On special zeros of $p$-adic $L$-functions of Hilbert modular forms

0.49 MB·English

by  Michael Spiess

ON SPECIAL ZEROS OF p-ADIC L-FUNCTIONS OF
HILBERT MODULAR FORMS
2
BY MICHAEL SPIESS
1
0
2

l
u
J Abstract. LetEbeamodularellipticcurveoveratotallyrealnumber

0 field F. We prove the weak exceptional zero conjecture which links a
1 (higher) derivative of the p-adic L-function attached to E to certain p-
adicperiodsattachedtothecorrespondingHilbertmodularformatthe

] places above p where E has split multiplicative reduction. Under some
T
mildrestrictionsonpandtheconductorofE wededucetheexceptional
N
zero conjecture in the strong form (i.e. where the automorphic p-adic
h. periods are replaced by the L-invariants of E defined in terms of Tate
t periods) from a special case proved earlier by Mok. Crucial for our
a
method is a new construction of the p-adic L-function of E in terms of
m
local data.
[

1
v Contents
9
8 Introduction 2
2
2 1. Generalities on distributions and measures 6
.
7
1.1. Distributions and measures 6
0
2 1.2. p-adic measures 8
1
: 2. Local distributions attached to ordinary representations 8
v
Xi 2.1. Gauss sums 8
r 2.2. Ordinary representations of PGL (F) 10
a 2
2.3. Universal tamely ramified representations of PGL (F) 11
2
2.4. Distributions attached to elements of Cα(F,M) 12
2.5. Local distributions 13
2.6. Extensions of the Steinberg representation 14
2.7. Semi-local theory 16
3. Special zeros of p-adic L-functions 17
3.1. Rings of functions on ideles and adeles 18
3.2. Computation of ∂((log ◦N)k) for k = 0,...,r 20
p
3.3. p-adic L-functions attached to cohomology classes 30
2000 Mathematics Subject Classification. Primary: 11F41, 11F67, 11F70; Secondary:
11G40.
Key words and phrases. p-adicL-functions, Hilbert modular forms, p-adicperiods.
1
2 BYMICHAELSPIESS
3.4. Integral cohomology classes 31
3.5. Another construction of distributions on G 33
p
4. p-adic L-functions of Hilbert modular forms 37
4.1. p-ordinary cuspidal automorphic representations 37
4.2. Adelic Hilbert modular forms 38
4.3. Hecke Algebra 39
4.4. Cohomology of GL (F) 41
2
4.5. Eichler-Shimura map 45
4.6. p-adic measures attached to Hilbert modular forms 46
5. Exceptional zero conjecture 48
5.1. Automorphic L-invariants 48
5.2. Main results 51
References 53
Introduction
Let E be a modular elliptic curve over a totally real number field F
and let p be a prime number and such that E has either good ordinary
or multiplicative reduction at all places p above p. Attached to E are the
(Hasse-Weil) L-function L(E,s) (a function in the complex variable s) and
a p-adic L-function L (E,s) (here s ∈ Z ). Both are linked by the interpo-
p p
lation property which expresses the p-adic measure associated to L (E,s) in
p
terms of twisted special L-values L(E,χ,1). A special case is the formula
L (E,0) = e(α ,1) ·L(E,1).
p p
p|p
Y
Here e(α ,1) is certain Euler factor defined in terms of the reduction of E
p
at p (see Prop. 4.10 for its definition). It is = 0 if and only if E has split
multiplicative reduction at p. Let S be the set of primes p of F above p
1
where E has split multiplicative reduction, let S be the set of all primes
p
above p and let S = S −S . Thus we have L (E,0) = 0 if S 6= ∅. In [Hi]
2 p 1 p 1
it has been conjectured that
(1) ord L (E,s) ≥ r : = ♯(S );
s=0 p 1
dr
(2) L (E,s)| = r! L (E) · e(α ,1) · L(E,1).
dsr p s=0 p p
pY∈S1 pY∈S2
Here the L-invariant L (E) is defined as L (E) = ℓ (q ))/o (q )
p p p E/Fp p E/Fp
where q is the Tate period of E/F , ℓ = log ◦N and o =
E/Fp p p p Fp/Qp p
ord ◦N .
p Fp/Qp
In this paper we prove (1) unconditionally and (2) under the following
assumptions (see Theorem 5.10): (i) p ≥ 5 is unramified in F; (ii) E has
ON SPECIAL ZEROS OF p-ADIC L-FUNCTIONS OF HILBERT MODULAR FORMS 3
multiplicative reduction at a prime q ∤ p, or r is odd, or the sign w(E) of
the functional equation for L(E,s) (i.e. the root number of E) is = −1.
The statements (1) and (2) are known as exceptional zero conjecture. In
the case F = Q it was formulated by Mazur, Tate and Teitelbaum [MTT]
and proved by Greenberg and Stevens [GS] and independently by Kato,
Kurihara and Tsuji. In the case r = 1, (2) was proved by Mok [Mo] under
the assumption (i), by extending the method of [GS].
Toexplain ourprooflet π betheautomorphicrepresentation of GL (A )
2 F
associated to E. Thus π has trivial central character and the local factor
π is discrete of weight 2 at all archimedean places v. The Hasse-Weil
v
L-function of E is then equal to the automorphic L-function L(s − 1,π).
2
Moreover L (E,s) is solely defined in terms of π (thus we write L (s,π) for
p p
L (E,s)).
p
In section 5.1 we shall introduce a second type of L-invariant L (π). It
p
is defined in terms of the cohomology of (S -)arithmetic groups. We show
p
that L (π) does not change under certain quadratic twists, i.e. we have
p
L (π ⊗ χ) = L (π) for any quadratic character χ of the idele class group
p p
I/F∗ of F such that the local components χ of χ at infinite places and at
v
v = p are trivial. We prove an analogue of (2) above (unconditionally) with
thearithmetic L-invariants L (E) replaced bytheautomorphic L-invariants
p
L (π), i.e. we show
p
dr
(3) L (s,π)| = r! L (π) · e(α ,1) · L(1,π).
dsr p s=0 p p 2
pY∈S1 pY∈S2
InthecaseF = QtheseL-invariantshavebeenintroducedbyDarmon([Da],
section 3.2). He showed thatthey areinvariant undertwists and alsoproved
(3). Also if the narrow class number of F is = 1 a different construction of
L (π) has been given in [Gr].
p
To deduce (2) from (3) it is therefore enough to show L (π) = L (E) for
p p
all p ∈ S . In future work [GIS] we plan to give an unconditional proof of it
1
(and thus of (2)) by comparing L (π) to the (similarly defined) L-invariant
p
of an automorphic representation π′ of a totally definite quaternion algebra
– which corresponds to π under Jacquet-Langlands functoriality – and by
using p-adic uniformization of Shimura curves (compare also [BDI] where a
similar proof has been given in the case F = Q under certain assumptions
on π).
However if p satisfies the conditions (i) above and E satisfies (ii) then we
can deduce the equality L (π) = L (E) for fixed p ∈ S by comparing the
p p 1
formulas(2)and(3)inthecaser = 1forcertainquadratictwistsofE andπ.
More precisely, by a result of Waldspurger [Wa], we can choose a quadratic
character χ such that the arithmetic and automorphic L-invariants at p do
not change under twisting with χ, L(1,π⊗χ) does not vanish and p is the
2
onlyplaceabovepwherethetwistedellipticcurveE hassplitmultiplicative
χ
reduction. Then by Mok’s result and (3) we can express both L (E) and
p
L (π) by the same formula.
p
4 BYMICHAELSPIESS
Thep-adicL-functionattached toπ istheΓ-transformofacertain canon-
ical measure µ on the Galois group G of the maximal abelian extension of
π p
F which is unramified outside p and ∞, i.e. it is given by
L (s,π) = hγisµ (dγ)
p π
ZGp
(for the definition of hγis see section 3.3).
Crucial for the proof of (1) and (3) is a new construction of µ 1. We
π
shall briefly explain it (for details see 4.6). Heuristically, we defineµ as the
π
x 0
direct image of the distribution µ ×Wp d×x under the reciprocity
πp 0 1
(cid:18) (cid:19)
map I = F∗×Ip → G of class field theory. Here the first factor µ is the
p p πp
productdistribution on F∗ = F∗ of certain canonical distributions µ
p p∈Sp p p
on F∗ attached to each local factors π , p ∈ S . Moreover d×x denotes the
p Q p p
Haar measure on the group of S -ideles Ip = ′ F∗ and Wp is a certain
p v∤p v
Whittaker function of πp = π (it is the product of local Whittaker
v∤p v Q
functions).
N
Toputthisconstructiononasoundfoundationconsiderthemapφ given
π
by
ζxp 0
φ (U,xp) = µ (ζU) Wp
π πp 0 1
ζ∈F∗ (cid:18) (cid:19)
X
where the first argument U is a compact open subset of F∗ and the second
p
an idele xp ∈ Ip. Then φ (ζU,ζxp)= φ (U,x) for all ζ ∈ F∗. Thus if we set
π π
φ (x ,xp): = φ (x U,xp) then φ can be viewed as a function on the idele
U p π p U
class group I/F∗ (so the map U → φ is a distribution on F∗ with values
U p
in a certain space of functions on I/F∗).
For a locally constant map f : G → C there exists a compact open
p
subgroup U ⊂ U = O∗ ⊂ F∗ such that f ◦ρ : I/F∗ → C factors
p p∈Sp p p
through I/F∗(U ×Up) (here ρ : I/F∗ → G denotes the reciprocity map).
Q p
Then f(γ)µ (dγ) is given by
Gp π
R
f(γ)µ (dγ) = [U :U] f(ρ(x))φ (x)d×x.
π p U
ZGp ZI/F∗
By using properties of the cohomology groups of arithmetic subgroups of
GL (F) we show that µ is bounded (i.e. it is a p-adic measure in the sense
2 π
of section 1.2 below) and so any continuous map Z → C can be integrated
p p
against it.
One way to describe the local distribution µ for p ∈ S is that it is
p p
the image of a certain Whittaker functional of π under a canonical map –
p
denoted by δ – from the dual of π to the space of distributions on F∗. We
p p
will give the definition of δ in the case p ∈ S , or equivalently, when π is
1 p
the Steinberg representation St (i.e. π is isomorphic to the space of locally
p
constant functions P1(F ) → C modulo constants). For c ∈ Hom(St,C) we
p
1In principle our construction is related to Manin’s [Ma]. However in our set-up the
measure µ is build in a simple mannerfrom local distributions µ at each place v of F
π πv
ON SPECIAL ZEROS OF p-ADIC L-FUNCTIONS OF HILBERT MODULAR FORMS 5
define δ(c) by f(x)δ(c)(dx) = c(f˜). Here for a locally constant map with
F
compact support f : F → C we define f˜: P1(F ) → C by f˜(∞) = 0 and
R p p
f˜([x : 1]) = f(x). Thus in the case π = St, the target of δ is the space of
p
distributions on F .
p
In particular the local contribution µ of µ at p ∈ S is actually a
p π 1
distributiononF (andnotonlyonF∗). Therefore,allowedasfirstargument
p p
in φ (U,xp) are not only compact open subsets U of F∗ but also of the
π p
larger space F × F∗. This fact is crucial for our proof that the
p∈S1 p p∈S2 p
vanishing order L (s,π) at s = 0 is ≥ r. The map δ and distributions µ
p p
Q Q
will be introduced in sections 2.4 and 2.5 respectively.
Chapter 3 is the technical heart of this paper. It provides an axiomatic
approach to study trivial zeros of p-adic L-function which can be applied
in other situations as well (e.g. to the case of p-adic L-functions of totally
real number fields [Sp], [DC]). We consider arbitrary two-variable function
φ: (U,xp) 7→ φ(U,xp)(U ⊂ F × F∗ compactopenandxp ∈ Ip)
p∈S1 p p∈S2 p
satisfying certain axioms and attach a p-adic distribution µ on G as above.
p
Q Q
By ”integrating away” the infinite places we obtain a certain cohomology
class κ ∈ Hd(F∗,D) associated to φ (where d = [F : Q]− 1, F∗ denotes
+ +
the group of totally positive elements of F and D is a certain space of
distributions on the adelic space F × F∗× ′ F∗) and the
p∈S1 p p∈S2 p v∤p∞ v
distribution µ can be defined solely in terms of κ. The space D contains a
Q Q Q
canonicallysubspaceDb (consisting–inacertainsense–ofp-adicmeasures)
andµ is ap-adicmeasureprovided thatκlies intheimage of Hd(F∗,Db) →
+
Hd(F∗,D) (see section 3.4).
+
In this case we define L (s,φ) as the Γ-transform of µ and show that
p
L (s,φ) has a zero of order ≥ r at s = 0. Furthermore we give a descrip-
p
tion of the r-th derivative dr L (s,φ)| as a certain cap-product. More
dsr p s=0
precisely, we associate to any continuous homomorphism ℓ : F∗ → C a co-
p p
homology class c ∈ H1(F∗,C (F ,C )) (for its definition and the notation
ℓ + c p p
see 3.4). If S = {p ,...,p } we will show
1 1 r
dr (r)
(4) dsrLp(s,φ)|s=0 = (−1) 2 r! (κ∪cℓp1 ∪...∪cℓpr)∩ϑ.
Here ϑ is essentially the fundamental class of the quotient M/F∗ where
+
M is a certain d + r-dimensional manifold on which F∗ acts freely (see
+
section 3.2). If U = O × O∗ and φ (x): = φ(x U ,xp) for
0 p∈S1 p p∈S2 p 0 p 0
x = (x ,xp) ∈ F∗×Ip = I, we will also prove
p p Q Q
(5) ZI/F∗ φ0(x)d×x = (−1)(r2) r! (κ∪cop1 ∪...∪copr)∩ϑ.
Inchapter4wewillverifythatthetheorydevelopedinthepreviouschap-
ter can be applied in the case φ= φ . The difficult part is to show that the
π
cohomology class κ attached to φ comes from a class in Hd(F∗,Db). This
π π +
isachieved byshowingthatitliesintheimageofaspecificcohomology class
κ ∈ Hd(PGL (F),A) under a canonical map ∆ : Hd(PGL (F),A) →
π 2 ∗ 2
Hd(F∗,D) (for the definition of the coefficients A and the map ∆ we refer
+ ∗
to section 4.4 and 4.5). The fact that any arithmetic subgroup of PGL (F)
b 2
6 BYMICHAELSPIESS
has the finiteness property (VFL) (introduced by Serre in [Se]) implies that
∆ factors through Hd(F∗,Db).
∗ +
Inthe last chapter 5 we will introducethe automorphic L-invariant L (π)
p
and deduce (3) from (4) and (5). The cohomology group Hd(PGL (F),A)
2
carries an action of a Hecke algebra and κ lies in the π-isotypic component
π
Hd(PGL (F),A) . Usingthefactthat classes c “come” fromcertain PGL
2 π ℓ 2
cohomology classes as well (they will be introduced in section 2.6) and the
b
factthatHd(PGL (F),A) isonedimensional(aresultsduetoHarder[Ha])
2 π
we show that the cup products κ∪c and κ∪c differ by a factor L (π)
ℓp p p
which is defined in terms of the cohomology of PGL (F).
2
Acknowledgement. I thank Vytautas Paskunas for several helpful conversa-
tions and Kumar Murty for providing me with the reference [FH]. Also I
am grateful to H. Deppe, L. Gehrmann, S. Molina and M. Seveso for useful
comments on an earlier draft.
Notation. The following notations are valid throughout this paper. A list
with further notations will be given at the beginning of chapters 2 and 3.
Unless otherwise stated all rings are commutative with unit.
We fix a prime number p and embeddings
ι :Q ֒→ C, ι :Q ֒→ C .
∞ p p
We let ord denote the valuation on C and Q (via ι ) normalized so that
p p p
ord (p) = 1. The valuation ring of Q will be denoted by O.
p
We denote the set of compact open subsets of a topological space X by
Co(X). If X and Y are topological spaces then C(X,Y) denotes the set of
continuous maps X → Y and C (X,Y) the subset of continuous maps with
c
compact support. If we consider Y with the discrete topology then we shall
also write C0(X,Y) and C0(X,Y) instead of C(X,Y) and C (X,Y).
c c
Put G: = PGL , and let B be the subgroup of upper triangular matrices
2
∗ 0
(modulo the center Z of GL ), T = /Z be the maximal torus
2 0 ∗
(cid:26)(cid:18) (cid:19)(cid:27)
t 0
of G in B. We write elements of G often simply as matrices (and
0 1
(cid:18) (cid:19)
neglect the fact that we consider them only modulo the center of Z). We
t 0
identift G with T via the isomorphism t 7→ . If R is a ring the
m 0 1
(cid:18) (cid:19)
determinant induces a homomorphism det :G(R) → R∗/(R∗)2.
1. Generalities on distributions and measures
1.1. Distributions and measures. Let X be a totally disconnected σ-
locally compact topological space (in practice X will be a e.g. profinite set
like an infinite Galois group or a certain space of adeles). For a topological
Hausdorff ring R we denote by C (X,R) the subring of C(X,R) consisting
⋄
of maps f : X → R with f(x) → 0 as x → ∞ (equivalently by setting
f(∞)= 0themap f extendscontinuously to theone-pointcompactification
ON SPECIAL ZEROS OF p-ADIC L-FUNCTIONS OF HILBERT MODULAR FORMS 7
of X). We have C0(X,R) ⊆ C (X,R) ⊆ C (X,R) ⊆ C(X,R). Note that if
c c ⋄
X = X ×X where X and X are σ-locally compact and if f ∈ C (X ,R),
1 2 1 2 1 ⋄ 1
f ∈ C (X ,R) then the map (f ⊗ f )(g ,g ): = f (g ) · f (g ) lies in
2 ⋄ 1 1 2 1 2 1 1 2 2
C (X,R).
⋄
Let M be an R-module. Recall that an M-valued distribution on X is a
homomorphism µ : C0(X,Z) → M. It extends to an R-linear map
c
(6) C0(X,R) −→ M, f 7→ f dµ.
c
ZX
WeshalldenotetheR-moduleofM-valueddistributionsonX byDist(X,M).
If X = X ×X , µ ∈Dist(X,M) and f ∈ C0(X ,R) then f 7→ f ⊗f dµ
1 2 1 c 1 2 1 2
is an M-valued distribution on X which will be denoted by 7→ f dµ i.e.
2 RX1 1
we have a pairing
R
(7) Dist(X,M)×C0(X ,R) −→ Dist(X ,M), (µ,f ) 7→ f dµ.
c 1 2 1 1
ZX1
Next we introduce the notion of a measure on X with values in a p-adic
Banach space. Assume that R = K is a p-adic field. By that we mean that
K is a field of characteristic 0 which is equipped with a p-adic value, i.e. a
nonarchimedian absolute value | | : K → R whose restriction to Q is the
usual p-adic value and such K is complete with respect to | |. We denote a
p-adic value often as | | and the corresponding valuation ring by O .
p K
A norm on a K-vector space V is a function k k : V → R such that (i)
kavk = |a| kvk, (ii) kv+wk ≤ max(kvk,kwk) and(iii) kvk ≥ 0with equality
p
iff v = 0 for all a ∈ K, v,w ∈ V. Two norms k k , k k are equivalent if
1 2
there exists C ,C ∈ R with C kvk ≤ kvk ≤ C kvk for all v ∈ V. A
1 2 + 1 2 1 2 2
normedK-vector space(V,k k)isa(K-)Banach space ifV iscompletewith
respect to k k. Recall that any finite-dimensional K-vector space admits a
norm, any two norms are equivalent and it is complete. The K-vector space
C (X,K) with the supremums norm kfk = sup |f(γ)| is a K-Banach
⋄ ∞ γ∈X p
space.
Let V be a K-vector space. Recall that an O -submodule L ⊆ V is a
K
lattice if aL = V and aL = {0}. For a given lattice L ⊆ V
a∈K∗ a∈K∗
the function p (v): = inf |a| is a norm on V. If k k is another norm
L v∈aL p
S T
then p is equivalent to k k if and only if L is open andboundedin (V,k k).
L
A lattice L ⊆ V is complete if V is complete with respect to p . Finally a
L
torsion free O -module L is said to be complete if L is a complete lattice
K
in L ⊗ K. For example the O -dual of a free module is a complete
OK K
torsionfree O -module.
K
Let (V,k k) be a Banach space. An element µ ∈ Dist(X,V) is a measure
(or bounded distribution) if µ is continuous with respect to the supremums
norm, i.e. if there exists C ∈ R, C > 0 such that | f dµ| ≤ kfk for
X p ∞
all f ∈ C0(X,K). We will denote the space of V-valued measures on X by
R
Distb(X,V). If L ⊆ V is an open and bounded lattice then Distb(X,V) is
the image of the canonical inclusion Dist(X,L)⊗ K → Dist(X,V). An
OK
element µ ∈ Distb(X,V) can be integrated not only against locally constant
functions but against any f ∈ C (X,K). In fact since C0(X,K) is dense in
⋄ c
8 BYMICHAELSPIESS
theBanach space(C (X,K),k k )the functional(6) extends to auniquely
⋄ ∞
to a continuous functional
(8) C (X,K) −→ V, f 7→ f dµ.
⋄
Z
If X = X ×X then we obtain as a refinement of the bilinear map (7) a
1 2
pairing
(9) Distb(X,V)×C (X ,K) −→ Distb(X ,V), (µ,f )7→ f dµ.
⋄ 1 2 1 1
ZX1
1.2. p-adic measures. Given µ ∈ Dist(X,C) we want to clarify what do
we mean by saying that µ is a p-adic measure. For simplicity assume that
X is compact. The distribution µ extends to C -linear map
p
(10) C0(X,C ) −→ C ⊗ C, f 7→ f dµ
p p Q
Z
and we denote its by V so that we can view µ as an element of Dist(X,V ).
µ µ
It is called a p-adic measure if V is a finitely generated C -vector space
µ p
and if µ ∈ Distb(X,V ). Equivalently, the image of µ (considered as a
µ
map C0(X,Z) → C) is contained in a finitely generated O-module. So if
µ ∈ Dist(X,C) is a p-adic measure (10) extends to continuous functional
C(X,C )−→ V , f 7→ f dµ.
p µ
R
2. Local distributions attached to ordinary representations
2.1. Gauss sums. Throughoutthis chapter F denotes a finite extension of
Q , O = O its ring of integers and p the maximal ideal of O. We denote
p F
by U the group of units of O and put U(n) = {x ∈ U| x ≡ 1 mod pn}. Let
q denote the number of elements of O/p. We fix an (additive) character
∗
ψ : F → Q such that Ker(ψ) = O and a generator ̟ of p. We denote by
|x| the modulus of x ∈ F∗ (i.e. |̟| = q−1) and by ord = ord the additive
F
valuation (normalized by ord(̟) =1). The normalized Haar measure on F
willbedenotedbydx(normalizedby dx = 1). Weputd×x = (1−1)−1dx
O q |x|
so that d×x = 1.
U R
LemmaR2.1. Let X ⊆ {x ∈ F∗ | ord(x) ≤ −2} be a compact open subset
such for all a ∈ X there exists n ∈ Z, 1 ≤ n ≤ −ord(a) − 1 such that
aU(n) ⊆ X. Then,
ψ(x)d×x = 0.
ZX
Proof. It is enough to consider the case X = aU(n) with 1 ≤ n ≤
−ord(a)−1. Choose b ∈ F∗ with ord(b)+ord(a) = −1. Hence ψ(ab) 6= 1
and ord(b) ≥ n and therefore
ψ(x)d×x = ψ(ax)d×x = ψ(a(1+b)x)d×x
ZX ZU(n) ZU(n)
= ψ(ax)ψ(abx)d×x.
ZU(n)
ON SPECIAL ZEROS OF p-ADIC L-FUNCTIONS OF HILBERT MODULAR FORMS 9
Since ord(abx−ab) = −1+ord(x−1) ≥ n−1≥ 0, we have ψ(abx) = ψ(ab)
for all x ∈U(n). It follows
ψ(x)d×x = ψ(ab) ψ(ax)d×x = ψ(ab) ψ(x)d×x,
ZX ZU(n) ZX
hence ψ(x)d×x = 0. (cid:3)
X
R
Recall that the conductor c(χ) of a quasicharacter χ : F∗ → C∗ is the
largest ideal pn of O such that U(n) ⊆ Ker(χ).
Lemma 2.2. Let χ : F∗ → C∗ be a quasicharacter of conductor pn,n ≥ 1
and let a ∈F∗ with ord(a) 6=−n. Then we have
ψ(ax)χ(x)d×x = 0.
ZU
Proof. 1. case ord(a) > −n: Choose b ∈ F∗ with max(−ord(a),0) ≤
ord(b) < n, 1+b ∈ U and χ(1+b) 6= 1. Then,
ψ(ax)χ(x)d×x = ψ(ax(1+b))χ(x(1+b))d×x =
ZU ZU
= χ(1+b) ψ(ax)ψ(abx)χ(x)d×x
ZU
= χ(1+b) ψ(ax)χ(x)d×x
ZU
hence ψ(ax)χ(x)d×x = 0.
U
2. caseRord(a) < −n: By 2.1 above we have
ψ(ax)χ(x)d×x = χ(b) ψ(x)d×x = 0.
ZU bU(n)X∈U/U(n) ZabU(n)
(cid:3)
WerecallthedefinitionoftheGausssumofaquasicharacter(withrespect
to the fix choice of ψ).
Definition 2.3. Let χ : F∗ → C∗ be a quasicharacter with conductor pn,
n ≥ 0 and a ∈F∗ with ord(a) =−n. We define the Gauss sum of χ by
τ(χ) = τ(χ,ψ) = [U :U(n)] ψ(x)χ(x)d×x.
ZaU
For a quasicharacter χ :F∗ → C∗ we define
(11) χ(x)ψ(x)dx: = lim χ(x)ψ(x)dx.
ZF∗ n→+∞Zx∈F∗,−n≤ord(x)≤n
Lemma 2.4. Let χ : F∗ → C∗ be a quasicharacter with conductor pf.
Assume that |χ(̟)| < q. Then the integral (11) converges and we have
1−χ(̟)−1
if f = 0;
χ(x)ψ(x)dx = 1−χ(̟)q−1
ZF∗ ( τ(χ) if f > 0.
10 BYMICHAELSPIESS
Proof. Firstly, we remark
1 if ord(a) ≥ 0;
(12) ψ(ax)d×x = − 1 if ord(a) = −1;
 q−1
ZU  0 if ord(a) ≤ −2;
for all a ∈F∗. Since (1−1/q)d×x = dx, we obtain
|x|
∞
χ(x)ψ(x)dx = (1−1/q)q−n χ(x)ψ(x)d×x.
ZF∗ n=−∞ Z̟nU
X
If f > 0 then by 2.2 we have
χ(x)ψ(x)dx = (1−1/q)qf χ(x)ψ(x)d×x = τ(χ).
ZF∗ Z̟−fU
On the other hand if f = 0 then by (12) we get
∞
q
χ(x)ψ(x)dx = (1−1/q) − + (χ(̟)q−1)n
(q−1)χ(̟)
ZF∗ n=0 !
X
1−χ(̟)−1
= .
1−χ(̟)q−1
(cid:3)
2.2. Ordinary representations of PGL (F). We introduce more nota-
2
tion. Let K = G(O). For an ideal c ⊂ O let K (c) ⊆ K denote the
0
subgroup of matrices A (modulo Z) which are upper triangular modulo c.
Let π : G(F) → GL(V) be an irreducible admissible infinite-dimensional
representation (where V is a C-vector space). Recall [Cas] that there exists
a largest ideal c(π) – the conductor of π – such that VK0(c) = {v ∈ V |
π(k)v = v ∀k ∈ K (c)} =6 0. In this case VK0(c) is one-dimensional.
0
The representation π is called tamely ramified if the conductor divides
p. This holds if and only if π = π(χ−1,χ) for an unramified quasicharacter
χ : F∗ → C∗ (see e.g. [Bu], Ch. IV). More precisely if the conductor is O ,
F
then π is spherical hence a principal series representation π(χ−1,χ) where
χ : F∗ → C∗ is an unramifiedquasicharacter with χ2 6= |·|. If c(π) = p, then
π is a special representation π(χ−1,χ) where χ is unramified with χ2 = |·|.
Definition 2.5. Assume that π = π(χ−1,χ) is tamely ramified. Then π is
called ordinary if either χ2 = |·| or if π is spherical and tempered and if
χ(̟)q1/2 is a p-adic unit (i.e. it lies in O∗).
Thusifπ = π(χ−1,χ)istamely ramifiedandifweputα: = χ(̟)q1/2 ∈ C
then π is ordinary if either α =±1 or if α∈ O∗ and |α| = q1/2. Note that α
determines π uniquely, i.e. thereexists aone-to-one correspondencebetween
theset (of isomorphism classes) of ordinaryrepresentations of G(F) andthe
set {α ∈ O∗| α = ±1 or |α| = q1/2}. We will call an element of the latter
set an ordinary parameter. We will denote the class corresponding to α by
π and define χ (x): = αord(x) (thus π = π(χ−1| · |−1/2,χ | · |1/2)). If
α α α α α