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)
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.
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
Given a perversity
, we define a
to be -allowable
We then define the intersection homology
by considering only the
-allowable locally finite
is the middle perversity,
is often written as
for short. Similarly one can define the intersection homology with compact support
by considering only the
- 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.
. 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.
be a perversity. Then in the bounded derived category
, we have
is the usual truncation functor:
. Inductively, let
- 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 .
be a local system on
be the dual local system. Let
be complementary perversity (i.e.,
is the top perversity). Then
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:
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
satisfying the axioms a, b, c2, c2' for any stratification
The uniqueness follows easily from the following lemma.
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.
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.
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 ).
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
) 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.
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,
. The support and cosupport axioms then follow exactly by the smallness assumption.
The above theorem can be generalized to the following:
is a blow-up along a point
. This is no longer perverse and the extra terms
measures the failure of
In general we have the following decomposition theorem.
(Beilinson-Bernstein-Deligne + Gaber)
is proper and surjective, then in
is a direct sum of DGM complexes shifted by certain degrees:
(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. )
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
) is a nonsingular variety such that there exists a closed holomorphic 2-form
is nonzero everywhere (i.e.,
be a nonsingular variety over
. Then the cotangent bundle
is naturally a holomorphic symplectic manifold. In fact, let
be the local coordinates on
be the dual basis of
in the cotangent space, then
is a non-degenerate closed 2-form.
be a proper surjective birational map. If
is holomorphic symplectic, then
(Hilbert scheme of surfaces)
be a surface. Then symmetric power
becomes singular when some of the
collide. However, the Hilbert scheme
(which remembers infinitesimal information of the collision as well) is nonsingular. Moreover, one can show that
is a symplectic resolution of
is a 3-fold, both the
Let be a semisimple connected linear group over . Fix a Borel subgroup . Let be the nilpotent cone. Then is singular.
. Define the invariant map given by the coefficients of the characteristic polynomial of
is the preimage of
, i.e., the set of all traceless matrices
, 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.
be the flag variety, parametrizing all the Borel subgroups of
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
. 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
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
has more than one points: it has an obvious point corresponding to
. It also has an extra
. We can compute the scheme-theoretical fixed points of
, we have
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 .
be the ring of power series in
be the field of Laurent sereis. Due to the completion process,
is larger than the naive base change
is not finitely generated.
We define an -family of lattices
is a finitely generated projective
. This is equivalent to the data
is a vector bundle over
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).
, we see
be a regular nilpotent operator on
, we have
is algebraically closed. Let
be a group scheme over
. We define the functor
is the set of pairs
is a trivialization
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)
, we define
to be the fixed points of
. 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
are parametrized by elements in the Weyl group
be the invariant map. Define the orbit associated to
to be the Schubert variety
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)
is finite. Define the Deligne-Lusztig variety
to be the subvariety of
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.
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.
, we have
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).
is a local field. For
. Define the orbital integral
Notice that the convergence of this orbital integral is already an issue.
Let us consider the case
is regular semisimple. In this case the centralizer
is the diagonal torus
is compact, to show the convergence of the orbital integral, it suffices to show the convergence of
By the Iwasawa decomposition
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
, we have
, we find the integrand is nonzero only when the above matrix has
-entries. This means that
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.
-part acts trivially on
is regular semisimple. Then
-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.
is proper and of finite type.
is a finite dimensional ind-scheme and is locally of finite type.
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
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
. 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,
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.
with characteristic polynomial
. Assume that
is elliptic (i.e.
is irreducible, equivalently
is an anisotropic torus). Then
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
. This total space
sits in the projective bundle
). We have two affine charts given by the two coordinates
is given by the
We define the spectral curve
to be the zero locus of
is a reduced curve. The compactified Picard (or Jacobian) stack
is defined to be the stack of torsion-free coherent sheaves of rank 1 on
. 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
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.
, we have
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.
be an automorphic cuspidal representation of
that unramified everywhere. Let
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.
. Define the moduli space of pairs of rank two vector bundles together with a morphism:
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
be the union of all such
Now we define an analogue of the invariant map to the Hitchin base and an analogue of Hitchin fibers.
be the the moduli space of pairs
be the moduli space of triples
. We have a natural map
We have an invariant map , given by , and . Let be the fiber of this invariant map above .
be an effective divisor on
(viewed as a
) be the fiber of
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).
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 .
be an etale double cover. Define
to be the moduli space
and the map
is an element of
is the nontrivial Galois involution. Since we only consider regular semisimple orbits, we further impose the non-degeneracy condition
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
be an effective divisor on
(viewed as a
) be the fiber of
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.
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.
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
Hence the orbital integral identity for non regular semisimple orbits follows by same finite group representations identity (Theorem 23)!
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.
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
, we define
to be the moduli space of shtukas associated to the convolution correspondence
, we define the Hecke stack
to be the moduli space of arrows
of vector bundles of rank
. Similarly define
using the condition that
to be the Hecke stack of upper (increasing) modifications
) 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.
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).
be an even integer. Let
-tuple of signs. Let
be the moduli space of shtukas associated to the convolution of
. In other words, an
corresponds to a
-tuple of modification of vector bundles
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).
, the Hecke stack simply consists of isomorphisms
consists of vector bundles on
, which must come from pullback of vector bundles on
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
, 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
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.
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 .
be two points in
. We define a degree
Hecke correspondence to be the collection of injections
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.
is a ring homomorphism.
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
is the intersection number of the Heegner-Drinfeld cycles
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.
It then makes sense to talk about the -isotypic component and using the Theorem 26 one can show that
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!
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 .
by the constant