Vanishing of special values and central derivatives in Hida families

0.32 MB·English

by  Matteo Longo

VANISHING OF SPECIAL VALUES AND CENTRAL DERIVATIVES
IN HIDA FAMILIES
MATTEO LONGOAND STEFANO VIGNI
Abstract. The theme of this work is the study of the Nekova´ˇr–Selmer group He1(K,T†)
f
attachedtoatwisted Hidafamily T† of Galois representationsand aquadraticnumberfield
K. Theresultsthatweobtainhavethefollowing shape: ifatwistedL-functionofasuitable
2
modular form in the Hida family has order of vanishing r ≤ 1 at the central critical point
1 then the rank of He1(K,T†) as a module over a certain local Hida–Hecke algebra is equal to
0 f
2 r. Under the above assumption, we also show that infinitely many twisted L-functions of
modular forms in the Hida family have the same order of vanishing at the central critical
n
point. OurtheoremsextendtomoregeneralarithmeticsituationsresultsobtainedbyHoward
a
whenK isanimaginaryquadraticfieldandalltheprimesdividingthetameleveloftheHida
J
family split in K.
4
2

]
T 1. Introduction
N
Fix an integer N 1, a prime number p ∤ 6Nφ(N) and an ordinary p-stabilized newform
.
h ≥
t f = a qn S Γ (Np),ωj
a n k 0
∈
m nX≥1 (cid:0) (cid:1)
[ of weight k 2 (where ω is the Teichmu¨ller character and j k (mod 2)) whose associated
≥ ≡
mod p Galois representation is absolutely irreducible and p-distinguished (see [20, p. 297]).
1
v Let F := Q (a n 1) be the finite extension of Q generated by the Fourier coefficients of
p n p
9 f and let de|not≥e its ring of integers. Write for the branch of the Hida family where f
F
5 O R
lives (see [20, 5.3]); then is a complete local noetherian domain, finite and flat over the
0
§ R
5 Iwasawa algebra Λ:= F[[1+pZp]], and the Hida family we are considering can be viewed as
O
. a formal power series f [[q]].
1 ∞
∈ R
0 For p Spec( ) let denote the localization of at p and set F := /p . Hida’s
p p p p
2 theory ([∈10], [11]R, [12])Rshows that if p Spec( ) isRan arithmetic prime R(in thRe sense of
1 ∈ R
[20, 5.5]) then F is a finite extension of F and the power series in F [[q]] obtained from f
v: by co§mposing witph the canonical map F is the q-expansion ofpa classical cusp form∞
p
i R →
X f ; in particular, f = f for a certain arithmetic prime ¯p of weight k. Furthermore, with
p ¯p
r f∞ is associated a Gal(Q¯/Q)-representation T which is free of rank 2 over and such that
a V := T F is isomorphic to (a Tate twist of) the Gal(Q¯/Q)-representatioRn attached to f
p R p p
⊗
by Deligne. The choice of a so-called critical character
Θ : Gal(Q¯/Q) Λ×
−→
allows us to uniformily twist T and obtain a Galois representation T† := T Θ−1 which is
⊗
free of rank 2 over and such that for every arithmetic prime p of the representation
R R
V† := T† F is a self-dual twist of V .
p R p p
⊗
The general theme of this article is the study of the -rank of Nekov´aˇr’s Selmer group
R
H1(K,T†) attached to T† and a quadratic number field K.
f
We first let K be an imaginary quadratic field K of discriminant prime to Np and define
e
the factorization N = N+N− by requiring that the primes dividing N+ split in K and the
2010 Mathematics Subject Classification. 11F11, 11G18.
Key words and phrases. Hida families, quaternionic big Heegnerpoints, L-functions, Selmer groups.
1
2 MATTEOLONGOANDSTEFANOVIGNI
primes dividing N− are inert in K. We assume throughout that N− is square-free. Moreover,
since the case N− = 1 was studied in [13], we also assume that N− > 1. We say that we are
in the indefinite (respectively, definite) case if N− is a product of an even (respectively, odd)
number of primes. For every arithmetic prime p of , the choice of Θ gives an F×-valued
p
R
character χ of A× whose restriction to A× is the inverse of the nebentype of f (here A is
p K Q p L
the adele ringof a numberfield L). Then the Rankin–Selberg L-function L (f ,χ ,s) admits
K p p
a self-dual functional equation whose sign controls the order of vanishing of L (f ,χ ,s) at
K p p
the critical point s = 1. We say that the pair (f ,χ ) has analytic rank r 0 if the order of
p p
≥
vanishing of L (f ,χ ,s) at s = 1 is r.
K p p
A simplified version of our main results can be stated as follows.
Theorem 1.1. Let K be an imaginary quadratic field. If there is an arithmetic prime p of
weight 2 and non-trivial nebentype such that (f ,χ ) has analytic rank one (respectively, zero)
p p
then H1(K,T†) is an -module of rank one (respectively, an -torsion module).
f R R
Proeposition 5.3 (respectively, Proposition 4.3) implies that if the pair (f ,χ ) has analytic
p p
rank one (respectively, zero) then we are in the indefinite (respectively, definite) case. Under
this assumption, we also show that (f ,χ ) has analytic rank one (respectively, zero) for all
p′ p′
but finitely many arithmetic primes p′ of weight 2; see Corollary 4.5 (respectively, Corollary
5.5) for details. These results represent weight 2 analogues (over K) of the conjectures on
the generic analytic rank of the forms f formulated by Greenberg in [8], and the reader is
p
suggested to compare them with the statements in [20, 9.4] (in particular, [20, Conjecture
§
9.13]), where conjectures for forms f of arbitrary weight are proposed. Theorem 1.1 is a
p
combination of Theorems 4.6 and 5.6, where more general formulations (involving the notion
of generic arithmetic primes of weight 2, cf. Definition 2.1) can be found. In the case of rank
one, the corresponding statement in the split (i.e., N− = 1) setting can be proved using the
results obtained by Howard in [14, 3].
§
The proof of Theorem 1.1 given in Sections 4 and 5 relies crucially on the properties of
the big Heegner points whose construction is described in [20], where we generalized previous
work of Howard onthe variation of Heegner points inHidafamilies ([13]). With aslight abuse
of terminology (cf. Definition 3.1), for the purposes of this introduction we can say that big
Heegner points are certain elements Z H1(K,T†) (indefinite case) and Z J (definite case)
∈ ∈
defined in terms of distributions of suitable Heegner (or Gross–Heegner) points on Shimura
curves, where J is an -module obtained from the inverse limit of the Picard groups attached
R
to a tower of definite Shimura curves with increasing p-level structures. Since the Galois
representation T† canbeintroducedusingananalogous tower ofindefinite(compact) Shimura
curves, the construction of the elements Z proceeds along similar lines in both cases. These
constructions are reviewed in Section 3. It turns out that the image of Z in the F -vector
p
space H1(K,V†) (indefinite case) or J F (definite case) as p varies in the set of (weight
p R p
⊗
2) arithmetic primes controls the rank of H1(K,T†). In fact, in light of Propositions 4.3 and
f
5.3, Theorem 1.1 tells us that knowledge of the local non-vanishing of the big Heegner point
e
Z at a single arithmetic prime p of weight 2 and non-trivial nebentype allows us to predict
the rank over of the big Selmer group H1(K,T†).
R f
Finally, by suitably splitting H1(K,V†) and considering twists of Hida families, one can
f pe
obtain results in the same vein as Theorem 1.1 for general quadratic fields K (not only
e
imaginary ones) from the results (over Q) proved in [14, Section 4]. This strategy does not
make use of the big Heegner points introduced in [20], and the input has to be an arithmetic
prime of trivial character. We provide details in Section 6 (see Theorems 6.3 and 6.4).
Convention. Throughout the paper we fix algebraic closures Q¯, Q¯ and field embeddings
p
Q¯ ֒ Q¯ , Q¯ ֒ C.
p p
→ →
SPECIAL VALUES AND CENTRAL DERIVATIVES IN HIDA FAMILIES 3
2. Hida families of modular forms
Define Γ := 1 + pZ (so Z× = µ Γ where µ is the group of (p 1)-st roots of
p p p−1 × p−1 −
unity) and Λ := [[Γ]]. Let denote the integral closure of Λ in the primitive component
F
O R
to which f belongs (see [20, 5.3] for precise definitions). Then is a complete local
K § R
noetherian domain, finitely generated and flat as a Λ-module, and if hord is Hida’s ordinary
∞
Hecke algebra over of tame level Γ (N), whose construction is recalled in [20, 5.1], then
F 0
O §
there is a canonical map
φ : hord
∞ ∞ −→ R
which we shall sometimes call the Hida family of f. For every arithmetic prime p of (see
R
[20, 5.5] for the definition) set F := /p . Then F is a finite field extension of F and
p p p p
we fi§x an F-algebra embedding F ֒ Q¯R, soRthat from here on we shall view F as a subfield
p p p
of Q¯ . Now consider the compositio→n
p
(1) φ :hord φ∞ πp F ,
p ∞ −→ R −→ p
where π is the map induced by the canonical projection ։ F . For every integer m 1
p p p
R ≥
define Φ := Γ (N) Γ (pm). By duality, with φ is associated a modular form
m 0 1 p
∩
f = a (p)qn S Φ ,ψ ωk+j−kp,F ,
p n ∈ kp mp p p
nX≥1 (cid:0) (cid:1)
of suitable weight k , level Φ and finite order character ψ : Γ F×, such that a (p) =
p mp p → p n
φ (T ) F for all n 1 (here T hord is the n-th Hecke operator). We remark that if
p n ∈ p ≥ n ∈ ∞
t is the smallest positive integer such that ψ factors through (1 + pZ )/(1 + ptpZ ) then
p p p p
m = max 1,t . See [20, 5.5] for details.
p p
{ } §
Let Q(µ ) be the p-adic cyclotomic extension of Q and factor the cyclotomic character
p∞
ǫ :Gal(Q(µ )/Q) ≃ Z×
cyc p∞ −→ p
asaproductǫ = ǫ ǫ withǫ (respectively,ǫ )takingvaluesinµ (respectively,
cyc tame wild tame wild p−1
Γ). Write γ [γ] for the inclusion Γ ֒ Λ× of group-like elements and define
7→ →
Θ := ǫ(k+j−2)/2 ǫ1/2 :Gal(Q(µ )/Q) Λ×
tame wild p∞ −→
(cid:2) (cid:3)
1/2
where ǫ is the unique square root of ǫ with values in Γ. In the obvious way, we shall
wild wild
also view Θ as ×-valued. We associate with Θ the character θ and, for every arithmetic
R
prime p, the characters Θ and θ given by
p p
θ :Z× ǫ−cy1c Gal(Q(µ )/Q) Θ ×,
p −−→ p∞ −→ R
Θ :Gal(Q(µ )/Q) Θ × πp F×,
p p∞ −→ R −→ p
θ : Z× ǫ−cy1c Gal(Q(µ )/Q) Θp F×.
p p −−→ p∞ −→ p
Accordingtoconvenience,thecharactersΘandΘ willalsobeviewedasdefinedonGal(Q¯/Q)
p
via the canonical projection. If we write [] for the composition of [] with × F× then
p p
· · R →
(θ )2 = [] , and for every arithmetic prime p of weight 2 the modular form f† := f θ−1
p|Γ ·p p p⊗ p
has trivial nebentype (see [14, p. 806]).
As in [14, Definition 2], we give
Definition 2.1. An arithmetic prime p of weight 2 is generic for θ if one of the following
(mutually exclusive) conditions is satisfied:
(1) f isthep-stabilization(inthesenseof[14,Definition1])ofanewforminS (Γ (N),F )
p 2 0 p
and θ is trivial;
p
(2) f is a newform in S (Γ (Np),F ) and θ = ω(p−1)/2;
p 2 0 p p
4 MATTEOLONGOANDSTEFANOVIGNI
(3) f has non-trivial nebentype.
p
Let Gen (θ) denote the set of all weight 2 arithmetic primes of which are generic for θ.
2
R
Observethat, sinceonly afinitenumberofweight 2arithmetic primesdonotsatisfy condition
(3)inDefinition2.1,thesetGen (θ)contains allbutfinitelymanyweight2arithmeticprimes.
2
Before concluding this subsection, let us fix some more notations. If E is a number field
denote A its ring of adeles and write art : A× Gal(Eab/E) for the Artin reciprocity
E E E →
map. Furthermore, let h be the -Hecke algebra acting on the C-vector space S (Φ ,C)
m F 2 m
O
and write hord for the ordinary -subalgebra of h , which is defined as the product of the
m OF m
local summands of h where U is invertible (therefore hord = limhord, cf. [20, 5.1]). If M
m p ∞ m §
is an h -module set Mord := M hord. Finally, define the←Θ−−1-twist of a Λ-module M
m ⊗hm m
endowed with an action of Gal(Q¯/Q) as follows: let Λ† denote Λ viewed as a module over
itself but with Gal(Q¯/Q)-action given by Θ−1 and set M† := M Λ†.
Λ
⊗
3. Review of big Heegner points
Let B denote the quaternion algebra over Q of discriminant N−. Fix an isomorphism of
Q -algebras i : B := B Q M (Q ) and choose for every integer t 0 an Eichler
p p p Q p 2 p
⊗ ≃ ≥
order R of level ptN+ such that i (R Z ) is equal to the order of M (Q ) consisting of
t p t Z p 2 p
⊗
the matrices in M (Z ) which are upper triangular matrices modulo pt and R R for all
2 p t+1 t
⊂
t. Let B and R denote the finite adelizations of B and R and define U to be the compact
t t t
open subgroup of R× obtained by replacing the p-component of R× with the subgroup of
b b t t
(R Z )× consisting of those elements γ such that i (γ) 1 b (mod pt).
t⊗Z p b p ≡ 0d b
Fix an integer t 0 and define the set (cid:0) (cid:1)
≥
(2) X(K) = U B× Hom (K,B) B×,
t t × Q
(cid:15)(cid:0) (cid:1)(cid:14)
where Hom denotes homomeorphisms ofbQ-algebras. See [20, 2.1] for details. An element
Q
§
[(b,φ)] X(K) is said to be a Heegner point of conductor pt if the following two conditions
∈ t
are satisfied.
e
(1) The map φ is an optimal embedding of the order of K of conductor pt into the
pt
O
Eichler order b−1R b B of B.
t
∩
(2) Let x denote the p-component of an adele x, define U to be the p-component of U ,
p t,p t
b
set := Z , := Z and let φ denote the morphism obtained
pt,p pt Z p K,p K Z p p
O O ⊗ O O ⊗
from φ by extending scalars to Z . Then
p
φ−1 φ ( × ) b−1U×b = × (1+pt )×.
p p Opt,p ∩ p t,p p Opt,p∩ OK,p
(cid:0) (cid:1)
For every t 0 let H be the ring class field of K of conductor pt and set L := H (µ ).
≥ t t t pt
3.1. Indefinite Shimura curves. For every t 0 we consider the indefinite Shimura curve
≥
X over Q whose complex points are described as the compact Riemann surface
t
e X (C) := U B× (C R) B×.
t t
× −
(cid:15)(cid:0) (cid:1)(cid:14)
The set X(K) can be identifieed with a subsebt of X (Kab) where Kab is the maximal abelian
t t
extension of K. For every number field E let
e e
:= Gal(Q¯/E).
E
G
Then acts naturally on the geometric points X (Q¯) of X .
E t t
G
e e
SPECIAL VALUES AND CENTRAL DERIVATIVES IN HIDA FAMILIES 5
3.2. Definite Shimura curves. For every t 0 we consider the definite Shimura curve
≥
X := U (B× P)/B×
t t
\ ×
where P is the genus 0 curve over Qedefined bby setting
P(A) := x B A x = 0, Norm(x) = Trace(x) = 0 A×
Q
∈ ⊗ | 6
(cid:8) (cid:9)(cid:14)
for every Q-algebra A(here Norm and Trace are the normand trace maps of B, respectively).
(K)
There is a canonical identification between the set X in (2) and the K-rational points
t
X (K) of X .
t t e
If E/K is an abelian extension then we define
e e
:= Gal(Kab/E);
E
G
inparticular, istheabelianization oftheabsoluteGalois groupofK. Weexplicitly remark
K
G
that this group is different from the group denoted by the same symbol in the indefinite case.
The set X(K) is equipped with a canonical action of K×/K× given by
t GK ≃
e [(b,φ)]σ := (bφˆ(a),φ) , b
(cid:2) (cid:3)
where φˆ: K ֒ B is the adelization of φ and a K× satisfies art (a) = σ.
K
→ ∈
3.3. Constbructibons of points. In this and thebnext subsection our considerations apply
both to the definite case and to the indefinite case. Recall that in the former case the symbol
stands for the Galois group Gal(Kab/E) of an abelian extension E of K, while in the
E
G
latter we use for the absolute Galois group of a number field E. Given a field extension
E
G
E/F (with E and F abelian over K in the definite case) and a continuous -module M, for
F
G
every integer i 0 we let
≥
(3) res :Hi( ,M) Hi( ,M), cor :Hi( ,M) Hi( ,M)
E/F F E E/F E F
G −→ G G −→ G
be the usual restriction and corestriction maps in Galois cohomology.
Denoting α : X X the canonical projection, [20, Theorem 1.1] shows the existence
t t t−1
→
(K)
of a family of points P X which are fixed by the subgroup of and satisfy
e t e∈ t GLt GK
e e U P = α tr P
p t−1 t Lt/Lt−1 t
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
(here, for a finite Galois extension eE/F, the symbol tr e stands for the usual trace map).
E/F
(K)
Via the Jacquet–Langlands correspondence, the divisor group Div X is equipped
t ⊗ZOF
with a standard action of the F-Hecke algebra ht (see [20, 6.3]).(cid:0)Defin(cid:1)e
O § e
ord
(K)
D := Div X
t t ⊗OF
(cid:16) (cid:17)
(cid:0) (cid:1)
(notice that here and below we are regardingeΘ as a character of via restriction). Taking
K
G
the inverse limit, we may define
D := limD ,
∞ t
←t−
which is naturally an hord-module. Finally, set
∞
D := D .
∞ hord
⊗ ∞ R
The canonical structure of Λ-module on these groups makes it possible to define the twisted
modules D†, D† and D†.
t ∞
Let H =H be the Hilbert class field of K, write
0
P H0( ,D†)
t ∈ GHt t
6 MATTEOLONGOANDSTEFANOVIGNI
for the image of P in D (cf. [20, 7.1]) and define
t t
§
e := cor (P ) H0( ,D†).
Pt Ht/H t ∈ GH t
Following [20, Definition 7.3], we give
Definition 3.1. The big Heegner point of conductor 1 is the element
:= limU−t( ) H0( ,D†)
PH p Pt ∈ GH
←t−
obtained by taking the inverse limit of the compatible sequence U−t( ) and then taking
p Pt t≥1
the image via the map D† D†. (cid:0) (cid:1)
∞
→
See [20, Corollary 7.2] for a proof that the inverse limit considered in Definition 3.1 makes
sense. Finally, set
(4) = := cor ( ) H0( ,D†).
K H/K H K
P P P ∈ G
3.4. Weight 2 arithmetic primes. We introduce some notations in the case of arithmetic
primes of weight 2. So fix an arithmetic prime p of of weight 2, level Φ and character ψ .
R mp p
In order to simplify notations, in this subsection put m := m and ψ := ψ . The action of
p p
hord on Dord factors through hord, which, by the Jacquet–Langlands correspondence, can be
∞ m m
identified with a quotient of the ordinary -Hecke algebra of level Φ . Define an F×-valued
F m p
O
character of A× by
K
χ (x) := Θ art (N (x))
p p Q K/Q
(cid:0) (cid:1)
and denote χ the restriction of χ to A×. Then the nebentype ψ ωk+j−kp of f is equal to
0,p p Q p p
χ−1 (see [14, 3] for a proof). In particular, since
0,p §
(Z/pZ)× (1+pZ )/(1+ptpZ ) (Z/ptpZ)×,
p p
× ≃
the conductor of χ−1 is ptp unless χ is trivial. For σ Gal(Kab/K) let x A× be such
0,p 0,p ∈ σ ∈ K
that art (x )= σ and define χ (σ) := χ (x ).
K σ p p σ
Since Θ factors through , we obtain the equality
p GQ(µpm)
(5) H0( ,D† )= H0( ,D ).
GLm m GLm m
Using (5), the point P H0( ,D† ) gives rise, by restriction, to a point res (P )
m ∈ GHm m Lm/Hm m
in H0( ,D ). Since is contained in the kernel of Θ , the map χ can be viewed
GLm m GQ(µpm) p p
as a character of Gal(L /K), so we may form the sum
m
(6) Pχp := res (P )σ χ−1(σ) H0( ,D F ).
m Lm/Hm m ⊗ p ∈ GLm m⊗OF p
X
σ∈Gal(Lm/K)
Via (5), we may also view Pχp as an element of H0( ,D† F ). Moreover, for every
m GLm m ⊗OF p
τ we have
K
∈ G
(Pχp)τ = Pστ|Lm Θ art (N (x )) χ−1(στ ) = Θ(τ)Pχp,
m m ⊗ p Q K/Q τ p |Lm m
σ∈GaXl(Lm/K) (cid:0) (cid:1)
so Pχp H0( ,D† F ).
Wme w∈ant tGoKexplmici⊗tlOyFrelapte Pχp and . In order to do this, we note that, by definition,
m m
P
there is an equality
Pχp = cor res (P ),
m Lm/K ◦ Lm/Hm m
SPECIAL VALUES AND CENTRAL DERIVATIVES IN HIDA FAMILIES 7
where the restriction and corestriction maps are as in (3) with E = L , F = H and
m m
†
M = D F . Therefore we get
m⊗OF p
Pχp = (cor cor cor res )(P )
(7) m H/K ◦ Hm/H ◦ Lm/Hm ◦ Lm/Hm m
= [L : H ]cor ( ).
m m H/K m
P
In the following we shall also make use of the module
D := D F ,
p R p
⊗
its twist D† = D† F and the image of in H0( ,D†) via the canonical map D D ,
p R p K p p
⊗ P G →
which we denote .
p
P
Remark 3.2. By extending Hida’s arguments in the proof of [12, Theorem 12.1] to the case of
divisors, it seems possible to show that the canonical map D D factors as
p
→
(8) D D F D .
−→ m⊗OF p −→ p
Then one would obtain from the above discussion the formula P = a (p)m[L : H ] in
p p m m p
P
H0( ,D†). Instead of deriving such an equality at this stage, we will obtain two related
K p
G
formulas (see Propositions 4.1 and 5.1) in specific situations where factorizations analogous
to (8) are available thanks to known cases of Hida’s control theorem.
4. Vanishing of central derivatives
Let the quaternion algebra B be indefinite and write p for a weight 2 arithmetic prime of
of level Φ and character ψ . Once p has been fixed, set m := m and ψ := ψ .
R mp p p p
4.1. Tate modules. For every integer t 0 let J be the Jacobian variety of X . Define
t t
≥
ord
e e
T := Ta J
t p t ⊗Zp OF
(cid:16) (cid:17)
(cid:0) (cid:1)
where, as usual, the upper index ord denotesethe ordinary submodule, then form the inverse
limit
T := limT .
∞ t
←t−
† †
We may also define the twists T and T . By [20, Corollary 6.5], there is an isomorphism
t ∞
(9) T T
∞ hord
⊗ ∞ R ≃
of Galois representations. We fix once and for all such an isomorphism, and freely use the
notation T for the Galois module appearing in the left hand side of (9). Now set
V := T F .
p R p
⊗
Then V† = T† F is (a twist of) the self-dual representation attached to f† by Deligne.
p R p p
⊗
Observe that the canonical map T V can be factored as
p
→
(10) T T F πm,p V .
−→ m⊗OF p −−−→ p
This follows from Hida’s control theorem [12, Theorem 12.1] (or, rather, from its extension to
the current setting given in [21, Proposition 2.17]). Following [13, p.101], for every t 0 one
≥
defines a twisted Kummer map
δ :H0(K,D†) H1(K,T†).
t t −→ t
Taking the limit as m varies and using the canonical map T† T†, we get a map
∞
→
δ : H0(K,D†) H1(K,T†).
∞
−→
8 MATTEOLONGOANDSTEFANOVIGNI
Recall the element introduced in (4) and define
P
Z := δ ( ) H1(K,T†).
∞
P ∈
Write Z for the image of Z in H1(K,V†). The map δ gives rise to a map
p p ∞
δ : H0(K,D†) H1(K,V†),
p p p
−→
and the square
H0(K,D†) δ∞ // H1(K,T†)
(cid:15)(cid:15) (cid:15)(cid:15)
H0(K,D†) δp // H1(K,V†)
p p
is commutative, so that Z is also equal to δ ( ) (in the notations of 3.4). We may also
p p p
P §
consider the class δ (Pχp) H1(K,T† F ), where Pχp H0(K,D† F ) is the
m m ∈ m ⊗OF p m ∈ m ⊗OF p
element introduced in (6) and δ is extended by F -linearity, and define X to be the image
m p p
of δ (Pχp) in H1(K,V†) via the map π appearing in (10).
m m p m,p
Proposition 4.1. The formula X = a (p)m[L : H ]Z holds in H0( ,T†).
p p m m p K p
G
Proof. We just need to review the above constructions. Thanks to factorization (10), the
class δ ( ) = Z is equal to the corestriction from H to K of the image of U−mδ ( )
p Pp p p m Pm ∈
H0( ,T† ) in H0( ,V†) under the map induced by the map labeled π in (10). Since
H m K p m,p
G G
the action of hord on V† is via the character φ defined in (1), one has
∞ p p
δ ( )= π cor a (p)−mδ ( ) .
p p m,p H/K p m m
P P
(cid:16) (cid:17)
(cid:0) (cid:1)
Finally, using (7) we find that
δ ( )= a (p)−m[L :H ]−1π δ (Pχp) = a (p)−m[L : H ]−1X ,
p p p m m m,p m m p m m p
P
and the result follows. (cid:0) (cid:1) (cid:3)
Corollary 4.2. Z = 0 X = 0.
p p
6 ⇐⇒ 6
Proof. Immediate from Proposition 4.1. (cid:3)
4.2. Non-vanishing results. Using the fixed embedding Q¯ ֒ C, we form the Rankin–
p
→
Selberg convolution L (f ,χ ,s) as in [15, 1]; note that if 1 is the trivial character then
K p p
§
L (f ,1,s) = L (f ,s). For the definition of the L-function L (f ,s) of f over K and of
K p K p K p p
its twists the reader is referred also to [4, 2.1]. The results obtained in this subsection are
§
the counterpart in an arbitrary indefinite quaternionic setting of those in [14, Propositions 3
and 4], where the case of the split algebra M (Q) is considered. Although many arguments
2
follow those in [14] closely, for the convenience of the reader we give detailed proofs.
Proposition 4.3. Suppose p Gen (θ) satisfies either (2) or (3) in Definition 2.1.
2
∈
(a) L (f ,χ ,s) vanishes to odd order at s = 1.
K p p
(b) L′ (f ,χ ,1) = 0 Z = 0.
K p p 6 ⇐⇒ p 6
Proof. Let h denote the algebra generated over Z by the Hecke operators T with (n,Np)= 1
n
and the diamond operators d with (d,Np) =1 acting on the C-vector space of cusp forms in
h i
S (Φ ,C). Write e for theidempotent in h F whichprojects onto the maximalsummand
2 m p Z p
⊗
on which every T acts as a (n) and acts as χ−1.
n p h·i 0,p
We apply the results in [15] with s = pm, c = pm and m = N. As recalled in [15, p. 829]
(cf. also [15, eqs. (2.5), (2.6), (2.8)]), the sign of the functional equation for L (f ,χ ,s) is
K p p
1, and this shows part (a).
−
SPECIAL VALUES AND CENTRAL DERIVATIVES IN HIDA FAMILIES 9
Let us prove (b). As in [15, 5.1], denote Hg the Hodge embedding defined in [27, 6.2].
§ §
We introduce the divisor
P := Pσ χ−1(σ) Div X(K) F ,
m⊗ p ∈ m ⊗Z p
σ∈GaXl(Lm/K) (cid:0)e (cid:1)
whose image in D under the canonical maps is P . After fixing an appropriate embedding
p p
K ֒ B, the image Hg(P) of our divisor P in e J (L ) F is equal to the point Q
p m m Z p χ,Π
→ ⊗
considered in [15, Theorem 4.6.2], hence (cid:0) (cid:1)
e
Hg(P) = 0 L′ (f ,χ ,1) = 0,
6 ⇐⇒ K p p 6
by [15, Theorem 4.6.2]. It remains to show that Hg(P) = 0 if and only if Z = 0, or
p
6 6
equivalently, thanks to Corollary 4.2, if and only if X = 0.
p
6
To prove this, we start by noticing that our hypotheses ensure that f is a newform of
p
level Npm, so that Eichler–Shimura theory and multiplicity one show that the summand
ord
e (T F ) = e Ta (J ) F is sent isomorphically onto V under the map in (10).
p m⊗OF p p p m ⊗Zp p p
Using [25, Corollary(cid:0)2.2 and Propos(cid:1)ition 2.3], it can be checked that the Kummer map
e
J (L ) F H1 L ,Ta (J ) F
m m ⊗Z p −→ m p m ⊗Zp p
(cid:0) (cid:1)
is injective, hence the abeove identifications yield an injecteion
e J (L ) F ord ֒ H1(L ,V ) H1(L ,V†).
p m m Z p m p m p
⊗ −→ ≃
(cid:0) (cid:1)
By definition, the imagee of Hg(P) under this map is equal to the image of Xp under the
restriction map H1(K,V†) H1(L ,V†), which is injective as its kernel is an F -vector
p m p p
space that is killed by [L :→K]. This concludes the proof. (cid:3)
m
Proposition 4.4. Suppose p Gen (θ) satisfies (1) in Definition 2.1.
2
∈
(a) L (f ,s) vanishes to odd order at s = 1.
K p
(b) L′ (f ,1) = 0 = Z = 0.
K p 6 ⇒ p 6
♯
Proof. In this case m = 1 and there is a normalized newform f of level Γ (N), with Fourier
p 0
coefficients a♯(p), whose p-stabilization is f . Moreover, χ is the trivial character 1. If ǫ
n p p K
♯
denotes the character of K then the L-functions of f and f are related by the equality
p p
p1−s p1−s
♯
(11) 1 1 ǫ (p) L (f ,s) = L (f ,s).
(cid:18) − a (p)(cid:19)·(cid:18) − K a (p)(cid:19)· K p K p
p p
The extra Euler factors on the left do not vanish at s = 1 because a (p) = 1 (since a (p) is
p p
6 ±
a root of the Hecke polynomial X2 a♯p(p)X +p = 0, its absolute value is √p), therefore the
−
♯
L-functions of f and f have the same order of vanishing at s = 1, which is odd in light of
p p
the functional equation recalled in [15, p. 829] (here we apply again the results of [15] with
s= p, c= p, m = N). This proves part (a).
ByCorollary4.2,toprovepart(b)onecanequivalentlyshowthatifX = 0thenL′ (f♯,1) =
p K p
0. Let X and X be the (compact) Shimura curves over Q associated with the Eichler orders
0 1
R and R , respectively, so that
0 1
X (C)= R× B× P B×
i i ×
(cid:15)(cid:0) (cid:1)(cid:14)
for i = 0,1. There are degeneracy mapbs α,βb : X1(C) X0(C), with α the canonical
projection and β corresponding to the action of an element→of norm p of R normalising R×
1 1
(see, e.g., [19, 3.1]). These maps induce homomorphisms by Picard functoriality
§
α∗,β∗ : J J
0 1
−→
where J and J are the Jacobian varieties of X and X , respectively. The maps α∗ and β∗
0 1 0 1
are compatible with the action of the Hecke operators T for (n,Np) = 1 and the diamond
n
10 MATTEOLONGOANDSTEFANOVIGNI
operators d for (d,Np) = 1. Write h for the Hecke algebra generated over Z by these
h i
operators and denote e the idempotent in h F associated with p, as in the proof of
p Z p
⊗
Proposition 4.3. Define
e Ta (J ):= e Ta (J ) F
p p i p p i ⊗Zp p
for i = 0,1. Since the form f is old at p(cid:0), the idempote(cid:1)nt e kills the p-new quotient
p p
J /(α∗(J ) β∗(J )) of J , hence by [2, 1.7] there is a decomposition
1 0 0 1
⊕ §
(12) e Ta (J ) α∗e Ta (J ) β∗e Ta (J ).
p p 1 p p 0 p p 0
≃ ⊕
The Hecke operator T acts on e Ta (J ) as multiplication by a♯(p) = a (p) and, by strong
n p p 0 n n
multiplicity one, T acts on it as multiplication by a♯(p). On the other hand, we have the
p p
relations
U α∗ = pβ∗, U β∗ = β∗ T α∗.
p p p
◦ ◦ ◦ −
A proof of the analogous formulas for classical modular curves is given in [3, Lemma 2.1], and
the arguments carry over mutatis mutandis to the case of Shimuracurves attached to division
quaternion algebras. For a precise reference, see [9, p. 93].
It follows that U acts on e Ta (J ) via decomposition (12) with characteristic polynomial
p p p 1
X2 a♯(p)X +p 2 = (X a (p))2 (X p/a (p))2
− p − p · − p
(cid:0) (cid:1)
and is diagonalizable. Furthermore, the projection to the a (p)-eigenspace corresponds to the
p
ordinary projection, because the other eigenvalue p/a (p) has positive p-adic valuation. By
p
multiplicity one,theF -vector spacese (Taord(J ) F )ande (T F )canbeidentified.
p p p 1 ⊗Zp p p 1⊗OF p
Thus, as in the proof of Proposition 4.3, it follows from the Eichler–Shimura relations that
the map T F V arising from (10) takes the summand e (T F ) isomorphically
1⊗OF p → p p 1⊗OF p
onto V . Since, as observed in the proof of Proposition 4.3, the Kummer map
p
J (L ) F H1 L ,Ta (J ) F
1 1 ⊗Z p −→ 1 p 1 ⊗Zp p
(cid:0) (cid:1)
is injective, using the abovee identifications we get an injeection
(13) e J (L ) F ord ֒ H1(L ,V ).
p 1 1 Z p 1 p
⊗ −→
(cid:0) (cid:1)
If Hg is the Hodge embedding eused in the proof of Proposition 4.3, the map in (13) sends
Hg(P ) to X . Taking into account the previous description of the ordinary projection and
p p
the injectivity of (13), it follows that
p
(14) X = 0 U e Q = 0,
p p p
6 ⇐⇒ (cid:18) − a (p)(cid:19) 6
p (cid:0) (cid:1)
e
where
Q := Pσ Div X(K)
1 ∈ 1
e σ∈GaXl(L1/K) e (cid:0)e (cid:1)
maps to P underthe ordinaryprojection (recall that the character χ is trivial in this case).
p 0,p
To complete the proof we shall relate e Q, and thus X , to classical Heegner points on X .
p p 0
Let P denote the image of P under the canonical map X X . This is a Heegner point
1 1 e 1 → 1
of conductor p on the indefinite Shimura curve X considered in [2, Section 2], and lives in
1
e e
X (H ). Define also Q to be the image of Q in X , so that
1 1 1
Q := Pσ =e[L :H ]P Div X (H )
1 1 1 ∈ 1 1
σ∈GaXl(L1/K) (cid:0) (cid:1)
with
P := Pσ Div X (H ) .
1 ∈ 1 1
σ∈GaXl(H1/K) (cid:0) (cid:1)