomology is simply

$$d \sum_{\Pi:\Pi_\infty=\pi} m(\Pi)\dim\Pi_f^{K_f}.$$

This is particularly effective for discrete-series representations and is one reason trace formulas work extremely well for Shimura varieties admitting discrete series.

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16. What changes for noncompact locally symmetric varieties?

This is the most important qualification.

Suppose

$$X=\Gamma\backslash G/K$$

has finite volume but is noncompact.

Then

$$L^2(\Gamma\backslash G) = L^2_{\rm disc} \oplus L^2_{\rm cont}.$$

The continuous part is generated by Eisenstein series attached to proper parabolic subgroups.

Consequently the naive integral

$$\int_{\Gamma\backslash G} K_f(x,x)\,dx$$

generally diverges.

Arthur introduces a truncation operator

$$\Lambda^T$$

and studies

$$J^T(f) = \int_{G(\mathbf Q)\backslash G(\mathbf A)} \Lambda^T K_f(x,x)\,dx.$$

The geometric and spectral expansions of $J^T(f)$ have polynomial-exponential dependence on $T$. After taking the appropriate constant term one obtains

$$\boxed{ J_{\rm geom}(f) = J_{\rm spec}(f). }$$

In the full invariant Arthur trace formula the geometric side contains weighted orbital integrals attached to Levi subgroups, while the spectral side contains automorphic representations of Levi subgroups together with intertwining operators and Eisenstein-series data. Arthur's foundational exposition describes precisely this progression from the compact kernel formula to truncation and the refined/invariant formula. (Clay Mathematics Institute)

Schematically,

$$\sum_M \sum_{\gamma\in M(\mathbf Q)} a^M(\gamma)J_M(\gamma,f) = \sum_M \sum_{\pi} a^M(\pi)J_M(\pi,f).$$

The $M=G$ part is the direct analogue of the compact formula. Terms with

$$M\neq G$$

encode parabolic/Eisenstein phenomena.

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17. Ordinary cohomology versus $L^2$-cohomology

For a noncompact arithmetic locally symmetric space, one must distinguish

$$H^\bullet(X,\mathcal E), \qquad H^\bullet_{(2)}(X,\mathcal E), \qquad H^\bullet_c(X,\mathcal E), \qquad H^\bullet_!(X,\mathcal E).$$

The automorphic discrete spectrum naturally computes $L^2$-cohomology:

$$H^\bullet_{(2)} \simeq \bigoplus_{\pi\in\Pi_{\rm disc}} m(\pi) H^\bullet(\mathfrak g,K;\pi_\infty\otimes\xi) \otimes\pi_f^{K_f}.$$

This is the noncompact Borel–Casselman version of Matsushima. (Princeton Math)

But ordinary cohomology also contains Eisenstein classes.

Franke's theorem says, roughly,

$$\boxed{ H^\bullet(X,\mathcal E_\xi) \simeq H^\bullet \left( \mathfrak g,K; \mathcal A(G)\otimes V_\xi \right), }$$

where $\mathcal A(G)$ is the space of automorphic forms of moderate growth. In other words, replacing all smooth functions by automorphic forms does not change the cohomology. (Encyclopedia of Mathematics)

Franke's theory decomposes ordinary cohomology schematically as

$$\boxed{ H^\bullet = H^\bullet_{\rm cusp} \oplus H^\bullet_{\rm Eis}. }$$

The Eisenstein part is built from Eisenstein series and their residues induced from cuspidal automorphic representations on Levi subgroups. (ScienceDirect)

Thus in the noncompact case the computational philosophy is

$$\dim H^q = \dim H^q_{\rm cusp} + \dim H^q_{\rm Eis}.$$

The trace formula handles the discrete/cuspidal part directly; the proper-Levi terms and Eisenstein cohomology supply the boundary contribution.

The modular-curve identity

$$H^1 = S_k \oplus\overline{S_k} \oplus E_k$$

is exactly the rank-one prototype.

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18. The general computational recipe

Suppose you are given

$$S_K = G(\mathbf Q)\backslash G(\mathbf A)/ K_\infty K_f$$

and an algebraic coefficient system $E_\xi$.

First determine which irreducible archimedean representations satisfy

$$H^q(\mathfrak g,K_\infty; \pi_\infty\otimes\xi)\neq0.$$

This is a local representation-theory problem; Vogan–Zuckerman theory classifies the irreducible unitary representations with nonzero $(\mathfrak g,K)$-cohomology.

For each such $\pi_\infty$, compute

$$d_q(\pi_\infty,\xi) = \dim H^q(\mathfrak g,K; \pi_\infty\otimes\xi).$$

Then the global problem is to determine

$$m(\pi)\dim\pi_f^{K_f}.$$

That is exactly the sort of quantity the trace formula controls.

Thus

$$\boxed{ \dim H^q = \sum_{\text{cohomological }\pi} \underbrace{m(\pi)}_{\text{global multiplicity}} \underbrace{\dim\pi_f^{K_f}}_{\text{level}} \underbrace{d_q(\pi_\infty,\xi)}_{\text{local cohomology}}. } \tag{15}$$

Formula (15) separates the problem beautifully:

$$\boxed{ \begin{array}{c} \text{representation theory at }\infty\\ +\;\text{automorphic multiplicities}\\ +\;\text{finite-level invariants}. \end{array}}$$

The trace formula is the device for computing or constraining the middle two factors.

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19. Euler characteristic versus individual Betti numbers

This distinction is worth emphasizing.

The Euler characteristic is especially trace-formula friendly:

$$\boxed{ \chi = J_{\rm geom}(f_{\rm EP}). }$$

Individual Betti numbers require more information:

$$\boxed{ b_q = \lim_{t\to\infty} J_{\rm geom}(k_{t,q}), }$$

or appropriately chosen cohomological pseudo-coefficients.

In favorable situations, representation theory tells you that cohomology occurs only in one degree. Then

$$\chi=(-1)^q b_q$$

and the Euler–Poincaré trace formula already gives the individual dimension.

In other situations, Poincaré duality, hard Lefschetz, known vanishing theorems, or knowledge of compact-dual classes can combine with $\chi$ to determine all Betti numbers.

In genuinely higher-rank examples, however, a single Euler-characteristic trace cannot determine every $b_q$.

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20. Where the full Arthur formula enters

For an anisotropic $G/\mathbf Q$, the adelic quotient is compact and the simple formula

$$\sum_\pi m(\pi)\operatorname{tr}\pi(f) = \sum_\gamma \operatorname{vol}(G_\gamma(\mathbf Q)\backslash G_\gamma(\mathbf A)) O_\gamma(f)$$

is already enough.

For groups such as

$$GL_n,\ SL_n,\ Sp_{2n}$$

over $\mathbf Q$, the quotient is generally noncompact. Then proper parabolic subgroups appear.

Arthur's trace formula replaces

$$\text{ordinary orbital integrals}$$

by

$$\text{weighted orbital integrals}$$

and replaces

$$\text{ordinary representation characters}$$

by

$$\text{weighted characters involving intertwining operators}.$$

This is why a fully detailed proof of the general Arthur–Selberg trace formula cannot realistically fit into one answer: the analytic truncation theorem alone is substantial, followed by the geometric expansion, spectral expansion, refinement, invariance, and often stabilization. Gelbart's lectures and Arthur's Clay notes are standard introductions to this architecture. (arXiv)

The important point for cohomology is that you usually do not need every term of the most general formula. A carefully chosen Euler–Poincaré or cuspidal test function often annihilates many proper-Levi contributions and produces a much simpler trace formula.

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21. The conceptual picture

The entire story can be compressed into the sequence

$$\boxed{ \begin{aligned} H^q(\Gamma\backslash G/K,E) &\overset{\text{Matsushima}}{\longleftrightarrow} \bigoplus_\pi m(\pi)H^q(\mathfrak g,K;\pi_\infty\otimes E) \\[1mm] &\overset{\text{test function}}{\longleftrightarrow} \sum_\pi m(\pi)\operatorname{tr}\pi(f) \\[1mm] &\overset{\text{trace formula}}{\longleftrightarrow} \sum_{[\gamma]} \operatorname{vol}(\Gamma_\gamma\backslash G_\gamma) O_\gamma(f). \end{aligned}}$$

For $f=f_{\rm EP}$,

$$\boxed{ \chi \longleftrightarrow \text{elliptic conjugacy classes}. }$$

For $f=k_{t,q}$,

$$\boxed{ b_q \longleftrightarrow \lim_{t\to\infty} \text{orbital integrals of the $q$-form heat kernel}. }$$

For $G=GL_2$ and a holomorphic-discrete-series projector,

$$\boxed{ \dim S_k \longleftrightarrow \text{class numbers}, }$$

which is the Eichler–Selberg formula.

That is the unifying answer: the trace formula converts the difficult spectral multiplicities appearing in Matsushima's cohomology formula into geometric orbital integrals.

For compact quotients this statement has the very explicit proof above. For noncompact quotients, Arthur truncation adds Levi/Eisenstein terms, and Franke's theorem explains why those same proper-Levi terms are exactly what ordinary Eisenstein cohomology requires.