The goal of this post is to give an exposition of trace formulas, which are indispensible tools across several areas of mathematics. They feature perhaps most prominently in the theory of automorphic forms, but before explaining how, we emphasize that the trace formula is at heart, a method that rests on more primitive scaffolding.
The recurring motif is that we construct an operator, compute its trace in two different ways, and thereby identify two kinds of information that are not visibly related. It begins with the finite-dimensional prototype, where the trace of an operator with kernel is the sum of the diagonal values . When the operator comes from a map of a finite set, this diagonal sum counts fixed points; when it comes from a finite group action, the same mechanism gives the finite group trace formula and Burnside’s lemma. The Grothendieck-Lefschetz trace formula then augments this principle from finite sets to varieties over finite fields: here, Frobenius-fixed points are rational points, and they are counted by the alternating trace of Frobenius on compactly supported étale cohomology.
The second half explains why the same idea becomes so central in the theory of automorphic forms. A compact quotient replaces the finite set, and a test function defines a convolution operator on . Computing spectrally gives a sum over irreducible representations occurring in ; computing it geometrically requires writing down the kernel, restricting to the diagonal, and unfolding the resulting integral into a sum over conjugacy classes and orbital integrals. This yields the compact automorphic trace formula. In the special case of compact hyperbolic surfaces, it becomes the Selberg trace formula, relating eigenvalues of the Laplacian to lengths of closed geodesics. The final sections explain how this compact picture points toward the full adelic and noncompact theory, where Arthur’s trace formula introduces truncation, weighted orbital integrals, Eisenstein series, and the comparison of trace formulae used in Langlands transfer.
The prototype
Let be a finite set, and let be the vector space of complex-valued functions on . For each , let be the delta function such that and for all . Then, is a basis of .
A function defines a linear operator given by That is, is the matrix coefficient of in row , column . Now, the map induces an isomorphism of vector spaces
Indeed, since is finite (say ), both vector spaces have dimension . We show first that the map is injective. Suppose . Then, for every , Evaluating at , we get Thus, for all , so . Hence, the map is injective, and since the dimensions agree, it is an isomorphism.
Now, we prove the most basic form of the trace formula.
Theorem (Diagonal trace formula). For every kernel ,
Proof The matrix of in the basis has entries The trace of a finite matrix is the sum of its diagonal entries. Therefore,
Fixed points as traces
Now, let be any map of finite sets. It induces on a pullback operator defined by
A foundational observation is that . Indeed, we compute the matrix of in the delta basis. For , Thus, if and only if . Therefore, So, the coefficient of in is if , and otherwise. The diagonal coefficient corresponding to is therefore precisely when The result follows.
This is the finite-set Lefschetz trace formula, and the key thrust is that fixed points of a map correspond to the trace of the induced operator on functions.
Finite group trace formula
Let be a finite group acting on a finite set . Again, let . The action of on gives rise to a representation via
Let . Then, we note that , furthering the paradigm described at the end of the previous section. To this end, the operator acts on the delta basis as follows: for , This equals exactly when , i.e. . Thus, In words, the matrix of is the permutation matrix corresponding to the permutation of .
A permutation matrix has diagonal entry in position exactly when , and it is otherwise. Therefore,
Next, we describe the averaging operator in the group algebra. Let Define More generally, if is any representation of , define
Theorem (Finite group trace formula). Let act on a finite set , and let be the vector space of complex-valued functions on . By Maschke’s theorem, where is the set of irreducible complex representations of , and Then, for every we have
Proof First, compute spectrally. Since decomposes into irreducible representations, the operator decomposes as Hence, Now, compute the same trace another way. By linearity,
From our prior discussion, Therefore, Equating the two expressions for yields the desired result.
As a corollary, we obtain Burnside’s lemma:
Proof Let Then, is the projection onto the -invariant functions. Indeed, for , This average is -invariant. If , then for all , so .
Therefore, But, is the space of functions on the orbit set , so
On the other hand, by the trace formula, Hence,
Lefschetz trace formula over finite fields
We may turn the previous theorem arithmetic when the finite set is the set of geometric points of a variety, and when the map is the Frobenius.
Let be a scheme of finite type over a finite field . Write The geometric Frobenius acts on -adic cohomology , where . First, we discuss the finite étale case. Suppose is finite étale. Then, we claim that
Proof Because of the finite étale condition, is a finite discrete set of points. Write Then, Moreover, for , because is a finite discrete space in the étale topology.
The rational points of are exactly the geometric points fixed by the Frobenius. By the trace formula for finite sets, as desired.
Theorem (Grothendieck-Lefschetz trace formula). Let be separated of finite type. Then,
The full proof requires the étale Lefschetz-Verdier fixed-point formula, which is roughly the assertion that for a correspondence satisfying appropriate transversality and properness hypotheses, where is a local intersection-theoretic contribution at the fixed point .
Now, apply this to and . As noted before, the fixed points of geometric Frobenius are exactly the rational points over . For Frobenius on a separated scheme of finite type over a finite field, the graph of the Frobenius intersects the diagonal with local multiplicity at every fixed point; the result follows. Here, the idea is that fixed points relate to the alternating trace on cohomology.
Now, we provide two examples. The first is the affine line.
Let . Then so
We compute the right-hand side of the Grothendieck-Lefschetz trace formula. Since is not proper, we use compactly supported étale cohomology. Over , we claim that and
Proof Consider the open immersion with closed complement The long exact sequence for compactly supported cohomology gives Now and The restriction map is an isomorphism. Hence and because
Continuing the long exact sequence, we have Since is a point, Therefore Finally, so
The geometric Frobenius acts on by multiplication by . Hence Therefore, Thus the Grothendieck-Lefschetz trace formula gives
Next, we turn to the projective line for our second example.
Let . Then so
Since is proper, compactly supported cohomology agrees with ordinary étale cohomology: The cohomology groups are and
Proof The group consists of locally constant -valued functions on the connected scheme , so The étale cohomology of projective space is for odd , and for . Hence and
The geometric Frobenius acts on by the identity, so It acts on by multiplication by , so Therefore, Thus, the Grothendieck-Lefschetz trace formula gives
Compact automorphic trace formula
In this section, we get a first glimpse of the utility of the trace formula in studying automorphic forms. We deal with the compact setting, as the noncompact case requires Arthur’s truncation and continuous spectrum.
Let be a connected real reductive Lie group, a discrete cocompact subgroup (sometimes referred to as a lattice), , a fixed right Haar measure on with induced quotient measure on , and .
Define the right regular representation by Then, define explicitly,
For and , define the automorphic kernel It is not difficult to check that this expression is independent of the choice of representatives .
Proof Suppose that and for some . Then As ranges over , the element also ranges over . Therefore, so is well-defined on .
We also need to know that the sum defining is locally finite. Let , which is compact. The summand indexed by is nonzero only if or equivalently, Since is compact and is discrete, the intersection is finite. Hence, for fixed , only finitely many terms contribute.
Moreover, if and vary in compact subsets , then any contributing lies in which is compact. Hence is finite. Since is compact, one may choose a compact set of representatives for all points of . It follows that the sum defining is locally uniformly finite. Since each summand is smooth, is a smooth function on .
Lemma (Kernel formula). For every smooth function on ,
Proof Let . Then By local finiteness, we may interchange the sum and the integral. We use the quotient integration formula for compactly supported continuous functions on . Apply this to Then Now set . Since is unimodular, , and therefore By definition, the right-hand side is Thus,
The diagonal trace
We now compute the trace of by restricting its kernel to the diagonal. Since is compact and is smooth, is a smoothing operator on a compact manifold. Hence it is trace class.
Lemma (Trace of a smoothing operator). Let be a compact smooth manifold with smooth measure . Let be a smoothing operator on with smooth kernel . Then is trace class and
Proof Choose a positive self-adjoint elliptic differential operator on , and let be an orthonormal basis of consisting of smooth eigenfunctions: Since is smoothing, for every , the operator is bounded on . Hence the matrix coefficients decay rapidly in and . In particular, so is trace class and
Using the kernel, Thus, By rapid convergence, we may interchange the sum and the integrals. Since as distributions, we obtain
Applying this to the automorphic kernel gives If , then Therefore,
Unfolding and conjugacy classes
We now rewrite the last expression as a sum over conjugacy classes in .
For , define and
Lemma (Conjugacy classes and centralizers). For fixed , the map defined by is a bijection.
Proof First, the map is well-defined. If with , then The map is clearly surjective. For injectivity, suppose Multiplying on the left by and on the right by gives Hence , which is equivalent to Thus the map is injective.
Lemma (Unfolding). Let be a subgroup. For a locally finite sum, one has
Proof The natural map has fiber over naturally identified with . Indeed, the points above are represented by where , and if and only if . Therefore, integration over may be computed by integrating over and summing over the fibers:
Theorem (Geometric expansion). For , where runs over -conjugacy classes.
Proof We have Decompose into conjugacy classes: Therefore, Hence, For fixed , define Then By the unfolding lemma, Thus,
For , define the orbital integral of at by whenever the integral converges.
Lemma (Orbital-integral form). Assume that has finite volume. Then
Proof The natural map has fiber . The function is left -invariant, since for , Thus the integral over factors as the volume of the fiber times the integral over : The last integral is .
The spectral side
Since is compact, the right regular representation decomposes discretely: where denotes the unitary dual of , is the Hilbert space of , and The multiplicities are finite.
For an irreducible unitary representation of , define
Theorem (Spectral expansion). For ,
Proof Under the discrete decomposition the operator decomposes as Since is trace class,
Theorem (Compact automorphic trace formula). Let be a connected real reductive Lie group, let be a cocompact discrete subgroup, and let . Then Equivalently, when the centralizer quotients have finite volume,
Proof Both sides are equal to . The first equality follows by comparing the spectral expansion with the geometric expansion. The orbital-integral form follows from the preceding lemma.
Compact hyperbolic surfaces
Let and let be torsion-free and cocompact. Then is a compact hyperbolic surface.
Let be -bi-invariant. Then preserves
Let be the positive hyperbolic Laplacian on . Since is compact, its spectrum is discrete: with . Write
For , define its spherical transform by where is the spherical function normalized by .
Theorem (Spectral side for a compact hyperbolic surface). For ,
Proof Since is compact, there is an orthonormal basis of consisting of smooth Laplace eigenfunctions: The operator commutes with the action of the algebra of -invariant differential operators on , hence with . Thus it preserves each Laplace eigenspace.
On the spherical representation generated by an eigenfunction with spectral parameter , the -fixed line is acted on by the scalar . Therefore Since is trace class,
Every nontrivial element of is hyperbolic. Indeed, a torsion-free subgroup contains no nontrivial elliptic elements, and a cocompact lattice in contains no parabolic elements.
For a hyperbolic element , let denote its translation length on .
Lemma (Centralizers in a cocompact Fuchsian group). Let be nontrivial. Then there is a unique primitive element and an integer such that and Moreover,
Proof The element is hyperbolic, hence has two fixed points on . Its axis is the geodesic joining these two points. If , then , so preserves the fixed-point set of , and hence preserves .
Thus acts by translations on . Since is discrete, the image of in the translation group of is a discrete subgroup of . Since it contains , it is nontrivial, hence infinite cyclic. Let be the generator translating in the same direction as with minimal positive translation length. Then and for some . Translation length is additive under powers along the same axis, so
The identity conjugacy class contributes For the standard Haar normalization, Thus the identity contribution is
For , the quotient has volume . The orbital integral computation for gives where Therefore the contribution of is
Theorem (Selberg trace formula, compact torsion-free case). Let be a compact hyperbolic surface. Let be an even Paley-Wiener function, and let Let be the eigenvalues of the positive Laplacian on , written as Then Here runs over nontrivial conjugacy classes in , and is the primitive element underlying .
Proof Choose whose spherical transform is . By the spectral-side computation, By the compact automorphic trace formula, The identity conjugacy class contributes Each nontrivial conjugacy class contributes Substituting these contributions into the trace formula gives
The adelic compact trace formula
Let be a number field, let , and let be a connected reductive group over . Suppose, for simplicity, that is compact. Let Define on .
For and , define The same proof as before shows that this is a well-defined kernel and that
Theorem (Compact adelic trace formula). Assume that is compact. Then where runs over irreducible automorphic representations occurring in , and
Proof The kernel computation gives Decompose into -conjugacy classes. For each , unfolding gives Since is left -invariant, this equals Thus the geometric side is
On the other hand, compactness gives the discrete spectral decomposition Hence Equating the two expressions gives the result.
Hecke operators as test functions
Let be a finite place of , and suppose is unramified at . Let The spherical Hecke algebra is with convolution If then defines a Hecke operator. For a decomposable global test function the operator is a global Hecke operator when the local factors are chosen in the appropriate local Hecke algebras.
Thus the trace formula computes traces of Hecke operators:
The noncompact case
If is not compact, the preceding argument fails in two places.
First, the diagonal integral may diverge.
Second, the spectral decomposition of contains continuous spectrum. The continuous spectrum is described by Eisenstein series attached to proper parabolic subgroups.
For example, for , the quotient is noncompact. The Eisenstein series initially converges for and admits meromorphic continuation to . These Eisenstein series account for the continuous spectrum of .
Arthur’s trace formula modifies the compact trace formula by replacing the divergent kernel integral with a truncated kernel integral. The truncation removes divergent contributions from proper parabolic subgroups.
Arthur’s trace formula
Let be a number field, let , and let be a connected reductive group over . Let Arthur defines two distributions and such that
In the compact case, and
In the noncompact case, the geometric side is a sum of weighted orbital integrals: Here ranges over Levi subgroups of , the element ranges over suitable conjugacy classes in , the coefficient is global, and is a weighted orbital integral. If , then , so this is an ordinary orbital integral.
The spectral side has the corresponding form Here denotes a suitable space of automorphic representations of , and the distributions are built from induced representations, Eisenstein series, and intertwining operators.
Theorem (Arthur trace formula, schematic form). Let be a connected reductive group over a number field . For every , Arthur defines distributions and such that The geometric side is expressed in terms of weighted orbital integrals. The spectral side is expressed in terms of automorphic representations, Eisenstein series, and intertwining operators. In the compact case, the formula reduces to the compact adelic trace formula.
Proof sketch Start from the formal kernel on . The integral need not converge. Arthur defines a truncation operator , depending on a truncation parameter , and studies This integral converges.
On the geometric side, one decomposes the kernel by conjugacy classes and parabolic subgroups. The truncation produces weight functions, and hence weighted orbital integrals.
On the spectral side, one decomposes into cuspidal data and Eisenstein series induced from Levi subgroups. The truncation produces terms involving normalized intertwining operators.
Both expansions have asymptotic expressions in . The equality of the constant terms gives
Transfer and the fundamental lemma
Let and be connected reductive groups over . A morphism of -groups is expected to induce transfer of automorphic representations from to .
The trace formula gives an indirect method. Suppose one can choose test functions and such that Then Arthur’s trace formula gives Thus equality of geometric distributions implies equality of spectral distributions.
At a nonarchimedean place , the required geometric equality is reduced to identities of orbital integrals. A typical local identity has the form Here is a stable orbital integral, is an ordinary orbital integral, and is a transfer factor.
The fundamental lemma asserts that, for canonical local test functions, these orbital integrals match. Since global orbital integrals factor into local orbital integrals, the fundamental lemma furnishes the local input needed to compare global trace formulas.
Shimura varieties, cohomology, and trace formulas
We now explain why Shimura varieties provide a natural setting in which the Grothendieck-Lefschetz trace formula and the Arthur-Selberg trace formula meet. A Shimura variety is attached to a Shimura datum , where is a reductive group over and is a suitable -homogeneous space. If is a compact open subgroup, the complex points of the associated Shimura variety are given by Thus a Shimura variety has two simultaneous descriptions. On the one hand, it is an algebraic variety, defined over a number field called the reflex field. On the other hand, its complex points are described by an adelic double quotient. This double nature is what makes Shimura varieties central in the comparison between geometry and automorphic forms.
The algebraic side gives Frobenius and étale cohomology. Let be the reflex field of , and suppose that has good reduction at a prime of . Let denote the geometric special fiber. The geometric Frobenius acts on the compactly supported étale cohomology groups Thus one can study the alternating trace where By the Grothendieck-Lefschetz trace formula, this alternating trace is expressed as a fixed-point count for on the special fiber.
The adelic side gives Hecke correspondences. If then the double coset defines a correspondence on . More generally, a compactly supported locally constant function defines a linear combination of Hecke correspondences away from . Combining this with Frobenius gives a Frobenius-Hecke correspondence on the special fiber. Hence one is led to study traces of the form By Grothendieck-Lefschetz, this trace is a fixed-point count for the correspondence
The representation-theoretic meaning of the same trace comes from the fact that the cohomology of Shimura varieties is governed by automorphic representations. Very schematically, automorphic representations of contribute to the cohomology through terms of the form where is the finite part of , is the archimedean part, and is relative Lie algebra cohomology. Thus the same Frobenius-Hecke trace should also admit a spectral expression in terms of automorphic representations.
This is the basic reason that Shimura varieties bring together the two trace formulas. The Grothendieck-Lefschetz trace formula gives with local multiplicities understood. The automorphic description of the same cohomology suggests that the same trace should be expressible by the spectral side of an Arthur-Selberg trace formula for a carefully chosen adelic test function Here records the Hecke correspondence away from , the factor records Frobenius and the geometry of the reduction at , and the factor is chosen to isolate the cohomological automorphic representations contributing to the Shimura variety.
The fixed-point side is already close to the geometric side of the Arthur-Selberg trace formula. The fixed points of the Frobenius-Hecke correspondence are described by group-theoretic data: rational conjugacy classes, local conjugacy classes away from , and -conjugacy classes at . In favorable cases, the Lefschetz fixed-point formula gives an expression of the form Here is an ordinary orbital integral away from , is a twisted orbital integral at , and is a global volume factor. Thus point-counting on the Shimura variety produces exactly the kind of orbital-integral expression that appears on the geometric side of the trace formula.
However, this comparison is not term-by-term at the level of the ordinary, unstabilized trace formula. The Lefschetz fixed-point formula naturally organizes the fixed-point data into stable or endoscopic packets. By contrast, the ordinary Arthur-Selberg trace formula is initially written in terms of ordinary conjugacy classes and ordinary orbital integrals. These two expressions are morally related, but they are not yet in the same language.
This is why stabilization is necessary. Stabilization rewrites the geometric side of the Arthur-Selberg trace formula in terms of stable orbital integrals and endoscopic contributions. Schematically, an ordinary geometric expression is replaced by a stable expression involving endoscopic groups : Here is a transfer of the test function to the endoscopic group , and denotes a stable geometric distribution.
The fundamental lemma is the local input that makes this stabilization possible. At a nonarchimedean place, it asserts that matching test functions have matching stable orbital integrals: Here is a stable orbital integral on an endoscopic group, is an ordinary orbital integral on , and is a transfer factor. Since global trace formulae factor into local orbital integrals, these local identities are precisely what allow the global geometric sides to be compared.
Thus Shimura varieties close the circle between the two trace formulas. The Grothendieck-Lefschetz trace formula counts fixed points of Frobenius-Hecke correspondences on the special fibers of Shimura varieties. The Arthur-Selberg trace formula organizes automorphic representations through orbital integrals. Stabilization, made possible by the fundamental lemma, is the mechanism that lets these two trace formulas be compared. In this way, point counts on Shimura varieties become a bridge from Frobenius and étale cohomology to automorphic representations and the Langlands program.
Thank you for reading!
Comments
Very informative! You’re like the Minbappe of math #woah….