These are my live-TeXed notes for the course *Math G6761: Perverse Sheaves and Fundamental Lemmas* taught by Wei Zhang at Columbia, Fall 2015. The final part of the course discusses the recent breakthrough *Shtukas and the Taylor expansion of L-functions* by Zhiwei Yun and Wei Zhang.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

(*Updated: 03/22/2017*: thank Tony Feng for helpful comments)

09/15/2015

##
Motivation

A *fundamental lemma* is an identity between orbital integrals on two different groups. For example, the *endoscopic fundamental lemma* arises from the stabilization of the Arthur-Selberg trace formula and endoscopic functoriality in the Langlands program. The *Jacquet-Rallis fundamental lemma* arises from the W. Zhang's relative trace formula approach to the Gan-Gross-Prasad conjecture for unitary groups. Both sides of the fundamental lemma can be thought of as counting the number of lattices satisfying certain properties.

In the equal characteristic case, the endoscopic fundamental lemma was proved by Ngo and the Jacquet-Rallis fundamental lemma was proved by Yun. These proofs use geometric methods (*perverse sheaves*) in an essential way. The advantage in the equal characteristic is one can endow the space of lattices in question with a *geometric structure* (e.g., the -rational points of an algebraic variety, usually realized as the fiber of an invariant map from a certain moduli space of vector bundles to an affine space). These -rational points then can be counted using Lefschetz trace formula.

In order to prove the identity between orbital integrals ("functions on orbits"), one instead proves the identity between perverse sheaves ("sheaves on orbits"). The miracle is that one can prove the identity between perverse sheaves by only verifying it over certain open dense subsets ("very regular" orbits), which can be easier. This reflects a uniqueness principle of orbital integrals: the "very regular" orbital integral determines the more degenerate ones. This principle, however, is hard to see directly from orbital integrals.

##
Intersection homology

The idea of perverse sheaves begins from Goresky-MacPherson's theory of intersection homology, which is a homology theory for *singular* manifolds with an analogue of *Poincare duality* and *Hodge decomposition*. More precisely, we define

To get a homology theory for pseudo-manifold with desired properties, one needs to consider only the chains that has nice intersection properties with the stratification.

To measure how "perverse" these intersections are, we introduce the

*perversity*, which is a function

such that

- ,
- .

Given a perversity

, we define a

-chain

to be

*-allowable* if

We then define the

*intersection homology* groups

by considering only the

-allowable locally finite

-chains. When

is the middle perversity,

is often written as

for short. Similarly one can define the

*intersection homology with compact support* by considering only the

-allowable finite

-chains.

09/17/2015

- All intersection homology groups are finitely generated, independent of the choice the stratification. It gives the usual homology groups for manifolds without singularities.
- if is connected.
- (intersection product) Suppose are perversities such that is still a perversity. There is an intersection product where is the real dimension of . When , we have a non-degenerate pairing In particular, when has even dimensional strata, we have the duality between and .

Let be the category of sheaves of -vector spaces on . Two important examples:

- The constant sheaf on , denoted by .
- Locally constant sheaves on . Recall that is
*locally constant* if for any , there exists an open such that the restriction map is an isomorphism. Locally constant sheaves with stalks of finite rank are called *local systems*.

Repeating the construction of -allowable -chains for each open we obtain a sheaf . We define a complex of sheaves given by (so is concentrated in the negative degrees). In particular, is concentrated in negative degrees.

Denote

. Then there is an isomorphism

It turns out that the IC sheaf (as an object in the derived category ) is topologically invariant. This topological invariance on the level of sheaves is more flexible and easier to prove.

09/22/2015

(Deligne, Goresky-MacPherson)
Let

. Let

(

). Let

be a perversity. Then in the bounded derived category

, we have

Here

is the usual truncation functor:

so that

for

. Inductively, let

and

Then

09/24/2015

References:

*An introduction to perverse sheaves*, Rietsch (brief)
*Intersection homology*, edited by Borel (detailed)

By checking at stalks, one finds that the cohomology of IC sheaves has support of dimension *in a particular range* depending on the perversity, It turns out that we can *characterize* Deligne's sheaf using the following four axioms:

- is constructible with respect to the given stratification.
- in .
- The stalk if and .
- The attachment map (given by ) is a quasi-isomorphism up to degree .

Our next goal is to rephrase axiom d as an axiom c' similar to c for the *dual* of and use Deligne's construction to prove the topological invariance of IC sheaves.

##
Operations on sheaves and duality

We would like to define a dualizing functor . We hope to satisfy the following properties. Suppose is a morphism. Then and Moreover, when is a point, we should have . Moreover, the natural transformation becomes a bi-duality when restricted the constructible sheaves .

It turns out there exists a dualizing sheaf such that (at least for constructible sheaves) It turns out that one can define to be the sheaf of locally finite -chains . In particular, when is a manifold, is quasi-isomorphic to .

Let

be a local system on

and

be the dual local system. Let

and

be complementary perversity (i.e.,

is the top perversity). Then

09/29/2015

##
Reformulation and topological invariance

Using the theorem that and the properties of the dualizing functor. We find the axiom d for can be formulated as a similar axiom on the *costalk* .

c'. if for .

In fact, by Remark 9 we have the exact sequence One sees that axiom d is equivalent to Let and , then Using the purity (since is a manifold), we have . Hence axiom d is equivalent to axiom c'.

Now suppose we are in the case of middle perversity. Let and We can (by shifting) translate the axioms c and c' to the axioms which don't mention the stratification:

c2. .

c2'. .

Notice that a, b still depend on the stratification . For example, when is a manifold and is the trivial stratification. Then the sheaves satisfying a, b, c2, c2' are of the form for local systems on . The shift by the complex dimension ensures that the dualizing functor preserves .

Given a local system

on an open

of real codimension at least 2, there exists a unique

such that

satisfying the axioms a, b, c2, c2' for any stratification

.

10/06/2015

The uniqueness follows easily from the following lemma.

Suppose

has codimension at least 2. If

is connected, then

is connected. Moreover, the natural map

is surjective. In particular, a local system on

has

*at most* one extension to

.

The connectedness of

follows from the long exact sequence in cohomology with compact support. The surjectivity follows from the connectedness and that any path can be deformed in codimension 2.

¡õ
We can use Deligne's construction to prove the existence of inductively. Suppose is a local system on . Then there exists a maximal open such that extends to . We define to be the maximal open of such that is a local system. Refining this idea, we obtain a coarser stratification by also requiring is also a local system. Iterating Deligne's construction, we obtain a stratification given by . The stratification thus obtained is the "coarsest" in some sense. This allows us to remove the dependence on the stratification and to show the topological invariance of IC sheaves.

##
Perverse sheaves

We now relax the axiom c2, c2' by allowing non-strict inequality. This modification is intended to include sheaves like for an IC sheaf on a closed (e.g., if ).

A sheaf

is

*perverse* if

and

The category of perverse sheaves is denoted by

.

- is an abelian category (the heart of perverse -structure on defined by the support and cosupport condition), stable under Verdier dual.
- All simple objects are DGM complexes.
- Every object in is a successive extension of simple objects.

A proper surjective morphism between two varieties

is

*small* (resp.

*semi-small*) if

(resp.

) for any

. In particular,

is a generically finite morphism. For example, when

, small is equivalent to finite. When

, small morphism can only have dimension 1 fibers above finitely many points. One can check that a morphism is semi-small if and only if

and small if and only

.

Small morphisms are compatible with IC sheaves.

If

is small and has generic degree 1, then

. In particular,

.

Consider the case

is smooth. Then

. So by Theorem

5 it suffices to check the support and cosupport axioms for

. By base change,

. If

, then

, i.e.,

. The support and cosupport axioms then follow exactly by the smallness assumption.

¡õ
10/08/2015

The above theorem can be generalized to the following:

Suppose

is small.

. Then

.

Suppose

is a blow-up along a point

with fiber

. Then

. This is no longer perverse and the extra terms

measures the failure of

being small.

In general we have the following decomposition theorem.

(Beilinson-Bernstein-Deligne + Gaber)
If

is proper and surjective, then in

we have

is a direct sum of DGM complexes shifted by certain degrees:

10/27/2015

(I was out of town and missed the two lectures on Oct 20 and 22. Thank Pak-Hin Lee for sending his notes to me. )

##
Springer fibers

Reference: PCMI 2015 lectures by Ngo, Yun and Zhu, transcription available on Tony Feng's website.

A important class of semi-small maps comes from holomorphic symplectic varieties.

A

*holomorphic symplectic variety* (over

) is a nonsingular variety such that there exists a closed holomorphic 2-form

satisfying that

is nonzero everywhere (i.e.,

is non-degenerate).

Let

be a nonsingular variety over

with

. Then the cotangent bundle

is naturally a holomorphic symplectic manifold. In fact, let

be the local coordinates on

and

be the dual basis of

in the cotangent space, then

is a non-degenerate closed 2-form.

(Symplectic resolution)
Let

be a proper surjective birational map. If

is holomorphic symplectic, then

is semi-small.

(Hilbert scheme of surfaces)
Let

be a surface. Then symmetric power

becomes singular when some of the

-points on

collide. However, the Hilbert scheme

of

-points on

(which remembers infinitesimal information of the collision as well) is nonsingular. Moreover, one can show that

is a symplectic resolution of

Notice when

is a 3-fold, both the

an

are singular.

Let be a semisimple connected linear group over . Fix a Borel subgroup . Let be the nilpotent cone. Then is singular.

Let

. Then

. Define the invariant map given by the coefficients of the characteristic polynomial of

:

. Then

is the preimage of

, i.e., the set of all traceless matrices

such that

. When

,

, which is simply a 2-dimensional cone, with a simple singularity at 0.

In general the nilpotent cone may have very bad singularities away from the regular nilpotent elements. Springer found a systematic way of resolving the singularities.

Let

be the flag variety, parametrizing all the Borel subgroups of

. Define

Then the natural map

is an isomorphism on the regular nilpotent elements. It turns out that the natural map

is a symplectic resolution, known as the

*Springer resolution*. In fact,

. The fiber

is called a

*Springer fiber*.

For

,

and

. The simple singularity of the 2-dimensional cone at 0 is resolved.

Using the fact that , one can check that is indeed semi-small. One can generalize the construction of and obtain a *small* map .

The

*Grothendieck-Springer fibration* is defined to be

The Springer fiber is always reduced. However, for the Grothendieck-Springer resolution , the fibers may be non-reduced. Therefore we have a commutative (but *not* Cartesian) diagram Explicitly, while

Let

. Let

be the complete flag associated to

. Then for the regular element

, the Springer fiber satisfies

while the Grothendieck-Springer fiber satisfies the weaker condition

. For example, when

is 2-dimensional,

. Let

, then

has more than one points: it has an obvious point corresponding to

. It also has an extra

because

. We can compute the scheme-theoretical fixed points of

on

: since

, we have

, hence

. Therefore

.

##
Affine Springer fibers

Let be a finite field. One can easily see that the number of -points in the Springer fiber is the same as fixed points of on the set , which can be rewritten as a simple orbital integral. Moreover, the finite set can be realized as the -rational points of the flag variety . Now we want to upgrade this to an "affine" version, i.e., for local fields of equal characteristic.

Let and . We want to geometrize the infinite set by an *affine Grassmannian* . The analogue of the full flag variety should be given by the *affine flag variety* whose points gives , where is the Iwahori subgroup.

In terms of moduli interpretation, when , is the set of -lattices in , where the identity coset corresponds to the standard lattice . is the set of chain of lattices in , where is length one -module such that .

Let

be a

-algebra. Let

be the ring of power series in

. Let

be the field of Laurent sereis. Due to the completion process,

is larger than the naive base change

when

is not finitely generated.

We define an

*-family of lattices* in

is a finitely generated projective

-submodule

such that

. This is equivalent to the data

, where

is a vector bundle over

of rank

and

is a trivialization of

over the punctured disk.

We define the functor . Then . has a reasonable geometric structure (though infinite dimensional).

is represented by an ind-scheme, i.e.,

, where each

is a projective scheme of finite type and each

is a closed immersion.

Here consists of -lattices (projective as -modules) such that Due to this boundness, it can be viewed as the set of quotient -modules of (projective as -modules). In other words, let (so ). Then is the set of quotient projective -modules of , such that the -action (given by a nilpotent operator ) satisfies . Observe that is nothing but a union of generalized version of Springer fibers : Here is a Grassmannian of one-step flags (instead of full flags).

For

, we see

. Let

be a regular nilpotent operator on

. When

, we have

In general

(compare:

).

10/29/2015

More generally,

(Affine Grassmannians)
Suppose

is algebraically closed. Let

be a group scheme over

. We define the functor

such that

is the set of pairs

, where

is a

-torsor over

and

is a trivialization

, where

is the trivial

-torsor. In other words,

is the moduli space of

-torsor together with a rigidification. This may remind you of the definition of a Rapoport-Zink space, which is the moduli space of certain

-divisible groups together with a rigidification.

Notice . So acts on by .

(Affine Springer fibers)
Let

, we define

to be the fixed points of

and

. It turns out that

is a closed subscheme of

.

We provide two more analogous constructions.

(Affine Schubert varieties)
By the Bruhat decomposition (over

)

The orbits of

on

are parametrized by elements in the Weyl group

. Let

be the invariant map. Define the orbit associated to

to be the

*Schubert variety*, so

These are locally closed subvariety of

(defined by incidence relations) . Then

sits in the Cartesian diagram

One can analogously define *affine Schubert varieties*. By the Cartan decomposition where consists of the dominant co-characters of , we have an invariant map Then for , define the affine Schubert variety by

(Affine Deligne-Lusztig varieties)
Assume

is finite. Define the

*Deligne-Lusztig variety* to be the subvariety of

given by

In other words, the Deligne-Lusztig variety sits in the Cartesian diagram

Deligne-Lusztig constructed all the irreducible representations of finite reductive groups in the cohomology (with local systems as coefficients) of Deligne-Lusztig varieties. The computation of the Deligne-Lusztig characters are naturally related to counting points of Springer fibers. Deligne-Lusztig varieties form one of the starting point of the geometric approach to representation theory initiated by Kazhdan and Lusztig.

One can then analogously define affine Deligne-Lusztig varieties using the point-wise condition. Affine Grassmannians and affine Deligne-Lusztig varieties are fundamental objects in geometric representation theory and in the study of local models of Shimura varieties.

11/05/2015

An alternative definition of affine Grassmannians uses loop spaces and arc spaces. Let be a field and . Let be a scheme. One would like to geometrize the sets and . We define the *loop space* functors Similarly we define the *arc space* (or *positive loop space*) functor These are presheaves under the fpqc topology.

When

, we have

. Therefore

is in fact represented by a scheme

, given by the leading coefficient

and rest of the coefficients

. The points

are more complicated: they are Laurent series of the form

When taking the reduced structure, we find that

is an infinite copies of

.

More generally, we have

- is represented by a scheme. It is affine if is affine.
- is represented by an ind-scheme.
- The affine Grassmannian (as the quotient sheave under the fpqc topology).

##
Orbital integrals

Suppose

is a local field. For

and

. Define the orbital integral

Notice that the convergence of this orbital integral is already an issue.

Let us consider the case

,

, and

is regular semisimple. In this case the centralizer

of

is the diagonal torus

. Since

is

-bi-invariant and

is compact, to show the convergence of the orbital integral, it suffices to show the convergence of

By the Iwasawa decomposition

where

is the unipotent radical of the Borel subgroup of

(i.e., group of upper triangular unipotent matrices for

), we know that the convergence is equivalent to

The key observation is that

and one easily compute

since each root group is an eigenvector under the adjoint action of the semisimple element

. For example, when

and

, we have

Take

, we find the integrand is nonzero only when the above matrix has

-entries. This means that

, hence

is bounded and the integral converges.

Ngo showed that the action of on is not faithful. The action factors through the quotient known as the *local Picard group*.

For

,

. Then

and

(i.e., the

-part acts trivially on

).

(Goresky-Kottwitz-MacPherson, Ngo)
Assume

is regular semisimple. Then

Namely, the

-points of the stacky quotient is in fact a stable orbital integral (i.e., a sum of orbital integrals over conjugacy classes

which are stably conjugate to

).

Instead of the loop group action, Kazhdan-Lusztig also considered the discrete action of the lattice on the affine Springer fiber.

(Kazhdan-Lusztig)
is proper and of finite type.

is a finite dimensional ind-scheme and is locally of finite type.

11/10/2015

##
Hitchin fibers

The quotient is a projective variety. It has singularities, but people still expect certain "purity" of its cohomology, which implies the fundamental lemma for regular semisimple elements in the maximal torus.

This purity is still unknown. To prove the fundamental lemma, one instead consider a global version of the affine Springer fibers. Suppose is a smooth projective curve over a finite field with . Consider . The global analogue of affine Grassmannian is , the moduli stack of rank vector bundles on . It represents the functor It has rational points

A pair

is called a

*Higgs bundle*. The Hitchin moduli space

is defined to be moduli space of Higgs bundles:

For the

*Higgs field* , define its invariants

. So

and

. We call the affine space

the

*Hitchin base*. We have the invariant map to the Hitchin base

More generally, we can replace

by any vector bundle

. The resulting moduli space of Higgs bundles

is called the

*-twisted Hitchin moduli space*. The affine space

is called the

*Hitchin base*.

When , then , which is a finite dimensional -subspace of . So one can view as a finite dimensional -subspace of an infinite dimensional -space . On the other hand, when varying (allowing more poles) these finite dimensional -subspaces will exhaust all elements of . More precisely, one can define a family version of the Hitchin base by considering Let be the open substack with . It turns out is the same as (the effective divisors of degree on ), hence is indeed a scheme. The complement of is isomorphic to (given by the zero section). More generally,

Define

The

*universal Hitchin base* is defined to be

a family of Hitchin bases over

.

The fibers of the invariant map (*Hitchin fibers*) are the global analogue of affine Springer fibers.

For

with characteristic polynomial

. Assume that

is elliptic (i.e.

is irreducible, equivalently

is an anisotropic torus). Then

11/12/2015

##
Spectral curves

Today we will discuss a bit more on the spectral curve mentioned last time. Starting next time we will do a concrete example: use the perverse continuation principle to prove Waldspurger's theorem for central values of -functions on in the function field setting.

Consider the total space of the line bundle

,

It is a

-fibration over

. This total space

sits in the projective bundle

(so

). We have two affine charts given by the two coordinates

and

Then

is given by the

. Let

We define the

*spectral curve* to be the zero locus of

.

Suppose

is a reduced curve. The

*compactified Picard (or Jacobian) stack* of

is defined to be the stack of torsion-free coherent sheaves of rank 1 on

. When

is smooth,

. By a theorem Altman-Iarrobino-Kleinman, for reduced curves with only

*planar singularities* (which by definition is satisfied by the spectral curves), the usual Picard scheme

is always open dense in

. Hence

is naturally a compactification of

. Notice that in general

may have singularities.

For a torsion-free coherent sheaf on , the pushforward under is a torsion-free sheaf of rank on the smooth curve , hence is indeed a vector bundle of rank . One can further construct an -linear endomorphism of with the given characteristic polynomial using the action of on . In this way one can describe a Hitchin fiber as the compactified Picard of the spectral curve.

For

, we have

.

11/17/2015

##
Waldspurger's formula via Jacquet's relative trace formula

Let be a quadratic extension of function fields, corresponding to an etale double cover of curves over a finite field . Let and be an anisotropic torus (with a fixed embedding ). *Waldspurger's formula* relates the toric automorphic period to central values of automorphic -functions on . We state a very special (unramified everywhere) case.

(Waldspurger)
Let

be an automorphic cuspidal representation of

that unramified everywhere. Let

, where

(so

is unique up to scaling). Then up to some explicit constants we have an equality

Now we use the well known procedure of relative trace formula to remove the dependence on the automorphic representations . Consider the distribution where and the kernel function is given by The kernel function has a spectral decomposition where runs over an orthonormal basis of level one cusp forms on . So we obtain the spectral decomposition where is the character determined by .

One can repeat the same story for the period on the anisotropic torus. Define Then similarly we have a spectral decomposition By the previous remark, Waldspurger's certainly implies the relative trace formula identity Conversely, using the linear independence of the automorphic representations, this identity is in fact also sufficient to prove Waldspurger's formula.

To prove this identity of two distributions, we use the geometric decomposition Notice the generic stabilizer is trivial and so the double integral is over and factors as a product of local orbital integrals. One has a similar geometric decomposition for .

We can parametrize the orbits and in a similar way. Consider the invariant map Then consists of exactly one orbit when . We call these *regular semisimple* and the corresponding orbital integral is automatically convergent (regularization process is needed for other ). Write It remains to compare the orbital integrals

##
Geometrization for the split torus

Let us ignore the quadratic character for the moment. So where . It is now convenient to lift the situation to and consider for , where is the diagonal torus in .

In order to geometrize this orbital integral, we define an analogue of Hitchin moduli space.

Let

such that

and

. Define the moduli space of pairs of rank two vector bundles together with a morphism:

where

are line bundles on

. For simplicity (since we only consider

*regular semisimple* orbits) we also impose the non degeneracy condition that

(which strictly speaking defines an open subset of

). Let

be the union of all such

's with

.

Now we define an analogue of the invariant map to the Hitchin base and an analogue of Hitchin fibers.

Let

be the the moduli space of pairs

, where

,

. Let

be the moduli space of triples

, where

. We have a natural map

given by

.

We have an invariant map , given by , and . Let be the fiber of this invariant map above .

Let

be an effective divisor on

of degree

. Let

(viewed as a

-subspace of

) be the fiber of

above

. Then

11/19/2015

Now sending a point in to defines a map By the non-degeneracy assumption on , this induces an isomorphism Now let the moduli space of triples such that . The we have a commutative diagram Here the right vertical map is induced by the addition map . In this ways the analogue of Hitchin moduli space becomes a simple construction using symmetric powers of the curve .

By the previous theorem, we would like to study Therefore we can forget about the orbital integrals and focus on the sheaf . At this stage one can also insert the character by taking a nontrivial local system on and then take . Here , is the local system on associated the the double cover and is the natural quotient map by .

Now it it remains to study the simpler object: where is the addition map. This is nothing but the push-forward of a local system under a finite map, a simplest example of a perverse sheaf (after shifting by the dimension).

- Since (the multiplicity free locus) is a Galois covering with Galois group , we know that is the middle extension (by the perverse continuation principle). Here and the local system on corresponds to the induced representation (of dimension ).
- To deal with the nontrivial coefficient, we need to go to the double covering to trivialize the local system. So we have a Galois covering which is Galois with Galois group . Here permutes in a natural way, in other words, is the wreath product . Let be the character that is nontrivial on the first factors and trivial on the last factors. The action of on has stabilizer exactly . Hence we can extend to . Then the local system on corresponds to the representation . It is irreducible of dimension (one check the irreducibility by computing the endormophism algebra to be a division algebra).

11/24/2015

##
Geometrization for the nonsplit torus

Today we will geometrize the distribution on the nonsplit torus as well and verify the identity for (at least for the regular semisimple orbits).

In order to geometrize the orbital integral , we define an analogue of the space .

Let

be an etale double cover. Define

to be the moduli space

, where

and the map

is an element of

where

is the nontrivial Galois involution. Since we only consider regular semisimple orbits, we further impose the non-degeneracy condition

where

.

Now sending a point in to and , we obtain an isomorphism (by an analogue of Hilbert 90) where the map is induced by the norm map We also have the invariant map induced by the norm map :

Analogous to Theorem 19, we have

Let

be an effective divisor on

of degree

. Let

(viewed as a

-subspace of

) be the fiber of

above

. Then

Similarly to the split case, we are now interested in the sheaf , where . When , is smooth, is a projective bundle and the norm map is smooth with kernel a Prym variety of dimension . Therefore is in fact smooth and hence is perverse (after shifting by the dimension). The local system underlying is the induced representation .

##
Orbital integral identity for regular semisimple orbits

Now we have tow invariant maps with a common base

The identity now becomes a statement purely about two perverse sheaves.

There is an isomorphism between perverse sheaves

This will follow from the even stronger claim.

By the perverse continuation principle, the proof of this theorem essentially boils down to representation theory of finite groups because the local system underlying both perverse sheaves have finite monodromy (trivialized after a finite covering). Namely,

This is much simpler statement to prove! Notice that both sides have dimension . By Frobenius reciprocity, it remains to show that there is a -equivariant embedding which can be explicitly written down.

12/01/2015

##
Orbital integral identity for non regular semisimple orbits

Now consider the case of non regular semisimple orbits, i.e., when the invariant . Let us only consider the case . The case corresponds to three -orbits, the identity orbit and two unipotent orbits represented by and . The case corresponds to one -orbit: the identity orbit (i.e., ) under the decomposition .

Now let us consider the moduli spaces for the non regular semisimple orbits. The moduli space for the nonsplit torus is again simpler. Let be the space as in Definition 22 but only requiring that are not zero simultaneously, i.e., . Our old nondegenerate moduli space is thus an open . By definition we have where is the closed locus where . Since , we have Now consider the invariant map When is sufficiently large, is smooth and is proper. By the same logic for the regular semisimple orbits, it remains to consider the norm map and check if is still perverse. Its restriction on is given by the norm map , whose fiber is certainly not finite (the kernel is the Prym variety of dimension ). But when is sufficiently large, this map is still *small*. In fact, the smallness in this case means , i.e, . Now by the perverse continuation principle for small maps, still decomposes as IC sheaves associated to the earlier finite group representation .

Now consider the moduli space for the split torus. The situation is slightly more complicated. In this case , which has infinitely many components (when , since there is only the zero section for a line bundle of negative degree).

Notice the the identity orbit gives no contribution to the orbital integral since we are inserting the nontrivial quadratic character . So we require the four sections in has at most one zero, which corresponds to the two unipotent orbits and . We impose further assumptions that if ; if ; if and if .

By these further assumptions if is nonempty, then . Again is smooth when is sufficiently larger and is proper. For a point in . Assume (so ), the fiber at is then The second term is finite (since the addition map is finite). Therefore the fiber has dimension . From this one can see that is no longer small:

Even though is no longer small, we can check that the sheaf in question still satisfies the strict support condition in Deligne's uniqueness principle.

Let

. Then

.

Notice

, where

from the finite part

. The claim follows from the fact that

By Remark

44, the right hand side has only one possibly nonzero term

, which becomes zero when

.

¡õ
Hence the orbital integral identity for non regular semisimple orbits follows by same finite group representations identity (Theorem 23)!

12/03/2015

##
Moduli spaces of shtukas

In the final part of the course, we are going to generalize the previous trace formula identity to *higher* derivatives. For this we need to introduce the moduli space of shtukas. We begin with a rather general construction.

Consider

defined over a finite field

(usually a certain moduli space). Suppose

is a correspondence. We define the the moduli space of shutaks associated to

to be the fiber product

More generally, suppose there are

correspondences

, we define

to be the moduli space of shtukas associated to the convolution correspondence

Consider

. For

, we define the Hecke stack

to be the moduli space of arrows

of vector bundles of rank

such that

. Similarly define

using the condition that

.

Define

to be the Hecke stack of

*upper (increasing) modifications*, i.e.,

(over

) is an injection such that

is a line bundle on the graph of a marked point

. Similarly define

to be the Hecke stack of

*lower (decreasing) modifications*.

We have two natural projections and also a natural map given by the location of modification.

Both

are representable and proper. When

, both have relative dimension

(Here 1 comes from the choice the location of modification and

comes from the choice of the modification with at a fixed location, i.e., a line in an

-dimensional vector space).

Let

be an even integer. Let

be a

-tuple of signs. Let

be the moduli space of shtukas associated to the convolution of

. In other words, an

-point of

corresponds to a

-tuple of modification of vector bundles

such that

.

Recall that (Remark 28) itself is only an Artin stack (which has a lot of automorphism). The moduli of shtukas has better properties.

(Drinfeld , Varshavsky in general)
- is a Deligne-Mumford stack, locally of finite type.
- The projection map is separated, smooth of relative dimension (in fact, an -iterated -bundle).

When

, the Hecke stack simply consists of isomorphisms

. So

consists of vector bundles on

such that

, which must come from pullback of vector bundles on

itself. Hence

is the discrete group

. This exactly puts us in the earlier situation of Waldspurger's formula when

. From this point of view, the study of automorphic forms (over function fields) is nothing but the study of degree 0 cohomology of the moduli of shtukas with

marked points.

When

, we have

given by the first line bundle

and the location of modification

. So we have the fiber diagram

Here the right vertical arrow is given by

. In particular, considering the degree zero part (and rotating the previous diagram) we obtain the fiber diagram

The right vertical arrow is exactly Lang's isogeny, whose kernel is the class group

of the function field

. This is a generalization of unramified geometric class field theory: when

, the etale map

has Galois group the class group

and realizes the Hilbert class field of

geometrically.

Next time we will introduce the Hecke algebra action on the moduli of shtukas and see how the equality of *higher* derivatives of -functions and intersection numbers of certain cycles on the moduli of shtukas becomes a refined structure on the perverse sheaves we constructed using Hitchin moduli spaces.

12/08/2015

##
Heegner-Drinfeld cycles and higher derivatives

Let , with an embedding . Fix a -tuple of signs . We have an induced morphism where is the etale double cover. This induces a map of moduli of shtukas and we have commutative diagram Notice the right vertical arrow has relative dimension , whereas left right vertical arrow has relative dimension 0 (generically etale with Galois group the class group). Though is not of finite type due to the instability, we can still talk about intersection number since is a *proper* smooth Deligne-Mumford stack (at least after dividing by ).

Now let us define Hecke correspondence on .

Let

and

be two points in

. We define a degree

Hecke correspondence to be the collection of injections

such that

and the natural diagram

commutes. The stack of such degree

Hecke correspondences on

is denoted by

.

We define a Hecke correspondence version of

by taking the fiber product

Then is indeed a correspondence on and thus defines a compactly supported cycle class of dimension In particular, acts on . One can similarly define a more refined correspondence for any effective divisor . Recall the spherical Hecke algebra is generated by , where runs over all effective divisors.

The map

is a ring homomorphism.

The map

induces a map

(like Heegner points are imaginary quadratic points of modular curves). We define the

*Heegner-Drinfeld cycle* to be the direct image of

in

under

.

Let

. Then

Here

is the intersection number of the Heegner-Drinfeld cycles

and

The ride hand side essentially corresponds to . After spectral decomposition it follows that the intersection number of the -isotypic component of the Heegner-Drinfeld cycle (turns out to be independent of the choice of ) is essentially the -th derivatives at the center. More precisely, even though that we don't yet know the action of on the entire Chow group is automorphic, we can consider the subspace of the Chow group generated by the Heegner-Drinfeld cycle. Let be its quotient by the kernel of the intersection pairing.

We have

.

It then makes sense to talk about the -isotypic component and using the Theorem 26 one can show that

For

an everywhere unramified cuspidal automorphic representation of

, we have up to a simpler factor

##
Orbital integral identity for higher derivatives

Our remaining goal is to prove that for sufficiently large, we have Notice the intersection number in question is given by the degree of the 0-dimensional scheme (in the proper intersection case) of the fiber product

The key observation is that this fiber product can be viewed in an alternative way involving the Hitchin moduli space . Look at the following commutative diagram:

Here all the vertical upward arrows are given by ). The bottom row shows the fiber products of the three columns and the right column shows the fiber product of the three rows. The intersection in question is the fiber product of the bottom row, which should also equal to the fiber product the right column! (Of course this needs extra work to check after defining the intersection number in the right way, like the change of order of integration). We denote this common fiber product by , which is a Hitchin version of moduli of shutaks. One can further decompose into pieces, i.e., the convolution of (consisting of only *2 by 2* diagram).

We have the following general Lefschetz trace formula for computing the intersection of a correspondence with the graph of the Frobenius morphism.

(Lefschetz trace formula)
Let be the invariant map to the Hitchin base. Notice a correspondence over defines an endomorphism . One can refine the Lefschetz trace formula relative to (take ):

This reduces the intersection number of Heegner-Drinfeld cycles to the study of the action of on the cohomology the Hitchin moduli spaces, which one can then compare to the -th derivative of the orbital integral on the split torus!

12/10/2015

Therefore,

Recall that where is a perverse sheaf on with generic rank . Now the final key thing is that each such perverse sheaf is an *Hecke eigensheaf* whose eigenvalue exactly matching up the extra factor in .

acts on

by the constant

.