## 1 Introduction, and the result

Let *g*. Let *X*, *S*, *N*.
Fix a prime *F* be the function field of
*X* over *F* unramified at each
closed point of *F*.
Fix an algebraic closure

The work [4] computes, and attempts to interpret, the
number *T* of isomorphism classes of the *n*-dimensional irreducible
*n*) unipotent monodromy at
each point of *S* (equivalently: the number of isomorphism classes of
rank *n* irreducible local *n* irreducible local

To compute *T*, using the automorphic-Galois reciprocity law in the function
field case ([13]) we relate *T* to the number of cuspidal representations of
*F* of *X* over *S*, where the component is Steinberg twisted by an unramified
character.

This number is related – using the correspondence from *F* which remains a
division algebra at the places of *S*, and is split elsewhere. In this
compact quotient case the trace formula can be used to carry out the computation.

The trace formula used in [4, Section 4] is based on a completely new, categorical approach, based on the notion of the mass of a category. But so far it has been developed only to count the cardinality of interest in [4, Section 4]. It is not yet clear if it can be extended to the non-compact quotient case or to deal with Hecke correspondences.

The aim of the present work is to compute the same number using the usual trace formula in the compact quotient case. As stated in [4] following Lemma 4.9, the equivalence between these two approaches seems to us interesting.

To understand *T*, [4, Proposition 6.28] shows that there exists a
virtual motive on the moduli stack *X* and disjoint sets *X*, such that *T* is the trace of the Frobenius on this virtual motive.

This leads to an understanding of how *T* varies with the base field

The fact that the cardinality *N* of *S* is at least 2 permits us to use a
simple – compact quotient – form of the trace formula. When *N* in [9]. The remaining cases being

The case of even *N* is reduced in [4, Section 5] to the
case dealt with here, which is the case of [4, Section 4].

The contents of this paper are as follows. We state our counting results in
this section. In Section 2 we recall what is meant by monodromy and examine the
behavior by extensions

The analytic part of Sections 3–7 can be read independently, it is just the motor behind the complete machine. For the full picture one needs the arithmetic-geometric avatar of Section 2. See [4, Sections 2 and 6].

To state our results, denote by

The fixed algebraic closure

Thus *t* such that

Write *X*. Let *F* be the function
field of *X* over *v*
(the residue field of *n* prime to *k* is the product of *r* distinct primes,

*The number of the isomorphism classes of the irreducible rank n local *

*where*

For example, suppose *X* is *S* consists of

If *S* consists of one point of degree 2 and two points of degree 1, then

If *S* consists of two
points of degree 2 each, then

If *S* consists of two points, of degrees 1 and 3, then

When

If *S* consists of two points of degree 1 (any

Let *F*. Let

Fix an idèle *F*. Put

A representation

is invariant under the Frobenius precisely when it extends to a representation,
say ρ, of the arithmetic fundamental group *F* in a fixed algebraic closure *S*. For any α in

which corresponds by CFT to a fixed idèle of degree 1 in
*v* in *S* in Theorem 1.1, as long as *S* is not empty. Any other
representation which restricts to

The number of *n*, with principal unipotent
monodromy at each

there are *n* possibilities for the character χ.
The representations *n* times that of Theorem 1.1.

Note that the automorphic counterpart, [4, Theorem 1.11 (ii)], asserts that if π is a
cuspidal automorphic *S* of closed points of *X*, or places of *F*, such that for each

We shall now rewrite the analytic expression *T* of Theorem 1.1 in a geometric
form. First we note that if *m*th roots of 1, we have that

is 0 if *m* if *S* by the multiset

Further, write *B* for the multiset consisting of the eigenvalues *B* is the multiset of eigenvalues of
the Frobenius

*We have*

*where*

Indeed, the first product over β is
*m* does not divide
*j* of

Proposition 6.28 of [4] concludes from this the following.

*The cardinality T is equal to the trace of the Frobenius *

## 2 Arithmetic perspective

To recall what we mean by monodromy, prepare to relating the arithmetic expression of Theorem 1.1 to analysis, and review the study of variation by base change in [4, Section 6], we leisurely review the objects under consideration, in relatively classical language.

The Galois group *q*
elements is isomorphic to the projective limit

The arithmetic Frobenius automorphism

Let *inertia subgroup*. The pullback of *Weil group*

of topological groups, where *G* by the closure of its commutator subgroup. Local class field theory
creates a canonical isomorphism between

Let *F* be a global field of characteristic *Weil group* of *F* and is denoted by *W*, as we fixed *F* and its
separable closure *W* is defined as in the local case. Denote by *F*, by *F* at a
place *v*. There is a natural class of embeddings of *W*, differing from each other by inner automorphisms of *W*.
Thus there is a natural homomorphism *v*,

coincides with

Denote by *X* the smooth projective model of the field *F*. Thus
*X* is a smooth projective absolutely irreducible curve over *F* coincides with the set *X*. Let *S* be a finite set of closed points of *X*.
Let *F* in *F*, which is unramified at each *v* in *v* in

For a geometric definition of *v* of *X* as
the scheme *v*, i.e. it is

Denote by *W* and *X*, or classes
of divisors on *X*, or the idèle classes
*D* is
denoted *v* in *v* with multiplicity one.
Unramified Class Field Theory (CFT) asserts that there is a unique
isomorphism *v* in

with

An *adic representation* of a topological group *G* is a
representation of *G* in a finite-dimensional vector space *V* over
*V*, the homomorphism *E* is a finite extension of

Class Field Theory establishes a natural bijection between one-dimensional representations
of

The automorphic-Galois global reciprocity theorem, proven by Lafforgue [13],
asserts, for all prime *n*-dimensional *S*,
such that the *n*-tuple of the roots of *n*-tuple of
Satake parameters of the unramified component *v* in *S*. We do not elaborate on this local correspondence,
but we now review the structure of the

There exists an epimorphism

Grothendieck showed that for an *N* is nilpotent and *U* which is a pro-

thus *N* is invariant under multiplication by *N* is nilpotent. We name *N* the logarithm
of the unipotent part of the monodromy of ρ.

If ρ is irreducible, then its kernel is open. Indeed, from
*N* is 0.
But *N* is nilpotent, hence

A *Weil–Deligne representation* (WD-representation) is a triple *V* is a finite-dimensional vector space over *N* is nilpotent and called the monodromy of ρ. Weil–Deligne representations
have the advantage of being defined algebraically, over any field, not equipped
with topology.

To associate a WD-representation to an *N* is the monodromy of ρ. The exp term
makes sense since *N* is nilpotent. Then

A WD-representation *semisimple*
if *semisimple* if the associated

The example which interests us here is

and

We express our computation in terms of the rational function *X*:

where

(see [16, Chapter VII, Section 6, Theorem 4]). It satisfies the functional equation

where *X*.
This implies the relation

since *q*,
are bounded by

It is dominated by

We would like to study the behavior of the number *q* suggest the appearance of eigenvalues of the Frobenius
on a compatible system of *n* prime to *j* increases. Consider each point of *S* as a
*X*. Then *S* can be
viewed as a set of *X*, *M* being independent of the base field

*Unless *

*r*-tuples

*n*-dimensional

*n*) monodromy at each point of

*S*, is

*For each i, *

## 3 Analytic perspective

The strategy of proof is to transform the arithmetic question to an analytic
one, concerning automorphic representations. Indeed, by the automorphic-Galois
global reciprocity law ([13]) we need to count the cuspidal representations
*F*, with Steinberg (twisted by an unramified character)
component at each

Let *D* be a central division algebra over *F* of rank *n* which is split at
each *n* is even, such a *D* exists provided *S* has even cardinality.

The Deligne–Kazhdan (or inner-forms) correspondence (for the main ideas,
in the number fields case, see [1, Section 25] (for the trace formula) and
[6, Section III] (for the correspondence), and [7] when there is a cuspidal
component; the case

The underlying *F*-group *G* is defined by the multiplicative group *D*. Put

Let *D* over *F* the *x* in

is a maximal compact open subgroup of

Consider the space

Since the group *G* is anisotropic (mod center), the quotient *automorphic representations*. The multiplicities are known to be one by the
Deligne–Kazhdan correspondence of these automorphic representations from
this group to *G* has no
proper *F*-parabolic subgroup, these

An irreducible representation π of *v* ranges over *v* – that is,
except for finitely many –

Our aim is then to compute the number of everywhere unramified
automorphic representations of

*Let N be the number of places in S. Suppose *

*as described in
Theorem 1.1.*

The number *n*. This is the number of *n*-dimensional ρ of

The unramified one-dimensional automorphic representations π of

Had we fixed a central character ω, the number of all π, one-dimensional or not, without taking orbits under twisting, is *n*, and divided
by

The number of one-dimensional unramified automorphic representations
of *n*-torsion subgroup *n* we get

Our proof of Theorem 3.1 is relatively elementary. It is independent of the
automorphic-Galois reciprocity law, and of the correspondence from

Note that each local component of the representations of the theorem, viewed
as representations of

Up to a scalar multiple, there is one nonzero *K*-fixed vector in
an unramified automorphic representation. So the number of unramified
automorphic representations whose central character is 1 at α,
is the cardinality of *X*,
a set with a structure of an Artin stack. Our technique relies on an evaluation
of the trace formula, but all we do is simply count the indicated cardinality.
However, the technique of the trace formula extends to deal with the cases

## 4 The trace formula

Our proof of Theorem 3.1 is based on an application of the trace formula.
In fact, we simply need to compute the cardinality of the finite set

So, choose a Haar measure *v* and *v*. Thus

is of finite rank hence
has a trace, which is equal to the sum over all irreducible π
in

Hence the trace is equal to the integral of the kernel over the diagonal, thus to

Since *G* is anisotropic (mod center), the elements of *F* of degree *n*. The centralizer
*m* with *F*-torus

plus

This is equal to the spectral side of the trace formula, which is
*v*. For the resulting function
*f*. We now evaluate it.

## 5 Scalars

Recall that *G* with GL(*n*) (over a separable algebraic
closure *F*) defines a differential form, also denoted ω, on
GL(*n*), which defines a Haar measure, also denoted *v*. The volumes of *G* and GL(*n*) are isomorphic over *F*. Thus we have

where

In this section we prove:

*The value of the contribution from the part of the geometric
side of the trace formula corresponding to the central elements is*

Consider the sum over γ in

To compute the volume of

The second factor here is equal
to

we conclude that

where

Note that

The *F*-group *G*
be an algebraic group over *F* of dimension *m*. Fix an invariant
*m*-form ω on *G*. For each closed point *v* in *v* in *G* is the generic fiber of a group scheme *X*. Thus for almost all
*v* in *G*
changes only a finite number of

Recall that the Tamagawa number of

Here *g* of *X*. Hence

Hence

Since

we obtain

Next we use the functional equation

for the ζ-function, at the values

As

and the value of the contribution from the part of the geometric side of the trace formula corresponding to the central elements is as asserted. ∎

## 6 Volumes

We still need to compute the factors

*Suppose *

*is equal to
*

Here *v*. Let *E*.

The cyclic algebra *D* can be realized as the algebra of matrices

thus

Then

thus

Superscript

*We have*

To see this, we count the number of entries in the *i*th entry in the first row – is

Then we count the number of second rows

which are not a multiple of the first row by *i*.

The entries

We need the reduced norm, which is in

This

To prove the proposition, we need to know the cardinality of *p*-Sylow subgroup of *p*-Sylow can be seen by arguing that any *p*-subgroup consists of
unipotents (if *x* is unipotent (*p*-group, hence maximal. Alternatively, by Bruhat decomposition
*q* otherwise.
Hence *q*. But *q*, so *p*-part of

To compute

We can now use a theorem of Gopal Prasad ([15, Theorem 2.3]). It asserts that the
quotient of *U* is the unipotent upper triangular subgroup
of SL(*n*)) by

## 7 Elliptic terms

The nonscalar elements are elliptic since *D* is a division algebra.
The contribution from these elements is the sum over the conjugacy classes
in *F* of degree *m*
dividing *n*. Fix a subfield *D*.

*If *

*m*divides

*n*which is prime to all

If *n* and

This γ lies in *D* if and only if *m* is prime to the residual exponent

Two

The sum then ranges over all *m* dividing *k* which is a
product of *r* distinct primes, and 0 elsewhere (see [14, Chapter V, Example 22],
where the word “monic” is missing in the definition of

We now compute the integrals

*When *

*S*-components, and

We have

*For *

For

For general γ in *m* over *F*. Note that

*If there are N places *

The volume

The last factor is

The factor *q* by

As for

This multiplied by

At the places *d*
over *m* over

For the last equality note that either *n* is even, and so is *N*,
or *n* is odd, so is then *d*, and

This work is based on preliminary notes to [4] that were not included in the published version of [4]. It was written up after the author gave a series of lectures on the subject at the University of Tokyo, at the kind invitation of Takayuki Oda. Warm thanks also to David Kazhdan, Dipendra Prasad, Eric Opdan, Elmar Große-Klönne, Thomas Zink for invitations to discuss this work at the Hebrew University, TIFR, University of Amsterdam, Humboldt-Universität zu Berlin, Universität Bielefeld, to Sasha Beilinson, Pierre Deligne, Vladimir Drinfeld, Maxim Kontsevich, Gopal Prasad for useful conversations and correspondence, and to the referee for careful reading.

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