CRITICAL p-ADIC L-FUNCTIONS
JOËL BELLAÏCHE
Abstract. We attach p-adic L-functions to critical modular forms and study them.
We prove that those L-functions fit in a two-variables p-adic L-function defined locally
everywhere on the eigencurve.
Contents
1.
Introduction
2
1.1.
Objectives and motivation
2
1.2.
Notations and conventions
3
1.3.
Reminder about modular forms and the eigencurve
3
1.4.
Results
4
1.5.
Method and plan of the paper
9
1.6.
2.
Perspectives
10
Modular forms, refinements, and the eigencurve
11
2.1.
The eigencurves
11
2.2.
Refinements
16
2.3.
Smoothness of the eigencurve and consequences
19
3.
Construction of the eigencurve through distributions-valued modular symbols
21
3.1.
Families of distributions
22
3.2.
Distributions-valued modular symbols and their families
26
3.3.
Mellin transform of distribution-valued modular symbols
34
3.4.
Local pieces of the eigencurve
36
4. p-adic L-functions
≤ν
Symb±
Γ (D(R))
40
as a module over
T±
W,ν
4.1.
The space
41
4.2.
Critical p-adic L-functions
42
4.3.
Two-variables p-adic L-functions
45
Key words and phrases. p-adic L-functions; modular forms; Eisenstein series; eigencurve.
During the elaboration of this paper, I was supported by the NSF grant DMS 08-01205. I thank Gaëtan
Chenevier for his invaluable help related to this article. I can say that most of sections 2 and 3 below arose
from discussions with him: not only did he give me many individual ideas or arguments, but I owe him my
whole vision of the subject. I also thank Glenn Stevens and Robert Pollack for giving me access to their
work even before it was made available on the web, and for many conversations about the themes of this
paper. Also the remarkable clarity of their arguments in the various papers by them in the bibliography
were a strong appeal for me to study this subject, and a great help when I did. I thank John Bergdall, Kevin
Buzzard, Francesc Castellà, Samit Dasgupta, Matthew Emerton and Loı̈c Mérel for useful conversations.
Finally, I thank an anonymous referee for his careful reading and his suggestions that helped me improve
the redaction.
1
2
J. BELLAÏCHE
4.4.
Secondary p-adic L-functions
References
48
49
1. Introduction
1.1. Objectives and motivation. The aim of this paper is to extend, and even in a
sense, to complete the works of Mazur-Swinnerton-Dyer ([M-SD]), Manin ([Man]), Visik
([V]), Amice-Vélu ([AV]), Mazur-Tate-Teiltelbaum ([MTT]), Stevens ([S]), Stevens-Pollack
([SP1] and [SP2]) on the construction of a p-adic L-function L(f, s) of one p-adic variable
s attached to a refined classical modular form f that is an eigenvector for almost all Hecke
operators, as well as the work of Mazur, Kitagawa ([Ki]), Greenberg-Stevens ([GS]), Stevens
[S2], Panchishkin ([PA]), and Emerton ([E]) on the construction of a two-variables p-adic Lfunction L(x, s) where the first variable x runs among the points in a suitable p-adic family
of refined modular forms, such that when x corresponds to a classical refined modular
eigenform f , the function s → L(x, s) is the classical one-variable p-adic L-function L(f, s).
Let us give a little bit more details. It has been known for more than three decades
how to attach p-adic L-functions to refined modular forms of non-critical slope. More
recently, Stevens and Pollack have constructed L-functions for certain refined modular
forms of critical slope, namely the ones that are not θ-critical. Their construction therefore
does not work for critical slope CM cuspidal forms and Eisenstein series, which are very
important for number-theoretic applications. In this paper, we construct in a unified way a
p-adic L-function for (almost) all refined modular forms. Our method uses the insights of
Stevens and Pollack-Stevens (p-adic L-functions coming from distributions-valued modular
symbols), together with a knowledge of the geometry of the eigencurve and of various
modules over it in the spirit of [BC2]. So we use family arguments even for the construction
of the L-function of an individual refined modular form.
Our second main result is the construction of a two-variables L-function L(x, s) interpolating the individual L-functions of refined modular forms (locally on x in the neighborhood
of (almost) any refined modular form in the eigencurve). The existence of such a function
restricted to the x’s of non-critical slopes has been shown by three different authors (firstly
Stevens in a work [S2] that unfortunately was not made available, then Panchishkin [PA]
and Emerton [E]). We extend their results by incorporating the missing x’s. Our method
is very close to what I believe is Stevens’ method, using families of distributions-valued
modular symbols over the weight space, but there is a new difficulty, which is that the
eigencurve is not étale over the weight space at the critical point x.
The interplay between the geometry of the eigencurve (absolute and relative to the weight
space) and the construction and properties of the L-functions is promising of interesting
phenomena in higher rank, when the geometry of the eigenvarieties can be much more
CRITICAL p-ADIC L-FUNCTIONS
3
complicated (see e.g. [B1]) and of number-theoretic significance (see [BC3, Chapter IX]).
For more on this, see 1.6 below.
In the remaining of this introduction, I will explain in more details my results and
their relations with the results obtained by the authors aforementioned, especially with the
results of Stevens and Pollack, which were a fundamental inspiration for me. For this I
need to introduce some notations and terminology.
1.2. Notations and conventions. Throughout this paper, we shall fix an integer N ≥ 1,
and a prime number p that we shall assume prime to N . In all normed extension of Qp we
shall use the normalized valuation vp (vp (p) = 1) and the normalized absolute value p−vp .
We fix embeddings Q̄ ,→ Q̄p and Q̄ ,→ C.
We shall work with the congruence subgroup Γ of SL2 (Z) defined by
Γ = Γ1 (N ) ∩ Γ0 (p).
†
†
For an integer k ≥ 0, we shall denote by Mk+2 (Γ) (resp. Sk+2 (Γ), Mk+2
(Γ), Sk+2
(Γ))
the Qp -space of classical modular forms (resp. cuspidal classical modular forms, resp.
overconvergent p-adic modular forms, resp. overconvergent cuspidal p-adic modular forms)
of level Γ and weight k+2. Those spaces are acted upon by the Hecke operators Tl for l prime
to N p, the Atkin-Lehner operator Up , and the diamond operators hai for a ∈ (Z/N Z)∗ , as
will be virtually all spaces and modules considered in this paper. To avoid repeating this
list of operators too often, we shall define H as the commutative polynomial algebra over
Z over the variables Tl (for l prime to N p), Up , and hai for a ∈ (Z/NZ)∗ , that is
H = Z[(Tl )l6 |pN , Up , (hai)a∈(Z/NZ)∗ ],
and simply say that the spaces Mk+2 (Γ) and Sk+2 (Γ) are acted upon by H.
In general, if M is a Qp -space on which H acts, by a system of H-eigenvalues appearing
in M we shall mean any character x : H → Q̄p such that there exists a non-zero vector v
in M ⊗ Q̄p satisfying hv = x(h)v for all h ∈ H. We call such a v (and also the 0 vector)
an x-eigenvector. If x is such a system, we shall call the eigenspace for the system of Heigenvalues x or shortly the x-eigenspace the space, denoted by M [x], of x-eigenvectors v
in M ⊗ Q̄p . We shall call generalized eigenspace for the system of H-eigenvalues x, denoted
by M(x) , or shortly the generalized x-eigenspace the space of v in M ⊗ Q̄p such that for all
h ∈ H, there exists an n ∈ N such that (h − x(h))n v = 0. Please keep in mind that we have
extended scalars to Q̄p , so that M [x] and M(x) are Q̄p -vector spaces. If N is another (or the
same) Qp -space of which H acts, and v ∈ N is an H-eigenvector, of system of eigenvalues
x, we shall write M [v] and M(v) instead of M [x] and M(x) .
1.3. Reminder about modular forms and the eigencurve. To state our results, and
to explain in more details their relations with the works mentioned above, we need to recall
the notions of a refinement of a modular form, and of a critical slope, θ-critical, and critical
refinement.
4
J. BELLAÏCHE
Let f =
P
n≥0
an q n be a modular form of level Γ1 (N ) and of weight k + 2 ≥ 2 with
coefficients in Q̄p . We shall say that f is a newform (of weight k + 2, level Γ1 (N ) ) if f
is eigenvector for the Hecke operators Tl (l prime to N ) and the diamond operators hai
(a ∈ (Z/N Z)∗ ) , and if there is no modular form of weight k + 2 and level Γ1 (N 0 ) where
N 0 is a proper divisor of N which is also an eigenform for all the Tl , (l, N ) = 1, and all the
diamond operators with the same eigenvalues as f .
If f is newform of weight k + 2, level Γ1 (N ), we have Tp f = ap f for some algebraic
integer ap and hpif = (p)f for some root of unity (p). Let us call α and β the roots of
the polynomial
(1)
X 2 − ap X + (p)pk+1 .
We shall assume throughout the paper without loss of generality1 and for simplicity that
α 6= β. As is well known, there are two normalized modular forms for Γ = Γ1 (N ) ∩ Γ0 (p),
of level k + 2, that are eigenforms for the Hecke operators Tl (l prime to pN ) and the
diamond operators with the same eigenvalues as f , and that are also eigenforms for the
Atkin-Lehner operator Up :
(2)
fα (z)
= f (z) − βf (pz)
(3)
fβ (z)
= f (z) − αf (pz).
Those forms satisfy Up fα = αfα and Up fβ = βfβ .
The forms fα and fβ are called refined modular forms, and they are the two refinements
of f . The choice of a root α or β of (1) is also called a refinement of f . Refinements are also
sometimes called p-stabilizations, but we shall not use this terminology. Refined modular
form are the natural objects to which one can attach a p-adic L-function, and they are
also (a coincidence due to our two-dimensional setting) the natural objects to put in p-adic
families: in particular a refined modular form fβ corresponds to a point x in the p-adic
eigencurve C of tame level Γ1 (N ). We call such points the classical points of C. Hence
classical points of C are in bijection with refinements of modular newforms of level Γ1 (N )
and arbitrary weight k + 2.
If fβ is a refined modular form as above, we shall say that it is of critical slope if
vp (β) = k + 1, is θ-critical if fβ is in the image of the operator θk+1 , and is critical if
†
Mk+2 (Γ)(fβ ) 6= Mk+2
(Γ)(fβ ) . We refer the reader to §2.2.2 for equivalent conditions and
more details. Let us just say now for the intelligence of our results that critical implies θcritical implies critical slope; that for a cuspidal form, critical and θ-critical are equivalent;
and that for an Eisenstein series, θ-critical and critical slope are equivalent; otherwise
implications are strict.
1.4. Results.
1This is conjectured in general, known in weight k + 2 = 2, and when this is not true we have v (α) =
p
vp (β) = (k + 1)/2 < k + 1, so the results we want to prove are already known.
CRITICAL p-ADIC L-FUNCTIONS
5
1.4.1. Modular symbols. To formulate our first result about the existence of a p-adic Lfunction, we shall adopt Stevens’ point of view of L-functions as Mellin transforms of the
values at the divisor {∞} − {0} of distributions-valued modular symbols. We begin by
recalling quickly this point of view.
Let D = D† [0] be the Qp -Frechet space which is the dual of the locally convex space
of locally analytic functions on Zp . For a fixed integer k, D is endowed with an action of
Γ0 (p), called the weight-k action,
which
is the dual of the action on the space of locally
a b
analytic functions given for γ =
by
c d
b + dz
(γ ·k f )(z) = (a + cz)k f
.
a + cz
For Γ = Γ1 (N )∩Γ0 (p), we define the space of modular symbols SymbΓ (Dk ) := HomΓ (∆0 , Dk )
where ∆0 is the abelian group of divisors of degree 0 on P1 (Q). This Qp -space is endowed
with a natural
H-action
and also with an involution ι (commuting with H) given by the
1 0
matrix
. We denote by Symb±
Γ (Dk ) the eigenspaces of this involution. For more
0 −1
details about those notions see below §3.2.1.
Our theorem has a technical condition, but fortunately it is very mild.
Definition 1. We shall say that a refined form fβ is decent if it satisfies at least one of
the three following assumptions:
(i) f is Eisenstein
(ii) fβ is non-critical.
(iii) f is cuspidal and Hg1 (GQ , adρf ) = 0
As we shall see in §2.2.4, all CM forms are decent, and there exist several general easy
to check criteria ensuring that a cuspidal non-CM form satisfies (iii) (so is decent) leaving
only a small number of residual forms out. Moreover, it is conjectured that every cuspidal
form satisfies (iii), and independently that every f which is cuspidal non CM satisfies (ii),
so cuspidal non-CM forms should be doubly decent.
Theorem 1. Let f be a newform of weight k + 2 and level Γ1 (N ), and let fβ a refinement
of f at p. We assume that fβ is decent, and if f is Eisenstein, that vp (β) > 0. Then for
both choices of the sign ±, the eigenspace Symb±
Γ (Dk )[fβ ] has dimension 1.
The theorem is proven in §4.2.1
This theorem was known by a work of Stevens-Pollack ([SP2]) in the case of forms fβ
that are not θ-critical and before in the case of forms of non-critical slope by a result of
Stevens ([S]). (Actually, they proved the stronger result that the generalized eigenspace
Symb±
Γ (Dk )(fβ ) is of dimension 1. This is not always the case when fβ is not θ-critical, as
we shall see: Theorem 4.)
The new cases provided by Theorem 1 are the θ-critical cases, which include all Eisenstein
series with their non-ordinary refinement, all CM forms such that p is split in the quadratic
6
J. BELLAÏCHE
field, with their non-ordinary refinement, and the cuspidal non CM fβ that are not known
to be non-θ-critical.
We note that the the one-dimensionality of the eigenspace Symb±
Γ (Dk )[fβ ] was observed
in several instances using numerical computations with a computer by Pasol, Pollack, and
Stevens. For example, for the modular form attached to the CM elliptic curve X0 (32) with
p = 5 ([SP2, Example 6.10], [SP1]), or for the Eisenstein series of level Γ0 (11) and weight
2 for p = 3 ([S3]). It was conjectured ([S3, ”Wild Guess”]) that the same result would
hold for Eisenstein Series for Γ0 (11) of all weights, and p = 3. Those observations and
conjectures were important motivations for the present work.
In the case where f is Eisenstein, the condition vp (β) > 0 is necessary. For if vp (β) = 0,
(fβ )
that is if fβ is the ordinary refinement of an Eisenstein series, we have dim SymbΓ
1 but
−(f )
SymbΓ β (Dk )[fβ ]
(Dk )[fβ ] =
= 0, where (fβ ) is a certain sign attached to fβ defined in §3.2.6.
This follows immediately from the definition of (fβ ) and Stevens’ control theorem (Theorem 3.14).
1.4.2. Construction of p-adic L-functions. The connection with the p-adic L-functions is as
±
follows: Let Φ±
fβ be generators of the eigenspaces SymbΓ (Dk )[fβ ] considered in Theorem 1.
Both generators are well defined up to multiplication by a non-zero scalar (in Q̄∗p ). We define
two p-adic L-function L± (fβ , σ) of the refined form fβ as analytic functions of a continuous
character σ : Z∗p → C∗p by
L± (fβ , σ) := Φ±
fβ ({∞} − {0})(σ).
Here by definition Φ±
fβ ({∞}−{0}) is a linear form on the space of locally analytic functions
on Zp and Φ±
fβ ({∞} − {0})(σ) is the value of that linear form on σ, which is seen as the
locally analytic function on Zp which is the character σ on Z∗p and 0 on pZp . The two
functions L± (fβ , σ) are each defined up to multiplication by a non-zero scalar, but note
that L± (fβ , σ) = 0 whenever σ(−1) 6= ±1, so the two functions have disjoint supports.
Therefore the zeros and poles of the following function
L(fβ , σ) := L+ (fβ , σ) + L− (fβ , σ)
are well-defined, even if L(fβ , σ) depends on the choice of two scalars.
When fβ is non-critical, this p-adic L-function is the same as the usual one, defined by
Mazur-Tate-Teitelbaum in the non-critical slope case, and by Stevens-Pollack in the critical
slope non-θ-critical case. This is clear by definition, at least if we use Steven’s reformulation
([S]) of the definition of the L-function in the non-critical slope case. In other words, our
construction is new only in the θ-critical case.
1.4.3. Properties of the p-adic L-functions. The p-adic L-functions enjoy the following
properties:
Analyticity: The function L(fβ , σ) is an analytic function of σ, in the following sense:
CRITICAL p-ADIC L-FUNCTIONS
7
Let q = p if p is odd, q = 4 if p = 2. We have a canonical isomorphism Z∗p = (Z/qZ)∗ ×
(1 + qZp ) and the group 1 + qZp is pro-cyclic. We fix a generator γ of that group and for
s ∈ Cp , |s − 1| < 1, we call χs the continuous character of 1 + qZp sending γ to s. All
continuous Cp -valued characters of 1 + qZp are of this form. Therefore, any continuous
character σ : Z∗p → C∗p can be written in a unique way σ = ψχs where ψ is a character of
(Z/qZ)∗ and χs a character of 1 + qZp , with |s − 1| < 1.
The assertion that L(fβ , σ) is analytic in σ is to be understood as follows: for every character ψ of (Z/qZ)∗ , the function s 7→ L(fβ , ψχs ) is given by a power series in s converging
on the open ball |s − 1| < 1.
Order: The function L(fβ , σ) is of order at most vp (β), in the sense that we have
±
L (fβ , ψχs ) = O((logp (s))vp (β) ), where for F and G two functions of s defined on the set
|s − 1| < 1, we have sup|s−1|<ρ |F (s)| = O(sup|s−1|<ρ G(s)) when ρ → 1− .
Interpolation: We know the value of L(fβ , σ) at all characters σ of the form t 7→ ϕ(t)tj ,
where ϕ is a character of finite order of Z∗p , and j is a rational integer such that 0 ≤ j ≤ k.
When vp (β) < k + 1, those values determine uniquely the function L± (fβ , σ) in view of the
above property. This is not true however in the critical-slope case vp (β) = k + 1.
If fβ is not θ-critical, the interpolation property is due to Pollack and Stevens (or earlier
works if vp (β) < k + 1):
(4)
L± (fβ , ϕtj ) = ep (β, ϕtj )
mj+1
j! L∞ (f ϕ−1 , j + 1)
j
(−2πi) τ (ϕ−1 )
Ω±
f
for all ϕ : Z∗p → Q∗p of finite order and conductor m = pν M (with M prime to p) satisfying
ϕ(−1) = ±1 and 0 ≤ j ≤ k, up to a scalar independent on ϕtj . Above, L∞ is the
−1
archimedean L-function, Ω±
) is the Gauss
f are the two archimedean periods of f , τ (ϕ
sum of ϕ−1 and
(5)
ep (β, ϕtj ) =
1
ϕ̄(p)(p)pk−j
ϕ(p)pj
(1 −
)(1 −
)
ν
β
α
α
Theorem 2. If fβ is θ-critical, then
(6)
L± (fβ , ϕtj ) = 0
for all ϕ : Z∗p → Q∗p of finite order and 0 ≤ j ≤ k, excepted perhaps if all the following
conditions hold: f is Eisenstein, fβ is not critical2, the sign ± is (fβ ).
For the proof of the properties above and of Theorem 2, see §4.3.3.
Let us note that the exception in the above theorem is really needed: see Example 4.10.
One can reformulate this theorem by introducing, as in [PO], the analytic function of
the variable s (on the open unit ball of center 1 and radius 1, that is |s − 1| < 1)
log[k] (s) :=
k
Y
logp (γ −j s),
i=0
2This condition is expected to always hold for f Eisenstein, see Prop. 2.13.
8
J. BELLAÏCHE
whose list of zeros, all with simple order, are all the points s = γ j ζ, where ζ is a p∞ -root
of unity and 0 ≤ j ≤ k. Then the theorem simply says that if fβ is θ-critical, then for all
tame characters ψ, the analytic function s 7→ L± (fβ , ψχs ) is divisible by log[k] (s) in the
ring of analytic functions on the open unit ball of center 1, excepted in the exceptional case
where f is Eisenstein, fβ non-critical, ± = (fβ ). As a consequence of this and the order
property, then L± (fβ , ψχs )/ log[k] (s) is an holomorphic function on the open ball |s−1| < 1
with a finite number of 0’s.
Let us recall that it is conjectured that L(fα , ψχs ) has infinitely many 0’s whenever
vp (α) > 0. This is known (cf. [PO]) in the case where the coefficient ap of the modular
form f is 0 (e.g. a modular form attached to a super-singular elliptic curve).
Corollary 1. If vp (β) = k + 1 (that is fβ of critical slope), then for all character ψ :
µp−1 → C∗p , the function s 7→ L± (fβ , ψχs ) has infinitely many zeros.
1.4.4. Two variables p-adic L-functions. We note C the eigencurve of tame level Γ1 (N ).
Theorem 3. Let fβ be as in theorem 1, and let x be the point corresponding to fβ in the
eigencurve C. There exists an affinoid neighborhood V of x in C, and for any choice of the
sign ±, a two-variables p-adic L-function L± (y, σ) for y ∈ V and σ a continuous character
Z∗p → C∗p , well defined up to a non-vanishing analytic function of y alone. The function
L± (y, σ) is jointly analytic in the two variables. If y corresponds to a refined newform fβ0 0 ,
this newform is decent and one has L± (y, σ) = cL± (fβ0 0 , σ) where c is a non-zero scalar
depending on y but not on σ.
1.4.5. Generalized eigensybols and secondary critical p-adic L-functions. The space Dk
admits as a quotient the dual Vk of the space polynomials of degree less k, and there is
±
an H-equivariant morphism ρ∗k : Symb±
Γ (Dk ) → SymbΓ (Vk ). For more details about those
notions see below §3.2.5.
Theorem 4. Let fβ be as in Theorem 1. The generalized eigenspaces Symb±
Γ (Dk )(fβ )
always have the same dimension for both choices of the sign ±, and this dimension is equal
to the degree of ramification e of the weight map κ at the point x corresponding to fβ on
the eigencurve C. The eigencurve C is smooth at x.
For every integer i, 0 ≤ i ≤ e, there exists a unique subspace Fi± of dimension of i of
±
±
±
Symb±
Γ (Dk )(fβ ) that is stable by H. One has SymbΓ (Dk )[fβ ] = F1 , and (Fi )i=0,...,e is a
flag.
Moreover the following are equivalent:
(i) fβ is critical.
(ii) One has e > 1.
(iii) One has
(TS± )
for both choices of the sign ±.
ρ∗k (Symb±
Γ (Dk )[fβ ]) = 0.
CRITICAL p-ADIC L-FUNCTIONS
9
If f is cuspidal, then if (TS± ) holds for one choice of the sign ± it holds for the other as
well. If f is Eisenstein, (TS−(fβ ) ) always holds if (fβ ) is the sign of fβ defined in §3.2.6.
Remark 1. We have e ≥ 2 if fβ is a critical-slope CM cuspidal form. We conjecture that
e = 2 in this case. In all other cases, it is conjectured that e = 1. (See Prop. 2.13)
In addition to the p-adic L-function L± (fβ , σ) that we shall also denote by L±
1 (fβ , σ),
we can define e − 1 secondary L-functions L±
i (fβ , σ) for i = 2, . . . , e, by setting, for all σ.
i−1 ±
∂ L (y, σ)
±
(7)
Li (fβ , σ) =
.
∂y i−1
|y=x
This make sense since C is smooth of dimension 1 at x.
Of course, since the two-variables L-function L± (y, σ) is only defined up to multiplication
by a non-vanishing function of y, the function L±
i (fβ , σ) is only defined up to multiplication
by a scalar and up to addition by a linear combination of the functions L±
j (fβ , σ) for
1 ≤ j ≤ i − 1. In other words, only the flag F1± ⊂ F2± ⊂ · · · ⊂ Fe± in the space of analytic
Pi
functions of σ is well defined, where Fi = j=1 Q̄p L±
i (fβ , σ).
The analytic functions L±
i (fβ , σ) for i = 0, . . . , e−1 enjoy the following properties (which
are all invariant under the indeterminacy in the definition of L±
i , as the reader can easily
check).
Connection with modular symbols: The application φ 7→ (σ 7→ φ({∞} − {0})(σ)
realizes an isomorphism from the flag of modular symbols (Fi± ) onto the flag on analytic
functions Fi± . In other words, the secondary L-functions of fβ are Mellin transform of
values at {∞} − {0} of generalized eigenvectors in the space of distribution-valued modular
symbols with the same system of eigenvalues as fβ , and those generalized eigenvectors are
well-defined up to the same indeterminacy as the functions Li (fβ , .)
Order: For all i = 1, . . . , e, L±
i (fβ ) is of order at most vp (β) = k + 1.
Interpolation: Assume that f is cuspidal. If 1 ≤ i ≤ e − 1, then L±
i (fβ , σ) satisfies the
interpolation property (6). If i = e, then L±
e (fβ , σ) satisfies the interpolation property (4).
Infinity of zeros: Assume that f is cuspidal. For all i = 1, . . . , e, and all ψ : (Z/qZ)∗ →
C∗p , the function s 7→ L±
i (fβ , ψχs ) has infinitely many zeros.
1.5. Method and plan of the paper. The idea of the proof of Theorem 3 is as follows:
over the weight space W, or at least over a suitable affinoid open subset W = SpR of W,
there exists a module of distributions-valued modular symbols. This module, that we note
SymbΓ (D(R)), or an analog to it, was already considered by other authors (e.g. [GS]):
its elements give rise, essentially by Mellin transform, to two-variables functions L(k, σ), k
being a variable on the open subset W of the weight space W. We need to see this module,
or the part of its Riesz decomposition corresponding to a Up -slope ≤ ν, as a module over
(the affinoid ring of a suitable affinoid subspace CW,ν of) the eigencurve. We have found no
other way to do so than to reconstruct the eigencurve inside it, that is to use that module
10
J. BELLAÏCHE
instead of Coleman’s module of overconvergent modular forms in the construction of the
eigencurve3. It is then easy to show using a theorem of Chenevier ([C2]) that the new
eigencurve thus constructed is essentially the same as the old one.
The second step is to study the structure of this module as a module over the affinoid
CW,ν . Here we use the fundamental fact that the eigencurve is smooth at all decent classical point. Near such a point, an elementary commutative-algebra argument shows that
our module is free (of rank two, each of its ±1-part being of rank one). This is the result
from which all the others are deduced: one first derives that the formation of the algebra
generated by the operators in H in the modules of modular symbols commute with specialization, which allows us to derive the structure of this algebra at a decent classical point
x: it always has the form Q̄p [t]/(te ) where e is degree of ramification of the eigencurve at
x over the weight e. This implies with a little supplementary work all results concerning
the one-variable p-adic L-function. As for the two-variables p-adic L-function, whose construction is not straightforward in the neighborhood of points that are not étale over the
weight space, we refer the reader to the corresponding section of the paper: §4.3.3
In section 2, we prove all results on the eigencurve and the notion of refinement that we
shall need later. In particular, we extend the result of [BC] that the eigencurve is smooth at
Eisenstein series to decent critical cusp forms, using a simple argument due to Chenevier.
In section 3, we construct the module of modular symbols SymbΓ (D(R)) (cf 3.1 and 3.2).
We then reconstruct the eigencurve in it (cf 3.4.1), We also study the Mellin transform in
this context (cf 3.3.1),
Section 4 is devoted to the proof of the results stated in this introduction.
1.6. Perspectives. As said earlier, the elegant and clearly exposed results of Stevens and
Pollack, and the remaining open questions appearing clearly from their work were a strong
motivation for this work. But there was a second motivation, that I wish now to explain.
As this explanation has little to do with the rest of the article, the reader may wish to skip
it.
From a larger point of view, I mean for automorphic forms for larger groups, especially
unitary groups, I believe it is very important, in connection to the Bloch-Kato conjecture,
to develop a theory of p-adic L-function for automorphic forms, and of the family thereof,
especially including the critical and critical slope cases. Let me explain why: Since 2002
a series of work by the author, Chenevier, Skinner and Urban have developed a strategy
to prove the lower bound on the Selmer group predicted by Bloch-Kato conjecture or its
p-adic variant. This strategy, which is part of the world of methods inspired by Ribet’s
proof of the converse of Herbrand’s theorem, can be summarized as follows: one picks up
3Such a construction was announced by Stevens at the Institut Henri Poincaré in 2000, but details never
appeared. We thus have to redo it from scratch, using of course many ideas from the published work of
Stevens.
CRITICAL p-ADIC L-FUNCTIONS
11
an automorphic form f for an unitary group (or a symplectic group) whose Galois representation is reducible (the automorphic form may be Eisenstein, or cuspidal endoscopic) and
contains the trivial character, and the Galois representation ρ of which we want to study
the Selmer group. One then deform it into a family of automorphic forms whose Galois
representations are irreducible, and thus obtain, by generalization of a famous lemma of
Ribet, extensions of 1 by ρ that give the desired elements in the Selmer group. So far, this
method has only allowed to construct one extension at a time ([BC],[SU1],[BC3]), that is
to prove a lower bound of 1 on the Selmer group4. However, two different sub-strategies
have been proposed (both during the Eigensemester in 2006) to get the lower bound on
the Selmer group predicted by the order of vanishing of the p-adic L-function of ρ. One,
by Skinner and Urban, would use for f an Eisenstein series on U(n + 2r) (where ρ has
dimension n and r is the number of independent extensions one expect to construct), the
other, by the author and Chenevier (see the final remarks in [BC3] for more details) uses for
f a cuspidal endoscopic non-tempered form for U (n + 2), and hopes that the number r of
independent extensions will be encoded into the singularity of the eigenvariety of U(n + 2)
at the point f , leading to the construction of r independent extensions by the results of
[BC3]. Both methods (at least it seems to me from my remembrance of Urban’s explanations of his and Skinner’s method at the eigensemester in 2006, since to my knowledge
those explanations never appeared in print) would need a good knowledge on the p-adic Lfunction on f , and of how it is related to the p-adic L-function of ρ. Such a knowledge is far
from being available, because in general little is known on higher rank p-adic L-function,
but also because in both methods the form f we work with is critical or θ-critical (and
there are fundamental reasons it is so, that it would take us too far to explain here), a case
which was not well understood even for GL2 . Therefore, understanding the case of GL2
critical p-adic L-function may be seen as a first, elementary step toward the general goal
of understanding the higher rank critical p-adic L-function.
2. Modular forms, refinements, and the eigencurve
2.1. The eigencurves.
2.1.1. The Coleman-Mazur-Buzzard eigencurve. Let W be the usual weight space (see
§3.1.3 below for more details). We shall denote by C the p-adic eigencurve of tame level
Γ1 (N ) (constructed by Coleman-Mazur [CM] in the case N = 1, p > 2, and in general
using similar methods by Buzzard [Bu]). The space C is equidimensional of dimension 1.
There exists a natural weight map κ : C → W, and it is finite and flat. We normalize κ
in such a way that if fx is a form of weight k + 2, then κ(x) is k.
4I observe that in [SU2], the proof of a lower bound of 2 when predicted by the sign of the Bloch-Kato
conjecture is announced. However, the proof has not been given yet (in particular, the crucial deformation
argument is missing, and does not appear either in Urban’s work in progress on eigenvarieties). Besides,
this method does not seem to be able to give lower bound greater than 2 – though the variant proposed at
the Eigensemester by the same authors does.
12
J. BELLAÏCHE
The operators Tl (l 6 |pN ), Up and the diamond operators, that is all elements of H define
bounded analytic functions on C.
If W = SpR is an affinoid subspace of W, and ν > 0 is a real number, we shall denote
by CW,ν the open affinoid subspace of the eigencurve C that lies above W and on which
one has vp (Up ) ≤ ν. By construction (cf. [Bu]), C is admissibly covered by the CW,ν for
certain pairs (W, ν) (those for which ν is adapted to W , see §3.2.4 below), for which CW,ν is
the rigid analytic spectrum of the affinoid sub-algebra of EndR (M † (Γ, W )≤ν ) generated by
the image of H, where M † (Γ, W ) denotes Coleman’s locally free of finite type R-module of
overconvergent modular forms of level Γ = Γ1 (N ) ∩ Γ0 (p), weight in W , and M † (Γ, W )≤ν
its sub-module of slope less than ν (cf. §3.2.4) which is finite flat over R. If κ ∈ W (Qp ) is
an integral weight k ∈ N, then there exists a natural H-isomorphism M † (W, Γ) ⊗R,κ Qp =
†
†
Mk+2
(Γ), where Mk+2
(Γ) is the Qp -space of overconvergent modular forms. This space
contains as an H-submodule the Qp -space of classical modular forms Mk+2 (Γ).
2.1.2. Cuspidal overconvergent modular forms and evil Eisenstein series. The following
definitions and results are natural generalizations of definitions and results stated in [CM]
in the case N = 1. Recall that Γ = Γ1 (N ) ∩ Γ0 (p).
†
Definition 2.1. An element f ∈ Mk+2
(Γ) is called overconvergent-cuspidal if it vanishes
†
at all cusps of X(Γ) in the Γ0 (p)-class of the cusp ∞. We call Sk+2
(Γ) the H-submodule
†
of Mk+2
(Γ) of overconvergent-cuspidal forms.
Definition 2.2. An Eisenstein series in Mk+2 (Γ) is called evil if it is overconvergent†
cuspidal as an element of Mk+2
(Γ).
Let us recall the well-known classification of Eisenstein series for the group Γ1 (M ). Let
χ and ψ be two primitive Dirichlet characters of conductors L and R. We assume that
χ(−1)ψ(−1) = (−1)k . Let
Ek+2,χ,ψ (q) = c0 +
X
qm
m≥1
X
(ψ(n)χ(m/n)nk+1 )
n|m
where c0 = 0 if L > 1 and c0 = −Bk,ψ /2k if L = 1. If t is a positive integer, let
Ek+2,χ,ψ,t (q) = Ek+2,χ,ψ (q t ) excepted in the case k = 0, χ = ψ = 1, where one sets
E2,1,1,t = E2,1,1 (t) − tE2,1,1 (q t ).
Proposition 2.3 (Miyake, Stein). Fix an integer M , and a Dirichlet character of
(Z/mZ)∗ .
The series Ek+2,χ,ψ,t (q) are modular forms of level Γ1 (M ) and character for all positives integers L, R, t such that LRt|M and all primitive Dirichlet character χ of conductor
L and ψ of conductor M , satisfying χψ = (and t > 1 in the case k = 0, χ = ψ = 1) and
moreover they form a basis of the space of Eisenstein series in Mk+2 (Γ1 (M ), ).
For every prime l not dividing M , we have
Tl Ek+2,χ,ψ,t = (χ(l) + ψ(l)lk+1 )Ek+2,χ,ψ,t .
CRITICAL p-ADIC L-FUNCTIONS
13
Proof — See Miyake ([Mi]) for the computations leading to those results, Stein ([St2]) for
the results stated as here.
From this description it follows easily that
Corollary 2.4. The Eisenstein series in Mk+2 (Γ1 (M )) that are new in the sense of the
introduction are exactly: the Eisenstein series Ek+2,ψ,χ,1 with M = LR (excepted of course
E2,1,1,1 which is not even a modular form); the Eisenstein series E2,1,1,M ’s when M is
prime. Those series generate their eigenspaces for the Hecke operators Tl , l does not divide
M , in Mk+2 (Γ1 (M ), ψχ).
We apply those results to the situation of our article:
Corollary 2.5. Let f be an Eisenstein series of weight k + 2 and level Γ1 (N ), eigenform
for the operator Tp , and let fα and fβ be its two refinement (forms for Γ = Γ1 (N ) ∩ Γ0 (p)).
(i) The valuation of α and β are, in some order 0 and k + 1.
(ii) If f is new, the H-eigenspace of fα and fβ are of dimension 1.
(iii) The form fβ is evil if and only if it is of critical slope, that is if vp (β) = k + 1.
Proof —
By Proposition 2.3, f lies in the same Tp -eigenspace as some Eisenstein series
Ek+2,χ,ψ,t , so if Tp f = ap f we have ap = χ(p) + ψ(p)pk−1 . Therefore, the roots α and β of
the polynomial (1) are χ(p) and ψ(p)pk+1 , and (i) follows.
Let us prove (ii). If f is new then f = Ek+2,ψ,χ,1 (q) with ψ and χ primitive of conductor
L and R, and N = LR by the corollary (we leave the exceptional case E2,1,1,M to the
reader). Applying Proposition 2.3 to M = N p show that the space of Eisenstein series
for Γ1 (N ) ∩ Γ0 (p) and character χψ (that is the space of Eisenstein series for Γ1 (N p) and
character χψ seen as a character of conductor N p) has dimension 2, and is generated by
Ek+2,χ,ψ,1 (z) = f (z) and Ek+2,χ,ψ,p (z) = f (pz). Since Up has two distinct eigenvalues α
and β on that space whose eigenvectors are respectively f (z) − βf (pz) and f (z) − αf (pz),
the result follows.
To prove (iii), let us assume k > 0, the case k = 0 needing trivial adjustments left to the
reader. Let us denote by V the subspace of Eisenstein series of Mk+2 (Γ, ε). We shall use
a slightly different basis (cf. [Mi, §7.1]) of V than the one introduced in Prop. 2.3, namely
the basis (Ek+2,L,R,χ,ψ ) indexed by quadruples (L, R, χ, ψ), where L, R are positive integers
such that LR|N p, χ is a non-necessary primitive character of modulus L and ψ a nonnecessary primitive character of modulus R, such that as characters of (Z/N pZ)∗ , we
P
P
have = ψχ, where Ek+2,L,R,χ,ψ (q) = c0 + m≥1 q m n|m (ψ(n)χ(m/n)nk+1 ) whith c0 = 0
if L > 1 and c0 6= 0 if L = 1. The Eisenstein series Ek+2,L,R,χ,ψ are eigenform for all Hecke
operators, including Up with eigenvalues χ(p) + ψ(p)pk+1 . Let us call V crit the linear span
of the form fβ where f is an Eisenstein series in Mk+2 (Γ1 (N ), ) which is an eigenform form
Up , and vp (β) = k + 1. We see easily that a basis of V crit is constituted of the Ek+2,L,R,χ,ψ
as above such that p|L (equivalently, χ(p) = 0). We deduce that dim V crit = dim V /2, since
14
J. BELLAÏCHE
we can establish a bijection between the quadruples (L, R, χ, ψ) as above that satisfies p|L
and those that do not, namely (L, R, χ, ψ) 7→ (L/p, R, χ, ψ) which makes sense because the
conductor of χ divides the conductor of , hence is prime to p, which allows us to see χ as
a character of modulus L/p, and because L/p is not divisible by p (since LQ|N p hence has
p-order at most one). Also, V crit is invariant by Γ0 (p) as is clear from [Mi, (7.1.17) and
(7.1.25)].
Now let us call V evil the space of evil Eisenstein series in Mk+2 (Γ, ε). If f is in V crit ,
f vanishes at ∞ and so does f|γ for all γ in Γ0 (p), hence f vanishes at all cusps that are
Γ0 (p)-equivalent to ∞, so f is evil. That is V crit ⊂ V evil .
Finally, we also note that dim V evil = dim V /2 since the number of cusps for Γ that
are Γp (p)-equivalent to ∞ is half the number of cusps for Γ = Γ1 (N ) ∩ Γ0 (p). Hence
V crit = V evil . This completes the proof.
Let us also note the following
Corollary 2.6 (Control theorem for cuspidal forms).
†
Sk+2
(Γ)<k+1 = Sk+2 (Γ)<k+1 .
Proof —
†
By Coleman’s control theorem [Co], any form f ∈ Sk+2
(Γ)<k+1 is classical. We
can assume that f is an eigenform, so the question is whether it is cuspidal or Eisenstein.
But if f was Eisenstein, it would be evil by definition, hence of critical slope k + 1 by point
(iii) of Corollary 2.5, which is absurd since it is of slope < k + 1.
2.1.3. The cuspidal eigencurve. Mimicking Coleman’s construction, it is easy to construct
a cuspidal analog S † (Γ, W ) of M † (Γ, W ), which is a Banach R-module (even satisfying
property (Pr) of [Bu]) with an action of H (and a H-equivariant embedding S † (Γ, W ) ,→
M † (Γ, W )) such that Up acts compactly, and with natural H-isomorphisms S † (Γ, W ) ⊗R,κ
†
Qp = Sk+2
(Γ), if κ ∈ W (Qp ) is an integral weight k. The general eigenvariety machine
([Bu]) then produces a cuspidal eigencurve C 0 of tame level Γ1 (N ), and by construction C 0
is equidimensional of dimension 1, provided with a map κ : C 0 → W which is locally finite
and flat, and is naturally identified with a closed subset of C.
2.1.4. Degree of the weight map and overconvergent modular forms.
Lemma 2.7. Let x be a point of C corresponding to a refined newform. Then in a neighborhood of C in x all classical points also correspond to refined newforms.
Proof — For N 0 a proper divisor of N let CN 0 denote the eigencurve of tame level Γ1 (N 0 )
constructed with the same algebra H we have considered so far, that is with no operators
Tl , for l a divisor of N/N 0 . By construction, there is a closed immersion CN 0 ,→ C.
CRITICAL p-ADIC L-FUNCTIONS
15
We claim that the image of this immersion does not contain x. If it did, there would be a
point x in CN 0 whose (semi-simple) Galois representation ρx would correspond to a newform
f of level Γ1 (N ). Hence the tame conductor of ρx would be N (this follows from the work of
Carayol [Ca] in the case f cuspidal, see [Li] for a definition of the tame conductor and this
result explicitly stated, and directly from the definitions in the Eisenstein case). On the
other hand, the conductor of the semi-simple Galois representation at classical points of CN 0
divides N 0 , and the same result follows at every point by semi-continuity of the conductor
(trivial from the definition). Hence N |N 0 , a contradiction which proves the claim.
Since the image of no CN 0 contains x, and since there is only a finite number of proper
divisors N 0 of N , there is a neighborhood of x which meets no CN 0 , and this implies the
lemma.
Lemma 2.8. Let f be a newform of level Γ1 (N ) and weight k + 2, and let α and β be
the two roots of the quadratic polynomial (1). Let fβ be one of its refinements, and x
the corresponding point in C, that we identify with the system of eigenvalues x : H → Q̄p
attached to fβ .
(i) The generalized eigenspace Mk+2 (Γ)(x) has dimension 1.
†
(ii) The generalized eigenspace Mk+2
(Γ)(x) has dimension e, where e is the length of
the affine ring of the scheme-theoretic fiber of κ : C → W at x.
If we assume in addition that f is cuspidal, or that f is Eisenstein and vp (β) > 0, then:
†
(iii) The point x belongs to C 0 and the generalized eigenspace Sk+2
(Γ)(x) has dimension
e0 , where e0 is the length of the affine ring of the scheme-theoretic fiber of κ : C 0 →
W at x. (We shall see in Cor. 2.17 that we always have e = e0 .)
Proof —
The first result is a well-known consequence of the theory of newforms in the
cuspidal case, and of Corollary 2.5(ii) in the Eisenstein case.
Let us prove the second result. First, we note that it reduces to (i) in the easy case where
†
κ is étale at x and vp (Up (x)) < k + 1, for in this case, e = 1, and Mk+2
(Γ)(x) = Mk+2 (Γ)(x)
by Coleman’s control theorem.
For a general x we can assume, shrinking W if necessary, that x is the unique geometric
point above k in the connected component X of x in CW,ν , and that vp (Up ) is constant on X.
Let us denote by M † (Γ, X) the R-submodule of M † (Γ, W ) corresponding to the component
X. This is a direct summand of M † (Γ, W ), hence it is a finite projective module over R.
One has by definition for every k 0 ∈ N ∩ W (Qp ) a natural isomorphism of H-modules
(8)
M † (Γ, X) ⊗R,k0 Q̄p = ⊕y∈X,κ(y)=k0 Mk†0 +2 (Γ)(y) .
The degree of the flat map κ|X : X → W is e by definition of e, so we have, for all points
0
0
k ∈ N ∩ W (Qp ) where κ|X is étale, e = |κ−1
|X (k )(Q̄p )|. If moreover k + 1 > vp (Up (x)),
0
0
then for any point y in κ−1
|X (k )(Q̄p ), such that y corresponds to a refined newform, one
has dim Mk†0 +2 (Γ)(y) = 1 by the easy case treated above. Therefore, by (8), for such a k 0 ,
16
J. BELLAÏCHE
the space M † (Γ, X) ⊗R,k0 Q̄p has dimension e. Since, in view of Lemma 2.7, we can find
integers k 0 ’s arbitrary closed in W to k satisfying the above conditions, we deduce that
†
†
the rank of MX
is e. By applying (8) at k, we see that dim Mk+2
(Γ)(x) = e since x is the
unique geometric point of X above k.
As for (iii), we note that it is clear that x belongs to C 0 if f is cuspidal; if f is Eisenstein,
†
the hypothesis vp (β) > 0 implies that fβ ∈ Sk+2
(Γ) by Corollary 2.5(iii). Therefore x
belongs to C 0 . The rest of the proof of (iii) is exactly as in (ii) replacing C by C 0 and M †
by S † everywhere.
Remark 2.9. The proof of this lemma is close to the proof of [BC2, Prop. 1(b)]. Note
that while [BC2] was written with the assumption that N = 1 (the only case the eigencurve
was constructed at that time), the arguments and results of this paper are still correct for
any N without any change if we simply use our Lemma 2.8 instead of [BC2, Prop. 1(b)]
and work with Eisenstein series that are newforms. In what follows, we shall use without
further comment the results of [BC2] for any level N .
2.2. Refinements.
2.2.1. The preferred Galois representation. As in the introduction, let f =
P
an q n be a
modular newform (cuspidal or eisenstein) of weight k + 2 and level Γ1 (N ),
Lemma 2.10. There exists a unique (up to isomorphism) Galois representation ρf : GQ →
GL2 (Q̄p ) such that
(i) The representation ρf is unramified outside N p and satisfies the Eichler-Shimura
relation tr (ρf (Frob l )) = al for all l prime to N p.
(ii) The restriction (ρf )|GQp is crystalline at p.
(iii) The representation ρf is indecomposable.
Moreover, the Hodge-Tate weights of (ρf )|GQp are 0 and k + 1 and the eigenvalues of the
crystalline Frobenius ϕ on Dcrys ((ρf )|GQp ) are the roots α and β of the quadratic polynomial
(1).
Proof —
In the case where f is cuspidal, the existence and uniqueness of ρf satisfying
(1) are well-known, as well as its properties at p, and ρf is irreducible in addition. In
the case where f is Eisenstein, what is well-known is the existence of a decomposable
representation χ1 ⊕ χ2 satisfying (i), crystalline at p, with say χ1 of Hodge-Tate weight 0
k+1
and χ2 of Hodge-Tate weight k + 1. In this case the quotient χ1 χ−1
(where is
2 is ω
the Dirichlet character of (Z/N Z)∗ defined by haif = (a)f ) and by the classification of
Eisenstein series, we have (−1) = 1, and moreover 6= 1 when k = 0 (since “E2 does not
exist”). The conjecture of Bloch-Kato, known by the work of Soulé for characters of GQ (cf.
[SO]), says that dim Ext1f (χ2 , χ1 ) = 1 and dim Ext1f (χ1 , χ2 ) = 0, where Ext1f parametrizes
the extensions that are crystalline at p. Therefore, there exists a unique representation ρf
CRITICAL p-ADIC L-FUNCTIONS
17
satisfying (i), (ii) and (iii) up to isomorphism, namely the extension of χ2 by χ1 defined by
any non-zero element in Ext1f (χ2 , χ1 ).
We call ρf the preferred representation attached to f . Note that since α 6= β (by
assumption) the choice of a refinement fα or fβ of f is equivalent to the choice of a
ϕ-stable line in Dcrys ((ρf )|GQp ), namely the eigenspace of the crystalline Frobenius ϕ of
eigenvalue α or β respectively. This line is called a refinement of ρf , and is the refinement
of ρf attached to the given refinement of f .
2.2.2. Refinements and the preferred Galois representation. Note that we have obviously
vp (α)+vp (β) = k+1 (recall that our forms are of weight k+2, not k), and 0 ≤ vp (α), vp (β) ≤
k + 1. We shall say that a refined modular form fβ is of critical slope if vp (β) = k + 1.
Then of course, for a given f as above, there is at least one of the two refinements fβ and
fα of f which is of non-critical slope, and exactly one if and only if vp (ap ) = 0 (in which
case f is called ordinary).
Proposition 2.11. For a refined form fβ , the following are equivalent.
(i) There exist non-classical overconvergent modular forms of weight k + 2 which are
generalized eigenvectors of H with the same eigenvalues as fβ . In other words,
†
the inclusion Mk+2 (Γ)(fβ ) ⊂ Mk+2
(Γ)(fβ ) is strict.
†
(i’) We have dim Mk+2
(Γ)(fβ ) ≥ 2.
(ii) We have vp (β) > 0 and the eigencurve C is not étale over the weight space at the
point x corresponding to fβ (see below)
(iii) We have vp (β) > 0 and the representation (ρf )|GQp is the direct sum of two characters.
(iv) The refinement of ρf attached to fβ is critical in the sense of [BC3], that is the
line of eigenvalue β in Dcrys ((ρf )|GQp ) is the same as the line defined by the weight
filtration.
Moreover if those property holds, vp (β) = k + 1, that is fβ is of critical slope.
In the case f is cuspidal, those properties are also equivalent to
†
†
(v) The modular form fβ is in the image by the operator θk : M−k
(Γ) → Mk+2
(Γ)
d k+1
which acts on q-expansion by (q dq
)
.
In the case f is Eisenstein, (v) is always true.
Proof —
All this is well known and we just give references or short indications of proof.
The equivalences between (i), (i’) and (ii) follow directly from Lemma 2.8.
The equivalence between (iii) and (iv) follows easily from the weak admissibility of
Dcrys ((ρf )|GQp ), as does the assertion after the moreover: see [BC3, Remark 2.4.6] for
details.
To finish the proof we separate the Eisenstein and cuspidal cases.
18
J. BELLAÏCHE
In the Eisenstein case, we only have to show that (i) is equivalent to (iii) to complete the
proof of the equivalence of (i) to (iv). But this is a theorem of the author and Chenevier
([BC2, Théorème 3]). We also have to check that (v) always hold. But the limit case
of Coleman’s control theorem (cf. [Co, Corollary 7.2.2]) states that the cokernel of θk in
(Mk+2 (Γ)† )(x) is isomorphic to Sk+2 (Γ)(x) , that is 0, or in other words, θk is surjective onto
(Mk+2 (Γ)† )(x) , so obviously fβ is in its image.
In the cuspidal case, we note that the equivalence between (v) and (iii) is a deep result of
Breuil and Emerton ([BrE, Theorem 1.1.3]), so it only remains to prove that (v) is equivalent
to (i’) to complete the proof. But again by the limit case of Coleman’s control theorem,
the image of θk has codimension 1 in Mk+2 (Γ)†(x) , so (i’) is equivalent to the assertion that
the image of θk is not 0. Therefore it is clear that (v) implies (i’). Conversely, since θk
is Hecke-equivariant up to a twist (by the character Tl 7→ (1 + l)k+1 , Up 7→ pk+1 ), if the
image of θk is non-zero there is a non-zero eigenvector in it. By looking at the eigenvalues,
this eigenvector has to be fβ , which proves (v).
Definition 2.12. We say that fβ is critical if conditions (i) to (iv) are satisfied and θcritical if condition (v) is satisfied.
2.2.3. Classification of critical and critical-slope refined modular form.
Proposition 2.13. Let f be a newform as above. Then there exists a refinement fβ of f
of critical slope in and only in the following cases (and in those cases there exist exactly
one critical-slope refinement)
(a) the newform f is cuspidal, not CM, and ordinary at p (that is vp (ap ) = 0).
(b) The newform f is cuspidal CM, for a quadratic imaginary field K where p is split.
(c) The newform is Eisenstein.
Remark 2.14. In any of the cases (a), (b) or (c) of the proposition, let fβ be the criticalslope refinement. Then
- In case (a), it is expected, but not known in general, that fβ is non-critical (hence
not θ-critical).
- In case (b), fβ is always critical (hence θ-critical).
- In case (c), fβ is always θ-critical, and it is expected, but not known in general,
that fβ is non-critical. It is known however that fβ is non-critical when p is a
regular prime, or for any given p and N that at most a finite number of fβ ’s are
critical.
Proof —
Since vp (β) + vp (α) = k + 1, and vp (β) ≥ 0, vp (α) ≥ 0, if one of those valuation
is k + 1, then the other is 0, and therefore we have vp (ap )vp (−β − α) = 0. This means that
f is ordinary at p. Of course, this is automatic for an Eisenstein series, and equivalent to p
being split in K for a CM form of quadratic imaginary field K. This proves the proposition.
CRITICAL p-ADIC L-FUNCTIONS
19
Let us justify the remark: it is clear that if f is cuspidal CM, (ρf )|GQp is the direct sum
of two characters, so fβ is critical by Prop. 2.11(iii). If f is cuspidal not CM, a well-known
conjecture asserts that (ρf )|GQp is indecomposable, and if this were true, the non-criticality
of fβ would follow. As for Eisenstein series, see [BC3, Théorème 3] and the following
remark.
2.2.4. Decent refined modular forms. We recall
Proposition 2.15. Let f be a cuspidal newform of level Γ1 (N ) and weight k + 2 and p a
prime number not dividing N . Then if any of the following holds:
(i) for any quadratic extension L/Q with L ⊂ Q(ζp3 ), (ρ̄f )|GL is absolutely irreducible;
(ii) the representation ρ̄f is not absolutely irreducible, and its two diagonal characters
are distinct after restriction to GQ(ζp∞ ) ;
(iii) the form f is not CM, and f is cuspidal or special at all primes dividing N (e.g.
f is of trivial nebentypus and N is square-free);
(iv) the form f is CM;
we have Hf1 (GQ , ad ρf ) = 0, and in particular fβ is decent for any choice of the root β
of (1).
Proof —
Cf. [K2] in cases (i) and (ii), and [W] in case (iii). In case (iv), it suffices
to show by inflation-restriction that Hf1 (GK , adρf ) = 0 where K is quadratic imaginary
field attached to f . But since the representation ρf is induced from K, (ρf )|GK is the
sum of two characters χ and χc (after extending the scalars to Q̄p ), conjugate under the
outer automorphism of GK defined by the non-trivial element in Gal(K/Q). Therefore,
Hf1 (GK , adρf ) = Hf1 (GK , χ⊗χ−1 )⊕Hf1 (GK , χ⊗(χc )−1 )⊕Hf1 (GK , χc ⊗χ−1 )⊕Hf1 (GK , χc ⊗
(χc )−1 ). The first and last terms are Hf1 (GK , Q̄p ) which are 0 by Class Field Theory. The
second and third terms are of the form Hf1 (GK , ψ) where ψ is a character of motivic weight
0, and their nullity result by standard arguments from the main conjecture for imaginary
quadratic fields proved by Rubin ([R]).
2.3. Smoothness of the eigencurve and consequences. The idea of the following
theorem is due to G. Chenevier (personal communication). In the case of an Eisenstein
series, it was proved earlier in [BC2].
Theorem 2.16. Let f be a decent newform of weight k + 2 and level Γ1 (N ) and fβ be a
refinement of f . If f is Eisenstein, assume that vp (β) 6= 0. Let x be the point on C 0 (see
Lemma 2.8) corresponding to f . Then C and C 0 are smooth at x.
Proof —
It suffices to prove the result for C, since x ∈ C 0 and C 0 is equidimensional of
dimension 1.
20
J. BELLAÏCHE
When f is Eisenstein, this result is the first main theorem of [BC2]. If fβ is non-critical,
then the eigencurve is étale at x by Lemma 2.8, so it is smooth. Therefore, since f is decent,
we can assume that f is cuspidal, and that fβ is critical. Thus it satisfies Hf1 (GQ , adρf ) = 0.
Since f is cuspidal, the Galois representation ρf of GQ,N p (the Galois group of the
maximal unramified algebraic extension outside N p and ∞) attached to it is irreducible.
Therefore, by a well-known theorem of Rouquier and Nyssen (cf. [Ro] or [Ny]), the Galois
pseudocharacter T : GQ,N p → Tx , where Tx is the completed local ring of the eigencurve C
at the point x corresponding to fβ (after scalar extension to Q̄p ) is the trace of a unique representation ρ : GQ,N p → GL2 (Tx ) whose residual representation is ρf . We have normalized
T and ρ so that 0 is a constant Hodge-Tate-Sen weight.
We consider the following deformation problem: for all Artinian local algebra A with
residue field Q̄p , we define D(A) as the set of strict isomorphism classes of representations
ρA : GQ → GL2 (A) deforming ρf such that
(i) The restriction (ρA )|GQp has a constant weight equal to 0;
(ii) the A-module Dcrys ((ρA )|GQp )ϕ=β̃ is free of rank one for some β̃ lifting β;
(iii) for l|N , the restriction (ρA )|Il to the inertia subgroup at l is constant.
By [K1, Prop. 8.7] (using the fact that β 6= α and that 0 6= k +1), and [BC3, Prop. 7.6.3(i)]
(for condition (iii)), we see that D is pro-representable, say by a complete Noetherian local
ring R.
We claim that ρ ⊗ Tx /I is, for all cofinite length ideal I in Tx , an element of D(Tx /I):
property (i) follows from our normalization and Sen’s theory, see e.g. [BC3, Lemma 4.3.3(i)];
property (ii) is Kisin’s theorem (see [K1] or [BC3, chapter 3]). For property (iii), let l|N . Let
gen
Nx and Ns(x)
be the special monodromy operator of ρ|Il at x and the generic monodromy
operator of T|Il at a component s(x) of the eigencurve through x (cf. [BC3, definition
7.8.16]). To prove (iii), by [BC3, Prop. 7.8.19 and Lemma 7.8.17] it is sufficient to prove
gen
that Nx ∼ Ns(x)
(see [BC3, §7.8.1] for the definition of the equivalence relation ∼ and the
pre-order ≺ on the set of nilpotent matrices). Assume first that f is not special at l, so
the monodromy operator Nx of (ρf )|Il is trivial. Then in a neighborhood of f , there is a
dense set of classical points that are newforms of the same level (Lemma 2.7) and same
nebentypus, hence that are not special either, and thus have a trivial monodromy operator.
gen
Hence by [BC3, Prop. 7.8.19(2)], Nx = Ns(x)
= 0. If f is special, then Nx is non-zero, but
gen
gen
since by [BC3, Prop. 7.8.19(3)] Nx ≺ Ns(x)
and we are in dimension 2, Nx ∼ Ns(x)
in this
case as well. This completes the proof of the claim.
Thus, ρ defines a morphism of algebras R → Tx . A standard argument shows that this
morphism is surjective. Since Tx has Krull dimension 1, if we prove that the tangent space
of R has dimension at most 1, it would follow that the map R → Tx is an isomorphism
and that R is a regular ring of dimension 1. This would complete our proof.
The tangent space of R, tD := D(Q̄p [ε]), lies inside the tangent space of the deformation
ring of ρf without local condition, which is canonically identified with H 1 (GQ , adρf ). Since
CRITICAL p-ADIC L-FUNCTIONS
Hf1 (GQ , adρf ) = 0, this space injects into
Q
l|N p
21
H/f1 (GQl , adρf ) (where H/f1 means H 1 /Hf1 )
and so does tD . The image of tD in H/f1 (GQl , adρf ) is 0 for l|N by (iii). Hence tD injects
in H/f1 (GQp , adρf ).
Since fβ is critical, (ρf )|GQp is by Prop. 2.11 the direct sum of two characters, χ1 and χ2 .
Both are crystalline, and say χ1 has weight 0 while χ2 has weight k +1. Since vp (β) = k +1,
β is the eigenvalue of the crystalline Frobenius on Dcrys (χ2 ). Then we compute:
−1
−1
−1
1
1
1
H/f1 (GQp , adρf ) = H/f1 (GQp , χ1 χ−1
1 )⊕H/f (GQp , χ2 χ1 )⊕H/f (GQp , χ1 χ2 )⊕H/f (GQp , χ2 χ2 ).
The condition on the constant weight 0 in our deformation problem D implies that the
image of tD in the first factor is 0. The condition on Dcrys (−)ϕ=β̃ implies that the image of
td in the third and fourth factors are 0 (the third factor is 0 anyway). Therefore tD injects
−1
in H/f1 (GQp , χ2 χ−1
1 ). Since χ2 χ1 is not trivial (its Hodge-Tate weight is not 0), local Tate
duality and Euler characteristic formula implies that H 1 (GQp , χ2 χ−1
1 ) has dimension 1,
−1
1
excepted if χ2 χ−1
1 is the cyclotomic character of GQp , in which case H (GQp , χ2 χ1 ) has
1
dimension 2 but H/f
(GQp , χ2 χ−1
1 ) has dimension 1. Therefore, tD has dimension at most
one, which is what remained to prove.
Corollary 2.17. Keep the notation and hypotheses of the theorem, but do not assume that
†
x is decent anymore. Then C and C 0 are locally isomorphic at x and one has Sk+2
(Γ)(fβ ) =
†
Mk+2
(Γ)(fβ )
Proof — If x is decent, then C is a smooth curve at x, so C 0 which is a union of irreducible
components of C and contains x, is isomorphic to C in a neighborhood at x. If x is not
decent, then in particular f is cuspidal so ρx is irreducible, and x cannot belong to an
irreducible component of C not in C 0 since on those Eisenstein components the Galois
representations are everywhere reducible. This proves the first assertion.
Applying Lemma 2.8(ii) and (iii) we deduce that with the notations of this lemma e = e0 ,
†
†
hence the inclusion Sk+2
(Γ)(fβ ) ⊂ Mk+2
(Γ)(fβ ) is an equality by equality of dimension. 3. Construction of the eigencurve through distributions-valued modular
symbols
In this section we use ideas of Stevens to give a construction of two versions C + and
C − of the eigencurve using families of overconvergent modular symbols. While Stevens has
those ideas since at least 2000, when he exposed them at the Insitut Henri Poincaré, there
is no preprint, nor circulating notes, giving the details of his method. Since the results have
been used several times since (e.g [SP2, Theorem 7.1]) or [PAR] ), we hope that providing
the details of the construction will be useful to the community.
±
Actually we shall only construct the standard local pieces CW,ν
of the eigencurves,
parametrizing modular symbols of weights belonging to a fixed affinoid subset W of the
weight space W, and of slopes bounded above by ν. Those local pieces are all what we
22
J. BELLAÏCHE
need (and all what is needed in most applications, like in [SP2] or [PAR]). Moreover the
global eigencurve can in principle be constructed by gluing the local pieces as explained in
[Bu] (see the forthcoming [B2] for details of the global construction).
±
The basic idea of the construction of CW,µ
is clear. We need to construct an R-module
(if W = SpR) of ”families over W of modular symbols”, which is orthonormalizable (so we
can apply Riesz theory for the operator Up , which needs to be compact) and which satisfies
a ”specialization theorem”, which says that its fibers at various weights w in W give back
the spaces of modular symbols of weight w (at least after restriction to the finite slope
subspaces). But defining and proving the properties of that module involve many technical
difficulties, that we overcome using in particular a big amount of the published results and
techniques on modular symbols of Pollack and Stevens (especially [S] and [SP2]). To our
surprise, our specialization theorem (Theorem 3.10) has an exception: when the weight
w is 0 (which corresponds to modular forms of weight 2), the fiber at w of the module of
modular symbols is just an hyperplane in the space of modular symbol of weight 0. This has
no consequence on our applications, but this implies that the statement of [SP2, Theorem
7.1] must be (very slightly) changed to be exception-free: see Theorem 3.27 below.
After the construction, we use a result of Chenevier to show that these eigencurves are
actually canonically embedded in the Coleman-Mazur eigencurve C and contain its cuspidal
locus C 0 .
3.1. Families of distributions.
3.1.1. Spaces of distributions. We recall from [S] and [SP2] the following definitions:
For r ∈ |C∗p | = pQ , we set, following Stevens,
B[Zp , r] = {z ∈ Cp , ∃a ∈ Zp , |z − a| ≤ r}.
This has a structure of an affinoid space over Qp , and we denote by A[r] (or by A[r](Qp ))
the ring of affinoid functions on B[Zp , r]. For f ∈ A[r], we set ||f ||r = supz∈B[Zp ,r] |f (z)|.
This norm makes A[r] a Qp -Banach algebra. Concretely an element f of A[r] is described
P∞
by the data, for any e ∈ Zp , of a formal expansion f (z) = n=0 an (e)(z − e)n converging
on the closed ball of center e and radius r, the sums of those formal series for various e
agreeing on B[Zp , r], and in particular on Zp . Hence an f ∈ A[r] determines in particular
a function f : Zp → Qp and in turn this function determines f ∈ A[r] (since it determines
the power expansion of f at any e ∈ Zp ).
We denote by D[r] (or D[r](Qp )) the continuous dual Hom(A[r], Qp ) of A[r]. It is a
Banach space over Qp for the norm ||µ||r = supf ∈A[r]
|µ(f )|
||f ||r .
It is also a module over the
ring A[r] by setting f · µ(g) = µ(f g) for f, g ∈ A[r], µ ∈ D[r].
There are natural restriction maps A[r1 ] → A[r2 ] for r1 > r2 that are clearly injective,
have dense image (since their images contain the dense subspace of polynomials in z), and
are compact (by e.g. [Co2, Prop. A.5.2]). So their transposed maps D[r2 ] → D[r1 ] are also
injective and compact (cf. [Sc, Lemma 16.4])
CRITICAL p-ADIC L-FUNCTIONS
Finally for any real number r ≥ 0, we set D† [r] = proj limr0 >r,
23
r 0 ∈pQ
D[r0 ]. In other
words D† [r] is the intersection of the D[r0 ] for r0 > r. The space D† [r] with the norms || ||r0
for r0 > r is a Frechet space (cf. [Sc, Prop. 16.10]; for the definition of a Frechet space over
Qp , see [Sc, §8]).
3.1.2. Modules of distributions. Let R be any affinoid Qp -algebra.
b Qp A[r]. As above, an element f ∈ A[r](R) can be seen as a
We set A[r](R) = R⊗
function f : Zp → R which around every e ∈ Zp admits a Taylor expansion converging on
a closed disc of radius r.
b Qp D[r]. Thus A[r](R) and D[r](R) are Banach R-modules, but
We set D[r](R) = R⊗
note that when R is not finite dimensional over Qp , D[r](R) is not the Banach dual of
A[r](R). Also, A[r](R) is an R-Banach algebra, and D[r](R) has a natural structure of
an A[r](R)-module (since the natural map R ⊗ A[r] → EndR (D[r](R)) is continuous and
thus extends into a map A[r](R) → EndR (D[r](R)), where EndR (D[r](R)) is the Banach
R-modules of bounded endomorphisms of D[r](R)).
By definition, the formation of A[r](R) and D[r](R) commutes with any base change, in
b R A[r](R) → A[r](R0 ) is
the sense that if R → R0 is a morphism of affinoid Qp -algebra, R0 ⊗
an isomorphism, and similarly for D[r](R0 ). The structure of A[r](R0 )-module on D[r](R0 )
is also compatible with the structure of A[r](R)-module on D[r](R) through those isomorphisms.
Lemma 3.1.
(i) The modules A[r](R) and D[r](R) are potentially orthonormalizable
Banach R-modules.
(ii) For r1 > r2 , the natural maps A[r1 ](R) → A[r2 ](R) and D[r2 ](R) → D[r1 ](R) are
compact.
Proof — Since the norm of every element of A[r] is an element of |Q∗p | (by our hypothesis
that r ∈ |Q∗p |), and since the same result holds, as a consequence, for D[r], those Banach
b
b
spaces are potentially orthonormalizable by [Se, Prop. 1]. Therefore, R⊗A[r]
and R⊗D[r]
are potentially orthonormalizable Banach R-modules by [Bu, Lemma 2.8], which proves
(i). The point (ii) follows form [Bu, Corollary 2.9].
For r ≥ 0, we define D† [r](R) as proj limr0 >r D[r0 ](R). Again, since the maps D[r2 ](R) →
D[r1 ](R) for r1 > r2 are injective, D† [r](R) is simply the intersection for r0 > r of the
D† [r0 ](R).
We shall need below the notion of completed tensor product of two Frechet spaces over
Qp , which is introduced in [Sc, §17], cf. in particular [Sc, Prop. 17.6]. We shall need
repeatedly the following observation, where for V a locally convex space, we denote by
c0 (V ) the locally convex space of sequences (vn )n≥0 of elements of V that converge to 0: if
b is naturally isomorphic to the Frechet space c0 (V ). Indeed,
V is a Frechet, then c0 (Qp )⊗V
one sees immediately that the obvious map c0 (Qp ) × V → c0 (V ) is separately continuous,
24
J. BELLAÏCHE
and has dense image, hence an isomorphism between the completion of c0 (Qp ) ⊗ V and
c0 (V ).
b † [r] → D† [r](R).
Lemma 3.2. We have a natural isomorphism of Frechet spaces R⊗D
0
b † [r] ,→ R⊗D[r
b
b † [r] ,→
Proof — The maps R⊗D
] for r0 > r induces an injective map R⊗D
proj limr0 >r D[r0 ](R) = D† [r](R). This map is obviously an isomorphism when R is finite
over Qp . Otherwise since R is orthonormalizable, we can choose an isomorphism R ' c0 (Qp )
and by the above observation, we only need to prove that the natural map c0 (D† [r]) →
proj limr0 >r c0 (D[r0 ]) = ∩r0 >r c0 (D[r0 ]) is an isomorphism, which is clear since a sequence
(vn ) in D† [r] goes to 0 if and only if ||vn ||r0 goes to 0 for all r0 > r.
We shall refrain from stating that the formation of D† [r](R) commutes with base change,
because this would suppose known the notions of Frechet R-modules and of the completed
tensor products of Frechet R-modules, which are not mentioned in our basic references ([Sc]
or [BosGuRe]), but a special case is proven in Lemma 3.5 below.
3.1.3. The weight space. Let W = Homcont,group (Z∗p , Gm ) be the standard one-dimensional
weight space. As is well known, W has a structure of rigid analytic space over Qp . If L is
any normed extension of Qp , a point w ∈ W(L) corresponds by definition to a continuous
morphism of groups w̃ : Z∗p → L∗ .
If k ∈ N, then the morphism k̃(t) = tk for all t ∈ Z∗p is a character Z∗p → Q∗p , and we
denote by k the corresponding point in W. Thus N ⊂ W(Qp ), and we call points in N of
W(Qp ) integral weights. For example, 0 ∈ W(Qp ) and this point corresponds to the trivial
character 0̃ : Z∗p → Q∗p , t 7→ 1.
Let q = p if p > 2, q = 4 if p = 2. For x ∈ Z∗p , let hxi ∈ Z∗p be the Teichmuller
lift of x (mod q) ∈ (Z/qZ)∗ . As is well known, there is a natural isomorphism Z∗p '
(Z/qZ)∗ × (1 + qZp ) given by x 7→ (x (mod q), x/hxi) and the group 1 + qZp is procyclic.
Choosing a generator γ of 1 + qZp identifies W with the disjoint union of φ(q) copies of the
rigid open ball of center 0 and radius 1.
If W = SpR is an admissible affinoid of W, the natural immersion SpR = W ,→ W
defines a canonical element K ∈ W(R), that is a continuous morphism of groups
K̃ : Z∗p → R∗ .
If w ∈ W (Qp ), we can see w as a ring morphism R → L, and we have formally
(9)
w̃ := w ◦ K̃.
Lemma 3.3. If w ∈ W(Qp ) then there is a positive real number r(w) > 0 such that the
map z 7→ w̃(1 + pz), Zp → Q∗p belongs to A[r(w)].
Similarly, if W = SpR is an affinoid of W, there exists an r(W ) > 0 such that the map
z 7→ K̃(1 + pz), Zp → R∗ belongs to A[r(W )](R).
CRITICAL p-ADIC L-FUNCTIONS
Proof —
25
We prove the second assertion in the case p > 2 (the first assertion is similar
and simpler, the case p = 2 needs only trivial modifications). Let us denote by | |R the
supremum norm on the affinoid R.
Consider the continuous function 1 + pZp 7→ R, u 7→ expp (logp (K̃(γ)) logp (u)/ logp (γ)).
It is clearly a character of u, which agrees with K̃ on the topological generator γ. Therefore
−1
it is K̃(u). For u ∈ Cp , |u − 1| < min(1/p, p p−1 | logp (γ)|/| logp (K̃(γ))|R ) the formal power
series expp (logp (K̃(γ)) logp (u)/ logp (γ)) in R[[u]] converges, and its value for such an u
which is moreover in 1 + pZp is, by the above, K̃(u). Taking u = 1 + pz, the results follows.
We end up this § with a technical definition:
Definition 3.4. We shall say that an admissible affinoid of W = SpR of W is nice if R
is a PID, the residue ring R̃ is a PID (where R̃ = R0 /pR0 , and R0 is the unit ball for the
supremum norm in R), and the points of Z ∩ W are dense in W .
For example, if k ∈ Z ⊂ W, and if W is the closed ball of center k and radius p−n for
n = 1, 2, . . . in the open ball of center 0 and radius 1 in which k lies (for any identification
of W with a union of such open balls defined by a generator γ of 1 + qZp as above), then
it is clear that W is a nice affinoid. In particular, every k ∈ Z has a basis of nice affinoid
neighborhood in W.
3.1.4. Actions of Σ0 (p). As in [S], let
a b
Σ0 (p) = {
∈ M2 (Zp ), p 6 |a, p|c, ad − bc 6= 0}.
c d
This is a monoid that contains the standard Iwahori group at p as a submonoid. If w ∈
W(Qp ) is a point on the weight space, and w̃ : Z∗p → Q∗p is the corresponding character,
then following Stevens ([S]) we define the weight-w actions of Σ0 (p) on A[r] and D[r], for
any positive real number r ≤ r(w) (where r(w) is as in Lemma 3.3) by
b + dz
(γ ·w f )(z) = w̃(a + cz)f
, f ∈ A[r]
a + cz
(µ ·w γ)(f ) = µ(γ ·w f ), µ ∈ D[r]
a b
where γ =
. Here, we see w̃(a + cz) = w̃(a)w̃(1 + ac z) as an element of A[r(w)],
c d
b+dz
hence by restriction of A[r], using Lemma 3.3; also, it is clear that a+cz
belongs to B[Zp , r]
b+dz
if z does, so f a+cz belongs to A[r].
When we think of A[r] and D[r] as provided with the weight w action of Σ0 (p), we
denote them by Aw [r] and Dw [r]. Note that when w = k is integral, the weight-k actions
are exactly the ones defined in the introduction (§1.4.1).
Finally, since the weight w actions on the various D[r] are obviously compatible with
the restriction maps, we can also define a weight-w action on D† [r] as soon as r < r(w)
26
J. BELLAÏCHE
(we need a strict inequality here, so that we have an action on the D[r0 ] for r < r0 < r(w)),
and we denote this space endowed with this action by D†w [r].
Now, let W = SpR be an affinoid of W. For r ≤ r(W ) (with r(W ) as in Lemma 3.3), we
shall define a right-action of Σ0 (p) on the R-modules D[r](R) that interpolates the various
weight w actions
for
w ∈ W = SpR.
a b
b Qp D[r] = D[r](R) by
Fix γ =
∈ Σ0 (p). We define a map γ : R ⊗Qp D[r] → R⊗
c d
sending r ⊗ µ, for r ∈ R and µ ∈ D[r] to
(r ⊗ µ)|γ = K̃(a + cz)(r ⊗ µ|0 γ ).
Here r⊗µ|0 γ ∈ R⊗Qp D[r] is seen as an element of D[r](R), and we use the natural A[r](R)module structure on D[r](R) to define the multiplication of this element by K̃(a + cz) ∈
A[r](W ) (cf. Lemma 3.3). Since this map is continuous, it extends uniquely to a continuous
map from D[r](R) to itself. Hence an action of Σ0 (p) on D[r](R).
Finally, we note that the preceding actions on D[r0 ](R) are compatible for various r, so
they define an action on the projective limit of those spaces for r0 > r, namely on what we
have called D† [r](R).
Lemma 3.5. If w ∈ SpR(Qp ) = Hom(R, Qp ) then there are canonical isomorphisms of
Qp -Banach spaces, compatible with the action of Σ0 (p),
D[r](R) ⊗R,w Qp
†
D [r](R) ⊗R,w Qp
→ Dw [r]
→ D†w [r]
Proof — The existence of the first isomorphism is just an instance of the compatibility of
the formation of D[r](R) with base change and we need only to check that it is compatible
with the actions of Σ0 (p). It is enough to do so on the dense subspace R ⊗ D[r] of D[r](R).
By R-linearity we need to check that the image of (1 ⊗ µ)|γ = K̃(a + cz)(1 ⊗ µ|0 γ ) in Dw [r]
is just µ|w γ . This image is (using the compatibility of the A[r](R)-structure on D[r](R)
with base change) is w(K̃(a+cz))µ|0 γ , that is w̃(a+cz)µ|0 γ using (9). But this distribution
is just µ|w γ , as one immediately checks by evaluating it on an f ∈ A[r]. This completes
the proof for the first isomorphism.
The existence of last isomorphism results from the first one by taking the projective limit
w
0 ×u
0
b
b
of the exact sequences 0 → (R⊗D[r
]) → (R⊗D[r
]) → D[r0 ] → 0, where u is a generator
0
b
of the ideal Ker w. Since those exact sequences are split (a section is D[r0 ] → R⊗D[r
],
b
µ 7→ 1⊗µ),
with compatible splitting for various r0 , the limit exact sequence is still exact
and gives the last isomorphism, which is clearly Σ0 (p)-compatible.
3.2. Distributions-valued modular symbols and their families.
CRITICAL p-ADIC L-FUNCTIONS
27
3.2.1. Reminder on modular symbols (after Stevens [S]). From now, we fix an integer N
prime to p and set Γ = Γ1 (N ) ∩ Γ0 (p) ⊂ GL2 (Q).
Let S0 (p) be the submonoid of Σ0 (p) consisting of matrices whose entries are rational
integers. Note that Γ ⊂ Γ0 (p) ⊂ S0 (p) ⊂ GL2 (Q).
Let ∆0 be the abelian group of divisors of degree 0 on P1 (Q). It is provided with a
natural action of GL2 (Q), hence by restriction, of S0 (p), and in particular of Γ.
For A any abelian group with a right action of S0 (p), the abelian group Hom(∆0 , A)
has a natural action of S0 (p) coming from the actions of S0 (p) on ∆0 and A (namely
Φ|γ (δ) = Φ(γ · δ)|γ for Φ ∈ Hom(∆0 , A) and all δ ∈ ∆0 .) Recall that one defines the
A-valued modular symbols with level Γ as
SymbΓ (A) := Hom(∆0 , A)Γ .
By its very definition as the abelian group of Γ-invariants in the larger abelian group with
S0 (p)-action Hom(∆0 , A), the abelian group SymbΓ (A) inherits an action of the Hecke
algebra of S0 (p) w.r.t Γ. This algebra contains the classical Hecke operators Tl for all
prime l not dividing N p, the diamond operators hai (a ∈ (Z/N Z)∗ ) as well as the classical
Atkin-Lehner operator Up , that is to say it contains H. Since the matrix diag(−1, 1) belongs
to S0 (p) and normalizes Γ, it also defines an involutive operator on SymbΓ (A) commuting
with the action of H. We denote by Symb±
Γ (A) the subgroup of SymbΓ (A) of elements
that are fixed or multiplied by −1 by that involution. If 2 acts invertibly on A, we have
−
SymbΓ (A) = Symb+
Γ (A) ⊕ SymbΓ (A). Note also that the functor SymbΓ (−) is obviously
left exact.
Of course, if A has a structure of module over a commutative ring B such that the
S0 (p)-action is B-linear, then SymbΓ (A) (resp. Symb±
Γ (A)) has a natural structure of Bmodule and the Hecke operators are B-linear applications. If B is a Banach algebra and
A is a Banach B-module, such that S0 (p) acts by continuous endomorphisms and Γ by
unitary endomorphisms, then SymbΓ (A) has a structure of Banach B-module defined by
||Φ|| = supD∈∆0 ||Φ(D)|| for Φ ∈ SymbΓ (A), on which the Hecke operators act continuously.
3.2.2. Families of distributions-valued modular symbols. Let us fix a nice (see Definition 3.4)
affinoid subdomain W = SpR of the weight space W, and consider the R-modules Symb±
Γ (D[r](R)),
for 0 < r ≤ r(W ). We summarize their properties in
Proposition 3.6. For any choices of sign ±,
(i) the modules Symb±
Γ (D[r](R)) are potentially orthonormalizable Banach R-modules;
(ii) the elements of H act continuously on those modules, and Up acts compactly.
(iii) For any two real number 0 < r, r0 ≤ r(W ), the modules Symb±
Γ (D[r](R)) and
0
Symb±
Γ (D[r ](R)) are linked in the sense of [Bu, §5]
Proof —
We have already seen that Symb±
Γ (D[r](R)) is a Banach R-module. Let (δi )i∈I
be a finite family of generators of ∆0 as a Z[Γ]-modules (such a family exists by Manin’s
28
J. BELLAÏCHE
theorem). The map
I
Symb±
Γ (D[r](R)) → D[r](R)
(10)
I
given by Φ 7→ (Φ(δi ))i∈I embeds Symb±
Γ (D[r](R)) as a closed subspace of D[r](R) which
is potentially othonormalizable by Lemma 3.1. Therefore, (i) follows from the general
lemma 3.7 below.
The continuity of the actions of elements of H is clear by the discussion on the topology
above. The compactness of Up follows from the fact that it factors through Symb±
Γ (D[r/p](R))
±
(see [S]), combined with the fact that the natural map Symb±
Γ (D[r/p](R)) → SymbΓ ([.r]R)
is compact, as it is a restriction to some closed subspaces of the natural map D[r/p](R)I →
(D[r](R))I (using (10)), and as the latter map is compact by Lemma 3.1(ii).
For the last point, we just note that the above proves that the modules Symb±
Γ (D[r](R))
0
0
0
and Symb±
Γ (D[r ](R)) are linked for any r such that r/p ≤ r ≤ r. Since to be linked is an
equivalence relation, (iii) follows.
Lemma 3.7. Let W = SpR be a nice affinoid of W, M be any potentially orthonormalizable Banach R-module, and N be any closed sub-module of M . Then N is potentially
orthonormalizable.
Proof —
We can change the norm of M so that M becomes orthonormalizable. In
particular, the norm of any element of M is a norm of an element of R, a property that
N obviously inherits. Therefore, to prove that N is orthonormalizable it is enough to
prove that Ñ is free as an R̃-module, where R̃ (resp. Ñ , M̃ ) is defined as R0 /pR0 (resp.
N 0 /pN 0 , M 0 /pM 0 ) and R0 (resp. N 0 , M 0 ) is the closed unit ball of R (resp. N , M ). But
the natural map of R̃-modules Ñ → M̃ is injective, since (pM 0 ) ∩ N = pN 0 obviously. So
Ñ is free as a sub-module of the module M̃ which is free (since M is orthonormalizable)
over the principal ideal domain R̃.
3.2.3. The specialization theorem.
Definition 3.8. The system of eigenvalues of E2 is the character H → L (L any ring) that
sends Tl to 1 + l, Up to p, hai to 1.
Lemma 3.9. Let W = SpR be a nice affinoid of W and 0 ≤ r < r(W ) If 0 6∈ W (Qp ) =
SpR(Qp ) = Hom(R, Qp ), then H0 (Γ, D† [r](R)) = 0. If 0 ∈ W (Qp ) = Hom(R, Qp ), then
we have a natural isomorphism of Qp -vector spaces H0 (Γ, D† [r](R)) = Qp . This isomorphism is an isomorphism of R-module if we give the RHS an R-module structure using the
morphism R → Qp corresponding to the character 0. Moreover, H acts on H0 (Γ, D† [r](R))
with the system of eigenvalues of E2 .
CRITICAL p-ADIC L-FUNCTIONS
29
Proof —
Let ∆ : D† [r] → D† [r] be the transpose of the map f (z) → f (z + 1) − f (z) and let
ρ : D† [r] be defined by ρ(µ) = µ(1). By [SP2, Prop. 3.1], there is an exact sequence of
Qp -Frechet spaces:
(11)
ρ
∆
0 −→ D† [r] −→ D† [r] −→ Qp −→ 0
We claim that the sequences of R-modules
(12)
ρ
∆
0 −→ D† [r](R) −→ D† [r](R) −→ R −→ 0
is still exact. Indeed, the completed tensor product by an orthonormalizable Banach space
is exact in the category of Frechet. To see this, it amounts to see that the functor c0
is exact, and only the right-exactness is not obvious. So let f : V → W be a surjective
morphism of Frechet space. We can assume that the topology of V is defined by semi-norms
(pk )k∈N such that pk ≤ pk+1 for all k. By the Open Mapping Theorem ([Sc, Prop. 8.6])
the topology of V 0 is the quotient of the topology of V , hence is defined by the semi-norms
p0k (v 0 ) := inf v∈V,f (v)=v0 pk (v). By [Sc, Prop 8.1 and its proof], the topology of V (resp.
V 0 ) is also defined by a distance d (resp. d0 ) such that d(v, 0) = supk
0
d
a
p0k (v 0 )
(v , 0) = supk 21k 1+p
0 (v 0 ) ). It follows easily that
k
0
0
sequence (vn ) in V such that d0 (vn0 , 0) goes to 0
0
0
1 pk (v)
2k 1+pk (v)
(resp.
0
d (v , 0) = inf v∈V,f (v)=v0 d(v, 0). Hence
can be lifted into a sequence (vn ) in V
such that d(vn , 0) goes to 0, which proves that c0 (V ) → c0 (V 0 ) is surjective.
Now we need to compute H 0 (Γ, D†K [r](R)) = D†K [r](R)/ID†K [r](R) where I is the augmentation ideal of Z[Γ]. We proceed asin [SP2,
Lemma 5.2], but ”in family”.
1 1
The ideal I contains (g −1) with g =
, which acts as ∆. Thus ID† [r](R) contains
0 1
Ker ρ, and we have an exact sequence
(13)
0 → ρ(ID† (R)) → R → H0 (Γ, D† [r](R)) → 0.
We now construct
in ρ(ID† (R)).
two explicit elements
1 + pN
pN
First, set γ =
∈ Γ1 (N p) ⊂ Γ. Let δ1 be the distribution that sends
pN
1 − pN
f to f (1). We compute
ρ((δ1 )|γ−1 )
= δ1 (K̃(1 + pN + pN z) − 1)
= K̃(1 + 2pN ) − 1 ∈ ρ(ID† (R))
Second, let a be an integer such that a (mod p) is a generator of (Z/qZ)∗ , and a ≡ 1
(mod N ) (remember that q = p if p is odd, q = 4 if p = 2). By an easy application of
a b
Bezout’s lemma there exists integers b, c, d such that γ :=
is in Γ. Let δ0 be the
c d
distribution f 7→ f (0). Then we compute:
ρ((δ0 )|γ−1 )
=
δ0 (K̃(a + cz) − 1)
=
K̃(a) − 1 ∈ ρ(ID† (R))
30
J. BELLAÏCHE
From the exact sequence (13) we deduce the existence of a surjective map
(14)
R/(K̃(1 + 2pN ) − 1, K̃(a) − 1) → H0 (Γ, D† [r](R)).
Since 1 + 2pN generates 1 + qZp , and a generates (Z/qZ)∗ , we see that R/(K̃(1 + 2pN ) −
1, K̃(a) − 1) is either Qp or 0, according to whether 0 ∈ W or not, and that in the former
case, the map R → R/(K̃(1 + 2pN ) − 1, K̃(a) − 1) = Qp is the one corresponding to the
trivial character 0. In this case, we have a natural map
H0 (Γ, D† [r](R)) → H0 (Γ, D†0 [r])
(15)
surjective by right exactness of H0 , hence H0 (Γ, D† [r](R)) has dimension at least 1 over
Qp , since by [SP2, Lemma 5.2 and its proof] H0 (Γ, D†0 [r]) has dimension 1. This shows
that the surjective maps (14) and (15) are both isomorphisms and that H0 (Γ, D† [r](R))
has dimension exactly 1 over Qp . Since (15) is H-compatible, and the action of the Hecke
operators on H0 (Γ, D†0 [r]) = Qp are easily computed, the lemma follows.
Theorem 3.10. Let W = SpR be a nice affinoid of W. Let w ∈ W (Qp ) = Hom(R, Qp )
and w̃ : Z∗p → Q∗p be the corresponding character. Let 0 ≤ r < r(W ) be a real number, then
there is a natural injective morphism of Qp -Banach spaces, compatible with the action of
H
±
†
Symb±
Γ (D [r](R)) ⊗R,w Qp ,→ SymbΓ (Dw [r]).
This map is surjective excepted when w = 0 and the sign ± is −1. In this case, the cokernel
is a space of dimension 1 on which H acts by the system of eigenvalues of E2 .
Proof — The last isomorphism of Lemma 3.5 may be reformulated as the exact sequence
×u
w
0 −→ D† [r](R) −→ D† [r](R) −→ D†w [r] −→ 0.
where u is a generator of the ideal Ker w in R. The long exact sequence of cohomology
with compact support attached to this short exact sequence is
×u
×u
†
0 → Hc1 (Γ, D† [r](R)) → Hc1 (Γ, D† [r](R)) → Hc1 (Γ, Dw
[r]) → Hc2 (Γ, D† [r](R)) → Hc2 (Γ, D† [r](R))
Using Poincaré duality for Γ, which is functorial and compatible with the Hecke operators
×u
in H we see that the last morphism is the same as H0 (Γ, D† [r](R)) → H0 (Γ, D† [r](R)).
The kernel A of this map is, by the lemma above, 0 excepted when κ is trivial in which case
it is Qp with action of H through the system of eigenvalues of E2 . The action of ι on A is
by −1 because the map z 7→ −z̄ on the Poincaré upper half plane, that induces the action
of ι, is orientation-reversing. Using the functorial isomorphism SymbΓ (V ) = Hc1 (Y (Γ), V )
of Ash-Stevens ([AS]), we get an exact sequence
×u
0 → SymbΓ (D† [r](R)) → SymbΓ (D† [r](R)) → SymbΓ (D†w [r]) → A → 0,
and the result follows.
CRITICAL p-ADIC L-FUNCTIONS
31
3.2.4. Bounded slope submodules. Let us begin by a reminder of the general definitions and
basic property of bounded slope submodules.
Let R be a reduced Qp -affinoid algebra, that we provide with its supremum norm | |
extending the norm of Qp . We set as usual vp (x) = − log |x|/ log p, so vp (p) = 1. Let us
P∞
call R{{T }} the ring of power series n=0 an T n , with an ∈ R, that converge everywhere,
that is such that vp (an ) − nν → ∞ for every ν ∈ R. For F (T ) ∈ R{{T }}, and ν ∈ R,
we write N (F, ν) for the largest integer n such that vp (an ) − nν = inf m (vp (am ) − mν).
A polynomial Q(T ) ∈ R[T ] is called ν-dominant if Q has degree N (Q, ν) and for all x ∈
SpR, N (Q, ν) = N (Qx , ν), where Qx is the polynomial in L(x)[T ] (L(x) being the filed
of definition of x) obtained by specializing Q at x. We say that the power series F is
ν-adapted if there exists a decomposition F = QG in R{{T }}, with Q(0) = G(0) = 1, and
Q is a ν-dominant polynomial of degree N (F, ν). In this case, the decomposition F = QG
is unique, and (Q, G) = 1 in R{{T }}. Moreover, if F is ν-adapted, then for any morphism
of reduced affinoid algebra R → R0 , the image F 0 of F in R0 {{T }} is also ν-adapted, and
the decomposition F 0 = Q0 G0 is the image of the decomposition of F . For any F and ν,
and a point x ∈ SpR, then any sufficiently small affinoid SpR0 ⊂ SpR containing x is such
that the images F 0 of F in R{{T }} are ν-adapted.
Let M be a Banach R-module which is orthonormalizable, or more generally satisfies
property (Pr) of [Bu, §2], and Up a compact operator of M . Then one can define (see [Bu])
the characteristic power series F (T ) ∈ R{{T }} of Up acting on M . If ν ∈ R, and if F
is adapted to ν, and thus has a decomposition F = QG ad above, then one defines the
submodule M ≤ν of M of slope bounded by ν as the kernel of Q∗ (Up ) in M . In this case we
also say that M is adapted to ν. This is a finite projective R-submodule of M , stable by
every operator on M commuting with Up . The formation of M ≤ν commutes with arbitrary
base changes of affinoids R → R0 .
Finally, if M 0 is another Banach R-module which satisfies property (Pr) and Up a compact operator of M 0 , then if M and M 0 are linked in the sense of [Bu, just before Lemma
5.6], we have that for any ν, M is adapted to ν if and only if M 0 is, and a link between M
and M 0 defines a canonical isomorphism M ≤ν ' M 0
≤ν
.
All the assertions above follow quite easily from the arguments in [Bu, §4 and §5] (see
also [C1]). For details, see [B2].
Now if W = SpR be a nice affinoid of W, then the theory above applies to the modules
SymbΓ (D[r](R)) for all 0 < r ≤ r(W ). In particular, for ν ∈ R, those modules are all
ν-adapted if one is, in which case we shall simply say that W = SpR is ν-adapted, and the
restriction maps define canonical isomorphisms between the modules SymbΓ (D[r](R))≤ν
for all 0 < r ≤ r(W ).
If 0 ≤ r < r(W ), the above result shows that, for a real r0 such that r < r0 < r(W ), the
intersection SymbΓ (D[r0 ](R))≤ν ∩ SymbΓ (D† [r](R)) taken in SymbΓ (D[r0 ](R)) (of which
SymbΓ (D† [r](R)) is a submodule because D† [r] ⊂ D[r0 ](R) and SymbΓ is left-exact) is
32
J. BELLAÏCHE
independent of the choice of r0 as a submodule of SymbΓ (D† [r](R)). It is natural to denote
this module by SymbΓ (D† [r](R))≤ν .
Proposition 3.11. Assume that R is ν-adapted. Then all the modules SymbΓ (D[r](R))≤ν
(for 0 < r ≤ r(W ) ) and SymbΓ (D† [r](R))≤ν (for 0 ≤ r < r(W )) are naturally isomorphic.
The above results extend to the case of families of modular symbols results of Stevens
for modular symbols over Qp (see [SP2, Lemma 5.3]). In this case, any ν is automatically
Qp -adapted, and if w ∈ W(Qp ), the modules SymbΓ (Dw [r])≤ν for 0 < r ≤ r(w) and
SymbΓ (D†w [r]) for 0 ≤ r < r(w) (defined similarly as above) are naturally isomorphic.
0
We can also define SymbΓ (Dw [r])<ν as the union of the SymbΓ (Dw [r])≤ν for ν 0 < ν,
and similarly for SymbΓ (Dw [r])<ν , and again, all those modules are naturally isomorphic.
It is thus clear that as far as we consider sub-modules with bounded slope, it doesn’t
matter if we work with D[r] or D† [r] nor which r we choose. We will therefore use the
following
Notation: D will stand for D† [0] (hence Dw for D†w [0], D(R) for D† [0](R), etc.)
With those notations, we can deduce the following corollary of Theorem 3.10
Corollary 3.12. Let ν ∈ R and W = SpR a nice affinoid of W which is ν-adapted. Let
w ∈ W (Qp ) = Hom(R, Qp ) and w̃ : Z∗p → Q∗p be the corresponding character. There is a
natural injective morphism of finite Qp -vector spaces, compatible with the action of H,
≤ν
≤ν
Symb±
⊗R,w Qp ,→ Symb±
.
Γ (D(R))
Γ (Dw )
This map is surjective excepted when w̃ is the trivial character 0̃, the sign ± is −1, and
ν ≥ 1. In this case, the cokernel is a space of dimension 1 on which H acts by the system
of eigenvalues of E2 .
≤ν
the image of the map
Definition 3.13. We shall denote by Symb±
Γ (Dw )g
≤ν
≤ν
Symb±
⊗R,w Qp ,→ Symb±
Γ (D(R))
Γ (Dw )
of the preceding corollary.
≤ν
The g should make think of ”global”, as Symb±
Γ (Dw )g is the space of modular symbols
≤ν
that comes by specialization from ”global” modular symbols, i.e. defined
in Symb±
Γ (Dw )
over W . The corollary shows that in almost all cases, the g may be dropped. Also, it is
≤ν
clear that Symb±
does not depend on the choice of the affinoid W .
Γ (Dw )g
3.2.5. Stevens’ control theorem and the Eichler-Shimura isomorphism. Let k ≥ 0 be an
integer. Following Stevens, we call Pk the set of polynomials of degree at most k in one
variable z, and we see it as a left GL2 (Qp )-module by setting
b + dz
a
k
) if γ =
(γ · P )(z) = (a + cz) P (
c
a + cz
b
.
d
CRITICAL p-ADIC L-FUNCTIONS
33
We call Vk the dual of Pk , provided with the right action deduce from the action of Pk :
for l ∈ Vk , and P ∈ Pk
l|γ (P ) := l(γ · P ).
The inclusion map Pk → Dk is compatible with the Σ0 (p)-actions, and so is its dual map,
ρ∗k : Dk → Vk .
Theorem 3.14 (Stevens’ control theorem). The morphism ρ∗k induces an isomorphism (of
Qp -spaces with an action of H and ι)
ρ∗k : SymbΓ (Dk )<k+1 → SymbΓ (Vk )<k+1
Proof —
See [S, Theorem 7.1] and [SP2, Theorem 5.4]
This theorem is usually coupled with the Eichler-Shimura isomorphism. Let us recall
that Mk+2 (Γ) (resp. Sk+2 (Γ)) denotes the Qp -spaces of modular forms (resp. cuspidal
modular forms) of weight k + 2 and level Γ, with coefficients in Qp , and let us denote
∨
Ek+2 (Γ) the quotient Mk+2 (Γ)/Sk+2 (Γ). All those spaces have an H-action, and by Ek+2
we denote the Qp -dual of Ek+2 (Γ) with its transposed H-action.
Proposition 3.15 (Eichler-Shimura, Manin-Shokurov). There exists an H-isomorphism
∼
es : SymbΓ (Vk ) −→
Sk+2 (Γ) ⊕ Sk+2 (Γ) ⊕ Ek+2 (Γ)∨ .
It splits according to the ι involution as a sum es = es+ ⊕ es− into two injective Hmorphisms
∨
es+ : Symb+
Γ (Vk ) ,→ Sk+2 (Γ) ⊕ Ek+2 (Γ) .
∨
es− : Symb−
Γ (Vk ) ,→ Sk+2 (Γ) ⊕ Ek+2 (Γ) .
whose images contains both Sk+2 (Γ). The inverse image by es of Ek+2 (Γ)∨ is the space of
boundary modular symbols, that is modular symbols that extend to Γ-invariant morphisms
from ∆ to Vk , where ∆ is the set of all divisors on P1 (Q), not necessarily of degree 0.
Proof —
This theorem is quoted in [Me] or in a preprint version of [SP2]. The cuspidal
part of the statement is due to Shokurov [Sho] (generalizing results of Manin in the level 1
case), and details can be found in [Me] and [St1]. One defines, for a cuspidal eigenform f ,
(es± )−1 (f ) as the ±-part of the Vk -valued modular symbol that to the divisor {b}−{a} ∈ ∆0
Rb
attaches the following linear form on Pk : P 7→ a f (z)P (z) dz, divided by the period Ω±
f
of the modular form (well-defined up to an algebraic unit).
Alternatively, following an argument due to Ash and Stevens, this cuspidal case can be
obtained by combining the classical Eichler-Shimura isomorphisms (relating Hc1 (Y (Γ), Vk )
with Sk+2 (Γ)2 cf. e.g. [H, §6.2]) with the isomorphism of Ash-Stevens (cf. [AS]) SymbΓ (Vk ) '
Hc1 (Y (Γ), Vk ).
For the Eisenstein part, it seems more difficult to find a complete proof in the literature.
There are some indication in [Me]. For a detailed proof, see the forthcoming [B2]. One
34
J. BELLAÏCHE
attaches to an Eisenstein series f the cohomology class [uf ] ∈ H 1 (Γ, Vk ) of the cocycle
R γx
uf : Γ → Vk , γ 7→ (P 7→ x f (z)P (z) dz) divided by a suitable power of (2iπ). Here x is a
chosen point in H, and the cocycle uf clearly depends on x only up to a coboundary. Using
Poincaré duality between H 1 (Γ, Vk ) and Hc1 (Y (Γ), Vk ), and the Ash-Stevens isomorphism,
one gets an injective map Ek+2 (Γ)∨ → SymbΓ (Vk ). By dimensions counting, one proves
that the image of this map is the space of boundary modular symbols, and finally one
constructs the map es.
Corollary 3.16. There exist canonical H-morphism
<k+1
es+ ◦ ρ∗k : Symb+
Γ (Dk )
,→
Sk+2 (Γ)<k+1 ⊕ Ek+2 (Γ)∨,<k+1
<k+1
es− ◦ ρ∗k : Symb−
Γ (Dk )
,→
Sk+2 (Γ)<k+1 ⊕ Ek+2 (Γ)∨,<k+1
whose images contain Sk+2 (Γ)<k+1 .
3.2.6. The sign of an Eisenstein series. Let f be a new Eisenstein series of level Γ1 (N ),
and fβ one of its refinement. We shall say that the sign of fβ is +1 (resp. −1) if the system
−
of eigenvalues of fβ belongs to Symb+
Γ (Vk ) (resp. SymbΓ (Vk )). By Prop. 3.15, the sign of
fβ is well-defined, and we shall denote if (fβ ).
Remark 3.17. It is easy to see that any Eisenstein series f of level Γ0 (N ) with N square
free has (fα ) = (fβ ) = 1. But in general, even for Γ0 (N ) with N = 9 and p = 2, there
are series f with (fα ) = −1.
3.3. Mellin transform of distribution-valued modular symbols. This subsection
will not be needed until §4.
3.3.1. Overconvergent distributions and Mellin transform in families.
Definition 3.18. We call R the Qp -algebra of analytic functions on W.
The space R is naturally a Frechet space. Elements of R can be seen as Cp -valued
functions of one variable σ belonging in W(Cp ).
Recall that D means D† [0]. Let W = SpR be a nice affinoid of W.
Definition 3.19. The Mellin transform is the application
Mel : D → R
R
that sends a distribution µ to the function σ 7→ Z∗ σ̃dµ for σ ∈ W(Cp ), σ̃ being as usual
p
the corresponding character Z∗p → C∗p , seen by extension by 0 on pZp as a function on Zp .
b → R⊗R
b defined
The Mellin transform in family is the application MelR : D(R) = R⊗D
b
as IdR ⊗Mel.
b can be seen as two-variables analytic functions on W × W. The
The elements of R⊗R
first variable, in W , is usually denoted by w or k, and the second variable by σ or s. We
CRITICAL p-ADIC L-FUNCTIONS
35
have the following obvious (from the properties of completed tensor products) compatibility
relation: If µ ∈ D(R), and if w ∈ W (Qp ) = Hom(R, Qp ), we denote by µw the image of µ
by the map D(R) → Dw induces by w. Then for all σ ∈ W (Cp ), we have
(16)
MelR (µ)(w, σ) = Mel(µw )(σ)
Fix ν > 0. Let W = SpR be a nice affinoid of W which is adapted to ν. In this §, we
use the Mellin transforms to define a map
ΛR : SymbΓ (D(R))≤ν
→
φ 7→
b
R⊗R
MelR (φ({∞} − {0}))
and for every w ∈ W (Qp ), a map
Λ : SymbΓ (Dw )≤ν
φ
→
R
7→ Mel(φ({∞} − {0}))
The two maps are compatible in the sense that for φ ∈ SymbΓ (D(R))≤ν , if φw denotes
its image in SymbΓ (Dw )≤ν , we have for all σ ∈ W(Cp ),
(17)
ΛR (φ)(w, σ) = Λ(φw )(σ).
This compatibility is obvious from the definitions and (16).
3.3.2. Properties of Λ. Let us recall the following well-known result:
Proposition 3.20 (Visik, Stevens). If φ ∈ SymbΓ (Dw )≤ρ , then Λ(φ) is, as a function of
σ, of order less that vp (ρ).
The definition of the order was recalled in the introduction.
We now turn to interpolation properties.
Lemma 3.21. Let k be a non-negative integer. Let φ ∈ SymbΓ (Dk ) be an eigensymbol
which is a generalized eigenvector for Up , of generalized eigenvalue β 6= 0. We assume
that ρ∗k ((Up − β)(φ)) = 0. Let µ = φ({∞} − {0}) ∈ D. Then for every integer such that
0 ≤ j ≤ k, every integer n ≥ 0, and every integer a such that 0 ≤ a < pn we have
Z
z j 1a+pn Zp dµ(z) = β −n φ({∞} − {a/pn })((pn z + a)j ).
(18)
(19)
Proof —
= β −n ρ∗k (φ)({∞} − {a/pn })((pn z + a)j ).
The equation (19) follows from (18) by definition of ρ∗k .
Let r be the smallest integer such that (Up − β)r φ = 0. We shall prove (18) by induction
on r.
If r = 1, φ is a true eigenvector for Up , and it is a relatively simple computation, done
in [SP1, §6.3] that
Z
(20)
z j 1a+pn Zp dµ(z) = β −n φ({∞} − {a/pn })((pn z + a)j ).
36
J. BELLAÏCHE
Let us assume that r > 1, and that (18) (and hence (19)) are known for all r0 < r.
We have (Upr − β r )φ = (Up − β)Q(Up )φ where Q is some obvious polynomial. Set φ0 =
(Up − β)Q(Up )φ. Then obviously (Up − β)r−1 φ0 = 0 and by assumption ρ∗k (φ0 ) = 0, which
implies a fortiori ρ∗k ((Up − β)φ0 ) = 0, so we can apply the induction hypothesis (19) to φ0 ,
which gives us, calling µ0 the distribution φ0 ({∞} − {0}):
Z
(21)
z j 1a+pn Zp dµ0 (z) = 0.
But independently the same computation as in [SP1, §6.3], starting with Upn φ − φ0 = β n φ
give us that
Z
Z
j
−n
n
z j 1a+pn Zp dµ0 (z) = β −n φ({∞} − {a/pn })((pn z + a)j ).
z 1a+p Zp dµ(z) − β
Using (21) we get (20).
Proposition 3.22. Let k be a non-negative integer. Let φ ∈ SymbΓ (Dk ) be an eigensymbol
which is a generalized eigenvector for Up , of generalized eigenvalue β 6= 0. We assume that
ρ∗k (φ) = 0. Then for all character ϕ : Z∗p → C∗p of finite image and all integers j with
0 ≤ j ≤ k, we have
Λ(φ)(ϕtj ) = 0
Proof — With our assumptions the above lemma tells us that for all integers j, n and a such
R
that 0 ≤ j ≤ k, n ≥ 0 and 0 ≤ a < pn , we have z j 1a+pn Zp dµ(z) = 0. But Λ(φ)(ϕtj ) =
R
ψ(z)z j dµ(z) is defined as a limit of Riemann sums that are linear combinations of terms
R
of the form z j 1a+pn Zp dµ(z) with j, n, a as above. So the Riemann sums, and their limit,
are all 0.
Proposition 3.23. Let k be a nonnegative integer, fβ the refinement of a cuspidal newform
∗
in Sk+2 (Γ). Let φ be a non-zero symbol in Symb±
Γ (Dk )(fβ ) such that ρk (φ) 6= 0. Then for
any special character of the form σ = ϕtj , with 0 ≤ j ≤ k, we have
Λ(φ)(σ) = ep (β, ψtj )
j! L∞ (f ϕ−1 , j + 1)
mj+1
,
(−2πi)j τ (ϕ−1 )
Ω±
f
j
up to a non-zero scalar depending only on φ, not on σ, with L∞ , Ω±
f ep (β, ϕt ), m as in
the introduction.
Proof —
Since ρ∗k (φ) 6= 0, and es± is an isomorphism, es± ◦ ρ∗k (φ) is a non-zero vector in
MΓ (k+2)(fβ ) . Since this space has dimension 1 (Lemma 2.8(i)), we see that es± ◦ρ∗k (φ) = fβ
up to a non-zero scalar. Also, (Up −β)fβ = 0 so we see that ρ∗k ((Up −β)φ) = 0, which allows
us to apply Lemma 3.21. The lemma then follows by a standard computation from (18)
combined with the explicit description of es± for a cuspidal form fβ recalled in the proof
of Proposition 3.15.
3.4. Local pieces of the eigencurve.
CRITICAL p-ADIC L-FUNCTIONS
37
3.4.1. Construction of the local pieces. We keep the notations of the preceding paragraphs,
in particular Γ = Γ0 (p) ∩ Γ(N ) with N an integer prime to p. Let ν ∈ R be a real and
let W = SpR be a nice affinoid subspace of the weight space W which is adapted to ν (cf.
§3.2.4).
Our aim is to construct by means of distributions-valued modular symbols an affinoid
±
±
space CW,ν
together with a finite flat “weight” morphism κ : CX,ν
→ W.
The construction follows the general method described in [Bu]: we consider the space
≤ν
Symb±
introduced in §3.2.4 This is a finite projective R-module on which H acts.
Γ (D(R))
±
≤ν
Definition 3.24. We call T±
) generated by
W,ν the R-subalgebra of EndR (SymbΓ (D(R))
the image of H.
The algebra T±
W.ν is obviously finite and torsion-free as an R-module, and since R is a
PID, it is also flat. As a finite algebra over R, it is an affinoid algebra, and we set
±
CW,ν
:= SpT±
W,ν .
The weight morphism
±
κ± : CW,ν
→ W = SpR
is simply the map corresponding to the structural morphism R → T±
W,ν and is therefore
finite and flat. This completes the construction.
3.4.2. The “punctual eigencurve”. Let w ∈ W (Qp ) = Hom(R, Qp ) be some point on the
weight space. We can perform the same construction in smaller at the point w:
±
≤ν
Definition 3.25. We define T±
w,ν as the Qp -subalgebra of EndQp (SymbΓ (Dw )g ) gener-
ated by the image of H.
≤ν
(For the definition of Symb±
Γ (Dw )g , see Definition 3.13.)
By construction, there is a natural bijection between the Q̄p -points of Spec T±
w,ν (that
is, the morphisms of algebras T±
w,ν → Q̄p ) and the systems of H-eigenvalues appearing in
≤ν
Symb±
(see §1.2). If x is such a point, or systems of H-eigenvalues, we denote
Γ (Dw )g
by Symb±
Γ (Dw )(x) the corresponding generalized eigenspace (we drop the
≤ν
which is
redundant since being in the (x)-generalized eigenspace implies being a generalized eigenvector for Up of generalized eigenvalue x(Up ) which by definition has valuation ≤ ν. We
drop the g for ease of notations, and because, as we shall soon see, it leads to no am≤ν
≤ν
and in Symb±
are the
biguity: the x-generalized eigenspace in Symb±
Γ (Dw )g
Γ (Dw )
same, since, when those two spaces are different, the system of eigenvalues of the line
≤ν
≤ν
≤ν
Symb±
/Symb±
does not appear in Symb±
Γ (Dw )
Γ (Dw )g
Γ (Dw )g , hence cannot be the
±
same as x – see Theorem 3.27.) Similarly we note (T±
w,ν )(x) the localization of Tw,ν ⊗Qp Q̄p
at the maximal ideal corresponding to x. Obviously we have
±
≤ν
±
Symb±
⊗Tw,ν
(T±
w,ν )(x) .
Γ (Dw )(x) = SymbΓ (Dw )
38
J. BELLAÏCHE
If w is an integral weight k ∈ N and if the system of eigenvalues x happens to be
the system of eigenvalues of a refinement fβ of a modular newform f of level Γ1 (N ) and
weight (necessarily) k + 2, recall (cf 1.2) that we sometimes write Symb±
Γ (Dw )(fβ ) instead
±
±
of Symb±
Γ (Dw )(x) and (Tw,ν )(fβ ) instead of (Tw,ν )(x) .
3.4.3. Comparison global-punctual. Let again w ∈ W (Qp ) = Hom(R, Qp ) be some point on
≤ν
≤ν
the weight space. Since by Definition 3.13, Symb±
⊗R,w Qp ' Symb±
Γ (D(R))
Γ (Dw )g ,
we have a natural specialization morphism of finite Qp -algebras
±
sw : T±
W,ν ⊗R,w Qp → Tw,ν
compatible with the morphisms from H.
Lemma 3.26. [C1, Lemme 6.6]. For all w ∈ W(Qp ), sw is surjective, and its kernel is
nilpotent. In particular, sw induces a bijection between the Q̄p -points of the fiber of κ± at
w and of SpT±
w,ν :
(κ± )−1 (w)(Q̄p ) ' SpT±
w,ν (Q̄p ).
Now if x is a Q̄p -point in Spec T±
w,ν , it corresponds to a Q̄p -point also denoted x in
±
CW,ν
such that κ(x) = w, and we can consider the rigid analytic localization (T±
W,ν )(x) of
T±
W,ν ⊗Qp Q̄p at the maximal ideal corresponding to x. This is naturally an algebra over R(w)
defined as the rigid analytic localization of R ⊗Qp Q̄p at the maximal ideal corresponding
to w.
Localizing at x the map sw gives rise to a a surjective local morphism of finite local
Q̄p -algebra with nilpotent kernel
±
sx : (T±
W,ν )(x) ⊗R(w) ,w Q̄p → (Tw,ν )(x)
(22)
3.4.4. Comparison with the Coleman-Mazur-Buzzard’s eigencurve. We keep the same notations as above.
Theorem 3.27.
(i) There exist canonical closed immersions
±
0
CW,ν
,→ CW,ν
,→ CW,ν
which are compatible with the weight morphism κ± and κ to the weight space W .
+
−
Moreover, the eigencurves are all reduced, and CW,ν = CW,ν
∪ CW,ν
.
(ii) For k ∈ Z ⊂ W(Qp ), there exist H-injections between the H-modules
†
†
ss,≤ν
Sk+2
(Γ)ss,≤ν ⊂ Symb±
⊂ Mk+2
(Γ)ss,≤ν ,
Γ (Dk )g
where ss means semi-simplification as an H-module.
(iii) A system of H-eigenvalues of finite slope (that is, with a non-zero Up -eigenvalue)
†
appears in SymbΓ (Dk ) if and only if it appears oi Mk+2
(Γ) except for the system
of eigenvalue of E2 which appears in SymbΓ (D0 ) but not in M2† (Γ).
†
(iv) if fβ is a cuspidal refined form, we have dim(Sk+2
(Γ))(fβ ) = dim(Symb±
Γ (Dk ))(fβ ) .
CRITICAL p-ADIC L-FUNCTIONS
39
Remark 3.28. We note that (iii) is very close to a theorem of Stevens ([S2], quoted and
used in [SP2, Theorem 7.1]) whose proof has not appeared yet. The differences are our
restrictions to finite slope (but this restrction was meant and forgotten in [SP2, Theorem
7.1]) and the situation of E2 , which is a very minor exception to Stevens’ theorem.
Proof —
We shall prove the theorem by interpolation from the situation for classical
modular forms and symbols. We shall do so by applying the work of Chenevier [C2] to our
preceding construction and to Coleman’s construction.
Let us first recall Chenevier’s relevant results, which take in our specific context (we
are working only with local pieces of the eigencurve, so all the rigid analytic concerns
disappear) a much simpler form, and have much simpler proof, than they have in their full
generality. Let X be a Zariski-dense set in W = SpR. Let us consider two finite projective
modules M1 and M2 over R, both with an action of H. Chenevier proves (cf Théorème
1 loc. cit.) that if for all x ∈ X, we suppose that there exists an injection of H-modules
ss
ss
ss
ss
M1,x
⊂ M2,x
, then we have more generally an injection of H-modules, M1,w
⊂ M2,w
for
every w ∈ W , and a canonical immersion C1 ,→ C2 above W and respecting H, where C1 ,
C2 denotes the eigencurve attached to M1 , M2 . Moreover, for i = 1, 2, if the H-modules
Mi,x are semi-simple for all x ∈ X, then the eigencurve Ei is reduced (cf. Proposition 3.9
loc. cit.).
We shall apply those results repeatedly.
†,≤ν
First we take for M1 Coleman’s SW
(cf. §2.1.1), and for M2 we take either of the two
≤ν
modules Symb±
. Let X be the set of integral weight k satisfying k > ν + 1 and
Γ (D(R))
k > 1 of W . Since W is a nice affinoid, X is Zariski-dense in W .
†
The fiber M1,k at an integral weight k ∈ X of M1 is Sk+2
(Γ)≤ν (remember our shift
†
in the weights for modular forms), which is the same as Sk+2
(Γ)≤ν by Coleman’s control
theorem, in the form given in Corollary 2.6.
≤ν
The fiber at k ∈ X of M2,k is Symb±
by Theorem 3.10 (recall that k ≥ 1, so we
Γ (Dk )
≤ν
don’t need the g), which by Stevens’ control theorem (Theorem 3.14) is Symb±
.
Γ (Vk )
The existence, for every x ∈ X, of an injective map of H-modules M1,x ⊂ M2,x (even
without semi-simplification) thus follows from Prop. 3.15.
Therefore, Chenevier’s result gives us two canonical embeddings of curves C0 ⊂ C + and
ss
ss
C0 ⊂ C − as well as embeddings of H-modules M1,w
⊂ M2,w
, that is in particular, using
Theorem 3.10 and Definition 3.13 embeddings
†
≤ν
Sk+2
(Γ)ss,≤ν ,→ Symb±
Γ (Dk )g ,
for any k ∈ Z ∩ W (Qp ). But since any integral weight k lies in a nice affinoid W ⊂ WQp ,
the results actually hold for all integers k ∈ N (and even k ∈ Z).
†,≤ν
Now, take for M3 Coleman’s MW
(cf. §2.1.1), and the same X as above, so for k in X,
using Coleman’s control theorem, the fiber M3,x is Mk+2 (Γ)≤ν . We note that by Prop. 3.15
40
J. BELLAÏCHE
≤ν
,→ Sk+2 (Γ)≤ν ⊕ Ek+2 (Γ)∨,≤ν . But
we have an injective morphism M2,x = Symb±
Γ (Vk )
since a semi-simple H-module is clearly isomorphic to its dual, the above morphism induces
ss
an injective morphism on semi-simplifications M2,x
,→ Mk+2 (Γ)≤ν,ss ' M3,x .
Applying again Chenevier’s result to M2 and M3 , using Prop. 3.15, gives the inclusion
C
±
,→ C and for every k ∈ Z,
(23)
†
ss,≤ν
,→ Mk+2
(Γ)ss,≤ν .
Symb±
Γ (Dk )g
So far, we have proved (i) and (ii) of the theorem, except for the assertion on the union
of C + and C − , and the assertion that the eigencurves are reduced. We apply again Ch≤ν
⊕
enevier’s result to M3 and M4 , with M4 defined as SymbΓ (D(R))≤ν = Symb+
Γ (D(R))
≤ν
Symb−
. We thus get an embedding of H-modules, for every k ∈ Z,
Γ (D(R))
†
Mk+2
(Γ)ss,≤ν ,→ SymbΓ (Dk )ss,≤ν
.
g
Combining this with (23) we see that any system of H-eigenvalues of finite slope (just
†
work with a ν greater than this slope) appears in Mk+2
(Γ) if and only if it appears in
SymbΓ (Dk )g . But by Theorem 3.10, the systems of eigenvalues appearing in SymbΓ (Dk )
are the same as those appearing in SymbΓ (Dk )g , plus the system of H-eigenvalues of E2 in
the case k = 0. But this system does not appear on M2† (Γ) by the main theorem of [CGJ].
Hence we get (iii). We have also shown in passing that any point on C was either on C + or
on C − , which completes the proof of (i), except for the reducedness of the eigencurves.
For this, we take a smaller set X 0 ⊂ X, defined as the k ∈ X such that k > 2ν − 1, which
is still Zariski-dense in W . It suffices to observe that for x ∈ X 0 , the H-modules M1,x ,
M2,x and M3,x are semi-simple, and from the injective morphisms between those modules
constructed above, it suffices to prove that all operators in H acting on Mk+2 (Γ)≤ν are
semi-simple for k ∈ X 0 . All the Hecke operators in H excepted Up are normal, hence semisimple. It is enough to prove that Up acts semi-simply on every generalized H-eigenspace
of Mk+2 (Γ)≤ν . For a generalized eigenspace which is new at p, this results from AtkinLehner’s theory. Consider therefore a generalized eigenspace Mk+2 (Γ)≤ν
(f ) which is old at
p, corresponding to a newform f for Γ1 (N 0 ) for some N 0 dividing N (possibly N 0 = N ),
of Tp -eigenvalue ap and nebentypus . Since X 2 − ap X + pk+1 (p) has a unique root
α such that vp (α) ≤ ν (because ν < (k + 1)/2), we have that g 7→ gα is an isomor≤ν
phism Mk+2 (Γ1 (N ))(f ) ' Mk+2 (Γ)(f
) , and that Up acts by the scalar α on our generalized
eigenspace. This complete the proof that the eigencurves are reduced.
Finally (iv) is easy and left to the reader.
4. p-adic L-functions
We keep the notations of §3.4.1.
CRITICAL p-ADIC L-FUNCTIONS
41
≤ν
as a module over T±
4.1. The space Symb±
Γ (D(R))
W,ν . We first prove an elementary
lemma of commutative algebra.
Lemma 4.1. Let R be a discrete valuation domain, T be a finite free R-algebra, and M
be a finitely generated T -module, free as an R-module. Suppose that T is also a discrete
valuation domain. Then M is free as a T -module as well.
Proof —
Since T is a discrete valuation domain, we can write as a T -module M =
Mfree ⊕ Mtors , with Mfree a finite free module over T of some rank r and Mtors a finite
torsion module over T . We claim that Mtors is torsion as an R-module: for any m ∈ Mtors ,
there is an element 0 6= x ∈ T such that xm = 0; if a0 is the constant term of a polynomial
in R[X] killing x, chosen of minimal degree, then a0 6= 0, and one sees easily that a0 m = 0,
which proves the claim. But since M is locally free as an R-module, it has no torsion, so
Mtors = 0, and M = Mfree is free over T .
Remark 4.2. The hypothesis that T is a discrete valuation domain cannot be weakened
to “T is local”: let R = k[[u]] where k is a field of characteristic 0, M = R2 , and T be
the subalgebra ofEndR (M
) generated over R by the endomorphism t of matrix (in the
0 u
canonical basis)
. Then t2 = u3 , and one sees that T = R[t]/(t2 − u3 ) as an
u2 0
R-algebra, so that T is a finite free R-module of rank 2, and as a ring is obviously local,
but not regular. However M is not free as a T -module, for if it were, it would be obviously
of rank one, but at the maximal ideal (t, u) of T , the fiber of M is M/(t, u)M = k 2 has
dimension 2.
However, it can happen that M is free over T without T being a discrete valuation
domain. For an example,
just
modify the preceding example by redefining t as the endo0 1
morphism of matrix
. Then T is the same algebra as before, but now M is free
u3 0
over T , obviously generated by the second vector of the standard basis.
±
Proposition 4.3. Let x be any smooth Q̄p -point in CW,ν
. Then the (T±
W,ν )(x) -module
Symb±
Γ (D(R))(x) is free of finite rank.
Proof —
Since x is smooth, and C is a reduced curve, the local ring T := (T±
W,ν )x is a
discrete valuation domain. Setting
M := Symb±
Γ (D(R)) ⊗T±
W,ν
T = Symb±
Γ (D(R))(x)
and replacing R by its localization at κ± (x) bring us to the exact situation and notations
of Lemma 4.1. This lemma tells us that M is free as a T -module.
±
Corollary 4.4. For any w ∈ W (Qp ), and x ∈ CW,ν
(Q̄p ) a smooth point such that κ(x) = w,
the morphism of Q̄p -algebras
±
sx : (T±
W,ν )(x) ⊗R(w) ,w Q̄p → (Tw,ν )(x)
42
J. BELLAÏCHE
is an isomorphism.
Proof —
±
Any operator t ∈ (T±
W,ν )(x) that acts trivially on SymbΓ (D(R))(x) ⊗R,w Q̄p =
±
±
Symb±
Γ (Dw )(x) must have 0 image in (TW,ν )(x) ⊗R,w Q̄p since SymbΓ (D(R))(x) is free. In
other words, the map sx is injective. Since it is surjective by Lemma 3.26 and the discussion
below it, it is an isomorphism.
±
Proposition 4.5. Let x be a point in CW,ν
. We assume that there exists a decent newform
f of weight k + 2 and level Γ1 (N ) and a refinement fβ of f , such that the systems of
eigenvalues of fβ for H corresponds to x. Then Symb±
Γ (D(R))(x) is free of rank one on
(T±
W,ν )(x) .
Proof —
±
By Theorem 2.16 and Prop 3.27, CW,ν
is smooth at x. Therefore, on a suitable
±
neighborhood of x in CW,ν
, all points are smooth, the map κ is étale at all points but perhaps
±
x (by the openness of étaleness and the fact that CW,ν
is equidimensional of dimension 1) and
all classical points correspond to refined newforms (by Lemma 2.7). Since the localizations
of a locally free finitely generated module have locally constant rank, we can evaluate the
±
rank r of the free module Symb±
Γ (D(R))(x) on (TW,ν )(x) by replacing x by any point in
such a neighborhood. Choosing a classical point y different from x, of weight k 0 , we have
±
that r is the rank of Symb±
Γ (D(R))(y) over (TW,ν )(y) , and since κ is étale at y, r is the
†
dimension of Symb±
Γ (Dk0 )(y) over Q̄p . By Prop. 3.27 this is the dimension of Mk0 +2 (Γ)(y) ,
which is 1 by Lemma 2.8(ii).
4.2. Critical p-adic L-functions. Let f be a modular newform of weight k + 2 ≥ 2, level
Γ1 (N ). Let fβ be one of its two refinements, so fβ is a modular form for Γ = Γ1 (N )∩Γ0 (p).
Assume that fβ is decent, and that vp (β) > 0 in case f is Eisenstein.
Let us fix a real number ν > k + 1 ≥ vp (β), and a nice affinoid subspace W = SpR of W
that contains k, sufficiently small to be adapted to ν. Let ± denotes any choice of the sign
+ or −. Then we know that fβ corresponds to a unique point x ∈ C 0 ⊂ C (Lemma 2.8(iii)),
and that this point is smooth (Theorem 2.16). By Prop 3.27, we see x as a point of C ±
±
and this point x actually belongs to the open set CW,ν
. We have κ± (x) = k.
Recall that R(k) is the rigid-analytic localization of R ⊗QP Q̄p at the maximal ideal corresponding to k. The ring (T±
W,ν )(x) is an R(k) -algebra. We call e the index of ramification
of κ± at x (it is independent of ± by Corollary 2.17), that is the length of the connected
component containing x of the fiber of κ± at x. In particular, κ± is étale at x if and and
only if e = 1.
4.2.1. Proof of Theorem 1.
Proposition 4.6. For some choice of uniformizer u ∈ R(k) there is an isomorphism of
±
R(k) -algebras R(k) [t]/(te − u) → (T±
W,ν )(x) sending t to a uniformizer of (TW,ν )(x) .
CRITICAL p-ADIC L-FUNCTIONS
43
Proof — The rings R(k) and (T±
W,ν )(x) are Henselian discrete valuation domains since they
are the rigid analytic local rings of rigid analytic curves at smooth points. The algebra
(TW,ν )(x) is finite and flat (hence free) over R(k) , since the morphism κ is. We are therefore
dealing with a standard extensions of DVR, which is totally ramified of degree e, and
moreover tamely ramified since we are in characteristic 0. It is a well-known result (see e.g.
e
[L, Chapter 2, §5]) that (T±
W,ν )(x) has the form R(k) [t]/(t − u) for uniformizers u and t. Theorem 4.7. The Q̄p -space Symb±
Γ (Dk )(fβ ) has dimension e and is a free module of rank
e
one over the algebra (T±
k,ν )(fβ ) . This algebra is isomorphic to Q̄p [t]/(t ).
Proof —
We have
(T±
k,ν )(fβ )
=
(T±
k,ν )(x) by definition
=
(T±
W,ν )(x) ⊗R(k) ,k Q̄p by Corollary 4.4
'
R(k) [t]/(te − u) ⊗R(k) R(k) /(u) by Prop. 4.6
=
Q̄p [t]/(te )
±
±
On the other hand Symb±
Γ (Dk )(fβ ) = SymbΓ (Dk )(x) is free of rank one over (Tk,ν )(x) by
Prop. 4.5. This proves the theorem.
e−1
Corollary 4.8. The H-eigenspace Symb±
Symb±
Γ (Dk )[fβ ] is t
Γ (Dk )(fβ ) and has dimen-
sion 1 over Q̄p . Its image by ρ∗k is 0 if e > 1 or if f is Eisenstein and the sign ± is −(fβ ).
Otherwise, that is when e = 1, and either f is cuspidal, or f is Eisenstein, and ± = (fβ ),
this image is a line.
Proof —
Generally speaking, the eigenspace for some commuting linear operators and
system of eigenvalues in a finite-dimensional space is the socle of the generalized eigenspace
(for the same operators and eigenvalues) seen as a module over the algebra generated by
those operators. In our case, the eigenspace we are looking at is the socle of Symb±
Γ (Dk )
e
as a module over (T±
W,ρ )x ' Q̄p [t]/(t ). This socle is one dimensional, generated by the
image of te−1 .
For the second part, we note that by Corollary 3.16, the maps es± ◦ ρ∗k induce Heckeequivariant maps
es+ ◦ ρ∗k : Symb+
Γ (Dk )(fβ )
→ Sk+2 (Γ)(fβ ) ⊕ Ek+2 (Γ)∨
(fβ )
es− ◦ ρ∗k : Symb−
Γ (Dk )(fβ )
→ Sk+2 (Γ)(fβ ) ⊕ Ek+2 (Γ)∨
(fβ )
Note that the target of both maps are one dimensional since fβ is a newform (see Lemma 2.8(i)),
and that (by Corollary 3.16 and §3.2.6) both maps are surjective except when ± = −(fβ )
and f is Eisenstein, in which case the map is 0.
44
J. BELLAÏCHE
On the other hand the sources of both maps have dimension e by the theorem above. If
e = 1, the inclusion
±
Symb±
Γ (Dk )[(fβ )] ⊂ SymbΓ (Dk )(fβ )
is an equality, so the maps es± ◦ ρ∗k are surjective from Symb±
Γ (Dk )[(fβ )] to their respective
targets (save the exceptional case, where the map is 0). Therefore they are an isomorphism
by equality of dimensions of the sources and targets, except when the sign ± is −(fβ ) and
f is Eisenstein, in which case the map is 0.
In general, for e ≥ 1, the operator t acts nilpotently on the source of es± ◦ ρ∗k , so
nilpotently on the target, but since this target has dimension at most 1, it acts by 0. So if
e > 1, then te−1 acts by 0 on Mk+2 (Γ)(fβ ) , and the image of our eigenspace is 0.
We note that we have now proved Theorem 1, which is the first sentence of Corollary 4.8, and also Theorem 4. Indeed, the first two paragraphs of that theorem follows
from Theorem 4.7. The equivalence between (i) and (ii) then follows from Prop. 2.11(ii),
and the equivalence with (iii), as well as the last assertion, from the second sentence of
Corollary 4.8.
4.2.2. Properties of the p-adic L-function and proof of Theorem 2.
±
Definition 4.9. We now choose a non-zero vector Φ±
fβ in SymbΓ (Dk )[fβ ] (well defined up
to a scalar by Corollary 4.8) and define the p-adic L-function
L± (fβ , σ) = Λk (Φ±
fβ )(σ),
which clearly amounts to the same definition as in the introduction.
We check the properties of those functions listed in the introduction: The analyticity
is clear, the estimation of the order of L± (fβ , ·) follows from the result of Visik recalled in
Prop 3.20.
The interpolation properties follow immediately from Corollary 4.8 and Prop. 3.22 in
±
∗
case ρ∗k (Φ±
fβ ) = 0 and from Prop 3.23 in case ρk (Φf,β ) 6= 0 and f is cuspidal.
Example 4.10. Take N = 1 and k > 0 even. As is well known, there exists one normalized
Eisenstein series f = Ek+2,1,1 of weight k + 2 and level 1. Let fβ (with β = pk+1 ) be its
P
P
k+1 n
critical-slope refinement, so fβ (z) =
)q . There is only one (up to
n≥1 (
d|n, p6|d d
a scalar) non-zero boundary eigensymbol τ ∈ Hom(∆, Vk )Γ for Γ = Γ0 (p) and of weight
k + 2 satisfying Up τ = pk+1 τ . It satisfies for 0 ≤ j ≤ k, τ (∞)(z j ) = δj,k , τ (0)(z j ) =
pk+1 −1
δ
pk (p−1) j,0
(f )
(where δ is Kronecker’s symbol). By Prop 3.15, we have τ = ρ∗k (Φfβ β ), and
it easy to see that actually (fβ ) = 1. Let µ = Φ+
fβ ({∞} − {0}) ∈ D. Then by (18),
R j
R j
δ
pk+1 −1
pk+1 −1
z dµ(z) = δj,k − pk (p−1) δj,0 , and z 1pZp dµ(z) = j,k
p − p2k+1 (p−1) δj,0 . Therefore,
R
R
k+1
−1)2
z k dµ(z) = (p − 1)/p is non 0, and so is Z∗ dµ(z) = − p(p2k+1 (p−1)
. Hence L+ (fβ , tk ) 6= 0
Z∗
p
and L+ (fβ , 1) 6= 0.
p
CRITICAL p-ADIC L-FUNCTIONS
45
Finally, it remains to prove Corollary 1, that is that for fβ of critical slope, L± (fβ , σ)
has infinitely many 0’s in every open unit disc (that is, connected component) of W. When
fβ is θ-critical, the interpolation properties already gives an infinity of 0’s. When it is
not, the proof follows exactly the one given in [PO] (cf. Theorem 3.3 there, attributed to
Mazur): For any fixed character ψ of µp−1 we observe from the interpolation properties
the existence of elements cn ∈ C∗p such that for n ≥ 1
Lp (fβ , ψχζn −1 )
=
cn
β n+1
cn
Lp (fα , ψχζn −1 ) =
αn+1
where α is the other root of (1), ζn is a primitive pn -root of 1 in Q̄p , and as in the
introduction χζn −1 is the character of 1 + pZp that sends the chosen generator γ of the
group to ζn − 1. As shown in [PO, Theorem 3.3], this implies that on each connected
component of W, either L(fβ , .) or L(fα , .) has an infinity of 0’s. But vp (α) = 0 since
vp (β) = k + 1, so L(fα , .) has order 0, and thus has only a finite number of 0’s. This
completes the proof.
4.3. Two-variables p-adic L-functions. We keep the notations of the preceding subsection.
4.3.1. Modular symbols over the eigencurve.
Proposition 4.11. Up to shrinking the affinoid neighborhood W = SpR of k in W , there
±
exists an affinoid neighborhood V = SpT of x ∈ CW,ν
which is a connected component of
SpTW,ν , an element u ∈ R ⊗ Q̄p such that u(k) 6= 0, and k is the only 0 of u on U , an
element t ∈ T such that t(x) 6= 0, and an isomorphism of R-algebras T ' R[X]/(X e − u)
sending t on X.
Proof —
This follows easily from Prop 4.6.
We consider the T -module
≤ν
⊗T±
M := Symb±
Γ (D(R))
T.
W,ν
±
≤ν
It is free of rank one. Note that T = T±
W,ν for some idempotent , so M := SymbΓ (D(R))
is a submodule and direct summand of Symb±
Γ (D(R)).
Now we set
N = M ⊗R T.
This R-module has two structures of T -module, one coming from the T -structure that M
already had (called below the first T -structure), the other coming from the right factor T
in the tensor product (called the second T -structure).
One might think of N as the module of distributions-valued modular symbols over the
eigencurve C (more precisely, over the open subspace V of C) as opposed as M which is a
46
J. BELLAÏCHE
part of the module of distributions-valued modular symbols over the weight space (more
precisely, over the open subset W of W).
For every point y ∈ V (Q̄p ), of weight κ(y) ∈ W (Q̄p ) we define a specialization map
spy : N → Symb±
Γ (Dκ(y) ) ⊗Qp Q̄p
by composing the following canonical morphisms
N
→
N ⊗T,y Q̄p
=
(M ⊗R T ) ⊗T,y Q̄p
=
M ⊗R,κ(y) Q̄p
⊂
≤ν
Symb±
⊗Qp Q̄p
Γ (Dκ(y) )
where the tensor product is for the second T -structure
This map is H-equivariant if we give M the first T -structure (and the corresponding action
of H) and Symb±
Γ (Dκ(y) ) its natural action of H.
4.3.2. Mellin transform over the eigencurve. Recall from 3.3.1 that there is an R-linear
map
b
ΛR : SymbΓ (D(R))≤ν → R⊗R.
By restriction this maps induces a map
b
Λ|M : M → R⊗R
from which we get a map
∼
b
b
ΛT := Λ|M ⊗ IdT : N = M ⊗R T → (R⊗R)
⊗R T −→ T ⊗R.
Note that elements of the target of this map are two-variables analytic functions, the first
variable being a point y ∈ SpT = V ⊂ C, the second variable being an element σ of W.
The morphism ΛT is obviously T -equivariant if we give N its second T -structure.
Lemma 4.12. If Φ ∈ N and y ∈ V (Q̄p ), then for all σ, we have
ΛT (Φ)(y, σ) = Λ(spy (Φ))(σ).
Proof —
Assume first that Φ is a pure tensor φ ⊗ t with φ ∈ M and t ∈ T . Then
spy (Φ) = spy (φ ⊗ t) = t(y)φκ(y) ∈ M ⊂ Symb±
Γ (Dκ(y) ) by definition of spy . Therefore
Λ(spy (Φ))(σ) = t(y)Λ(φ)(σ) for all σ ∈ W(Cp ). On the other hand ΛT (Φ) = ΛT (φ ⊗
∼
b
b
t) = ΛR (φ) ⊗ t ∈ (R⊗R)
⊗ T −→ T ⊗R.
Therefore ΛT (Φ)(y, σ) = (ΛR (φ) ⊗ t)(y, σ) =
t(y)Λ(φ)(κ(y), σ). The equality Λ(spy (Φ))(σ) = ΛT (Φ)(y, σ) then follows from (17) applied
with w = κ(y).
The case where Φ is a general element of N , that is an R-linear combination of pure
tensors as above, follows immediately if we observe that the two maps N → R given by
Φ 7→ ΛT (Φ)(y, ·) and Φ 7→ Λ(spy (Φ)) are both R-linear if we give N its natural R-structure
(this is the same for the two T -structures) and R its R-structure using the morphism
κ(y) : R → Q̄p .
CRITICAL p-ADIC L-FUNCTIONS
47
4.3.3. Proof of Theorem 3. Let us choose φ a generator of the free T -module M , and set
Φ=
e−1
X
ti φ ⊗ te−1−i ∈ N.
i=0
It is clear that the element Φ of N depends on the choices made (the isomorphism in
Prop 4.11, the generator φ of M ) only up to multiplication by an element of T ∗ for the
first T -structure on N .
Lemma 4.13. One has
(t ⊗ 1 − 1 ⊗ t)Φ = 0.
Consequently, the action of any element of T on Φ for the first or the second T -structure
on N are identical.
Proof —
We compute in N = M ⊗R T ,
(t ⊗ 1 − 1 ⊗ t)Φ
=
(t ⊗ 1 − 1 ⊗ t)
e−1
X
ti φ ⊗ te−1−i
i=0
=
e−1
X
ti+1 φ ⊗ te−1−i − ti φ ⊗ te−i
i=0
= te φ ⊗ 1 − φ ⊗ te
”telescopic cancellations”
= te (φ ⊗ 1 − φ ⊗ 1)
since te = u is in R
=
0
In particular, the element Φ is well-defined up to a multiplication by an element of T ∗
for the second T -structure as well.
Proposition 4.14. If y ∈ V (Q̄p ) corresponds to a refined modular form fβ0 0 of weight
∗
k 0 + 2, with k 0 = κ(y), then in Symb±
Γ (Dk0 ) ⊗Qp Q̄p we have, up to a scalar in Q̄p , the
equality
spy (Φ) = Φfβ0 0 ,
where Φfβ0 is any generator of the one dimensional Q̄p -space Symb±
Γ (Dk0 )(f 0 0 ) (see Corolβ
lary 4.8)
Proof —
M⊗
R,k0
It is enough, by Corollary 4.8 to prove that spy (Φ) is in the H-eigenspace in
0
Q̄p ⊂ Symb±
Γ (Dk0 ) for the same system of H-eigenvalues as fβ 0 . But the algebra
of Hecke operators acting on M ⊗R,k0 Q̄p is generated by a single operator t, and we thus
only have to show that t acts on spy Φ as it acts on fβ0 0 , that is with the eigenvalues t(y).
This is clear by Lemma 4.13 and the definition of spy .
48
J. BELLAÏCHE
Definition 4.15. We set L± := ΛT (Φ) ∈ T ⊗ R and call it the two variables p-adic Lfunction. It is an analytic function of two variables (y, σ) 7→ L± (y, σ) defined on V × W.
The function L± is well-defined up to multiplication by a non-zero analytic function on V
(that is an element of T ∗ ).
If now y is a point of V (Q̄p ) corresponding to a refined modular form fβ0 0 of weight k 0 ,
then
L± (y, σ)
=
ΛT (Φ)(y, σ) by Definition 4.15
=
Λ(spy (Φ)(σ)) by Lemma 4.12
=
Λ(Φfβ0 0 )(σ)
=
L± (fβ0 0 , σ) (up to a scalar)
(up to a scalar)
by Prop. 4.14
by Definition 4.9
This completes the proof of Theorem 3.
Remark 4.16. The construction we have given of the two-variables p-adic L-functions is
very pedestrian, but this feature will be very helpful in comparing this p-adic L-function
with the secondary p-adic L-functions defined below.
There is a quicker and more conceptual construction, which does not allow for easy
comparison with the secondary p-adic L-functions, but which makes apparent the proximity
of our construction with the method of Mazur-Kitagawa that we now give.
We start with the canonical R-linear map
b
Λ|M : M −→ R⊗R
defined in the beginning of §4.3.2, and which we recall is simply the composition of the
≤ν
inclusion M ⊂ Symb±
with the evaluation at {0} − {∞} and then the Mellin
Γ (D ⊗ R)
transform. Obviously, this map can be seen as an element
b ⊗R M ∨ = R⊗M
b ∨,
Λ|M ∈ (R⊗R)
where M ∨ is the dual module HomR (M, R), seen as a T -module. Now the crucial property
is that M ∨ is free of rank one as a T -module. This can be seen in two ways: first, we know
that T is Gorenstein since it is regular, and since M is free of rank one, so is its dual; or
one can applying directly Lemma 4.1 to M ∨ , which proves that M ∨ is free of some rank,
which is proved to be one as we did for M . In any case, choosing a T -generator of M ∨
b ∨ ' R⊗T
b and allows us to see Λ|M as an element of R⊗T
b
defines an isomorphism R⊗M
(well-defined up to an element of T ∗ ), that is a two-variables analytic function, and it is
not hard to see that this function is exactly the one we have defined above.
4.4. Secondary p-adic L-functions. We keep all notations and assumptions as above.
We assume moreover than e > 1, that is that fβ is critical. Since t is a uniformizer of V at
x, we can consider the two-variables p-adic L-function L(y, σ) with y in a neighborhood of
x as L(t, σ) with t in a neighborhood of 0.
CRITICAL p-ADIC L-FUNCTIONS
49
Definition 4.17. For i = 0, . . . , e − 1,
L±
i (fβ , σ) =
∂ i L± (t, σ)
∂ti
.
|t=0
This is in accordance with the definition given in the introduction.
To get more information on those secondary L-functions, we compute
±
L (t, σ)
=
(ΛT )(
e−1
X
ti φ ⊗ te−1−i )
by Definition 4.15
i=0
=
e−1
X
te−1−i ΛT (ti φ ⊗ 1)
since ΛT is T -linear for the second T -structure on N
i=0
from which we get:
L±
i (fβ , ·)
=
∂ i L(t, ·)
∂ti
t=0
= i!ΛT (te−1−i φ ⊗ 1)|t=0 by Leibniz’ rule
= i!Λ(spx (te−1−i φ ⊗ 1)) by Lemma 4.12
= i!(Λte−1−i φk )
where φk is the image of φ in M ⊗R,k Q̄p = Symb±
Γ (Dk )(x) . Since φ is an T -generator of M ,
±
φk is also a T ⊗R,k Q̄p = (T±
k,ν )(x) -generator of SymbΓ (Dk )(x) . Recall that by theorem 4.7,
±
e
(T±
k,ν )(x) is isomorphic to Q̄p [t]/t and the module SymbΓ (Dk )(x) is free of rank one over
that algebra. So the flag (te−1 φk , te−2 φk , . . . , φk ) is independent of the generator φk in
Symb±
Γ (Dk )(x) hence independent of φk . As we have seen, the image by Λ of the generalized eigensymbols te−1 φk , te−2 φk , . . . , φk are up to a scalar the secondary L-functions
±
L± (fβ , ·), L±
1 (fβ , ·), . . . , Le−1 (fβ , ·) we have proved the connection with modular sym-
bols of the secondary L-functions stated in the introduction.
We now prove the other properties of the L±
i (fβ , ·). The order properties follows directly
from 3.20.
The interpolation properties for L±
i (fβ , ·) for 0 ≤ i ≤ e − 2 follows from Prop. 3.22,
since that function is up to a scalar Λ(te−1−i φk ) and ρ∗k (te−1−i φk ) = te−1−i ρ∗k (φk ) = 0 for
i < e − 1 since t acts by 0 in Symb±
Γ (Vk )(x) (see the proof of Corollary 4.8). However, when
±
i = e − 1, one finds that L±
e−1 (fβ , ·) = Λ(φk ) and φk is a generator of SymbΓ (Vk )(x) , so
since ρ∗k is surjective with non 0 image, we have ρ∗k (φk ) 6= 0. The interpolation property
thus follows from Prop. 3.23.
The infinity of 0’s properties for Li (fβ , ·) for i < e − 1 is a trivial consequence of the
interpolation property, while for Le−1 (fβ , ·) it follows exactly as in the case of a non-critical
form, considering the pair of functions Le−1 (fβ , ·) and L(fα , ·).
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CRITICAL p-ADIC L-FUNCTIONS
[PA]
[PO]
[Ro]
[R]
[Sc]
[Se]
[Shi]
[Sho]
[SU1]
[SU2]
[SO]
[St1]
[St2]
[S]
[S2]
[S3]
[SP1]
[SP2]
[V]
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51
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E-mail address: jbellaic@brandeis.edu
Joël Bellaı̈che, Brandeis University, 415 South Street, Waltham, MA 02454-9110, U.S.A