NON-VANISHING MODULO p OF CENTRAL CRITICAL RANKIN–SELBERG
L-VALUES WITH ANTICYCLOTOMIC TWISTS
MILJAN BRAKOČEVIĆ
Abstract. We prove non-vanishing modulo p, for a prime ` 6= p, of central critical Rankin–Selberg L-values
with anticyclotomic twists of `-power conductor. The L-function is Rankin product of a cusp form and a
theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with
the case when the weight of the Hecke character is greater than that of the cusp form, so the L-value is
essentially different in nature from the one in the landmark work of Vatsal and Cornut–Vatsal on the same
theme.
1. Introduction
Studying non-vanishing of the central critical values of modular L-functions has had powerful and farreaching applications to important problems of Iwasawa theory including various proofs of the celebrated
Main Conjectures. Generalizing the method of Sinnott ([Si]) to the context of the theory of Shimura varieties,
Hida studied non-vanishing of the Hecke L-functions of CM fields in [Hi04] and [Hi07] and computed the µinvariant of the Katz p-adic L-function in [Hi10a]. In his dissertation [Sun], written under Hida’s supervision
in 2007, Hae-Sang Sun used this method to prove non-vanishing modulo (a rational prime) p of L-values
of the modular L-function associated to a level 1 Hecke newform f twisted by a product λχ of a fixed
arithmetic Hecke character λ of an imaginary quadratic field and χ varying through the family of finite order
anticyclotomic characters of `-power conductor, for a prime ` 6= p. Here a fixed λ was of ∞-type (k, 0),
where k denotes the weight of f .
Following the path paved by [Hi04] and [Sun], the purpose of this paper is to extend such non-vanishing
result to the case of a Hecke newform of an arbitrary level N ≥ 1 and nebentypus ψ, and arithmetic Hecke
characters λ of ∞-type (k + m, −m) for arbitrary m ≥ 0. The moral of the latter is that, after fixing a
Hecke character λ0 of ∞-type (k, 0) satisfying certain criticality condition stated below, the anticyclotomic
twists in our case are actually of the form χ0 χ, where χ0 is a fixed anticyclotomic Hecke character of ∞-type
(m, −m) and χ’s range through the family of finite order anticyclotomic characters of `-power conductor –
in other words, our fixed λ := λ0 χ0 .
The main ingredients in the proof are the Zarisky density of CM points on modular Shimura varieties
studied in [Hi04] and [Hi10a], and a recent computation of an explicit Waldspurger–type of formula in [Hi10b].
Our modest aim is to understand the passage between these deep works of quite different mathematical flavor.
To state the main theorem precisely, let M be an imaginary quadratic field of discriminant d(M ), and set
d := |d(M )|. We fix two embeddings ι∞ : Q̄ ,→ C and ιp : Q̄ ,→ Cp and write c for both complex conjugation
of C and Q̄ induced by ι∞ . Fix two rational primes p 6= ` such that p splits in M . Let G denote the algebraic
group GL(2)/Q . Let f be a normalized Hecke newform of level Γ0 (N ), N ≥ 1, weight k ≥ 1, nebentypus ψ,
and let f be its corresponding adelic form on G(Q)\G(A) with the central character ψ (see Section 2.1 for
definition). All reasonable adelic lifts of f are equal up to twists by a power of the everywhere unramified
character |det(g)|A , and f u (g) := f |ψ(det(g))|−1/2 is a unique one which generates a unitary automorphic
representation πf . We further take the base-change π̂f to ResM/Q G. Pick an arithmetic Hecke character λ
of M × \MA× of ∞-type (k + m, −m), for arbitrary m ≥ 0, and such that condition λ|A× = ψ −1 holds. Write
λ− := (λ ◦ c)/|λ| for the unitary projection. Under this condition, the L-value L(1/2, π̂f ⊗ λ− ), regarded
Date: April 26, 2012.
2010 Mathematics Subject Classification. Primary 11F67; Secondary 11G18.
Key words and phrases. Non-vanishing; Rankin–Selberg L-values; CM points; Modular curves.
This research work was partially supported by Prof. Haruzo Hida’s NSF grant DMS-0753991 through graduate student research
fellowship and by UCLA Dissertation Year Fellowship.
1
as that of the Rankin–Selberg L-function associated to f and the theta series θ(λ− ) of λ− , is critical in the
sense of Deligne
and central with respect to the functional equation.
Q
If N = l lν(l) is the prime factorization, we choose λ as above, requiring that it has a sufficiently deep
Q
conductor with respect to N , namely Cλ := l|N lν̃(l) , for a fixed ν̃(l) ≥ ν(l). The role of the latter is optimal
and two-fold. On one hand side, Hida’s explicit Waldspurger-type of formula requires depth of the conductor
of λ at such primes, for otherwise the period integral vanishes. On the other side, sufficient ramification at
these primes in the sense of Proposition 3.8 of [JaLa] and Theorem 20.6 of [Ja] contributes to +1 sign in the
functional equation for the central critical L-value. Let W denote the ring of Witt vectors with coefficients in
the algebraic closure F̄p of the finite field of p elements Fp , regarded as a p-adically closed discrete valuation
ring inside the p-adic completion Cp of Q̄p , and let P be the prime of W over p. Set W = ι−1
p (W ) which is
a strict henselization of Z(p) = Q ∩ Zp . Let Ω∞ ∈ C× be a Néron period of a CM elliptic curve over W. We
normalize the L-value as
L(N `d) ( 12 , π̂f ⊗ λ− )
Γ(k + m)Γ(m + 1)
1
E(1/2)
,
Lalg ( , π̂f ⊗ λ− ) := G
2(k+2m)
2
π k+2m+1
Ω∞
where we write L(N `d) (s, π̂f ⊗ λ− ) for the imprimitive L-function obtained by removing Euler factors at
primes dividing N `d from the primitive one, E(1/2) is the modification Euler factor given by (5.9), and G
is essentially a product of the Gauss sums given by (5.8). Let S(N, `) be a finite set of prime divisors of
elements in
{N, ` − 1} ∪ {l − 1 : prime l | N ` is split in M } ∪ {l − 1, l + 1 : prime l | N ` is inert in M } .
Then our result states:
Theorem 1.1. Let p 6= ` be two fixed rational primes such that p splits in M . Let f be a normalized Hecke
newform of level Γ0 (N ), N ≥ 1, weight k ≥ 1, and nebentypus ψ. Suppose that (N `, d(M )) = 1, ` - N ,
and that p > 2 is outside the above finite set of primes S(N, `). If 2 is inert in M , assume 2 - N `. Fix a
Hecke character λ of M of ∞-type (k + m, −m), m ≥ 0, such that λ|A× = ψ −1 and having sufficiently deep
conductor with respect to N . Then
1
Lalg ( , π̂f ⊗ (λχ)− ) 6≡ 0 (mod P)
2
for all but finitely many (finite order) anticyclotomic characters χ of `-power conductor.
The above finite set of primes p is excluded from consists of the primes which divide the fudge factors in
Hida’s Waldspurger-type of formula we use. The proof is actually valid under a condition milder than ` - N
(see Assumption 7.2). We also prove validity of the Theorem 1.1 when `kN and the local component πf ,` is
a special representation (see Proposition 7.3), but we hope to treat the general case when ` | N in a future
paper.
−
The L-value in the Theorem 1.1 is actually L( 21 , π̂f ⊗ (λχ| · |m
MA ) ), however the norm character has
trivial unitary projection. The weight k + 2m + 1 of the theta series θ(λχ| · |m
MA ) associated to the Hecke
character λχ| · |m
MA being strictly greater than the weight k of the cusp form f , incites essentially different
arithmetic nature from the L-value studied in the landmark work of Vatsal started in deep and beautiful
papers [Va02] and [Va03], and continued in joint work with Cornut [CoVa], where the comparison of the
weights is opposite. Thus, the period for the L-value there depends on f only, namely, it is Hida’s canonical
period from [Hi88], as opposed to a power of a CM period attached to M . The canonical Selmer group
and the Main Conjecture associated to our L-values are distinct. We note that, as an application of Hida’s
generalization of the Sinnott’s method in [Hi04], our proof fits in the Vatsal’s philosophy on how ergodic
rigidity principles underly the non-vanishing of L-values, as explained in his ICM 2006 report [Va06]. The
general principle in question here is the Hecke orbit principle due to Chai and related to the conjecture of
André and Oort ([Ch], see Section 6 for details).
In a recent preprint [Hs], the above result is studied over a totally real field setting, however under different
hypotheses and using an ostensibly different Waldspurger-type of formula. In particular, we do not impose
any restriction on the level N of f at the inert primes.
2
Acknowledgment. This paper is a part of my PhD thesis under Prof. Haruzo Hida at UCLA. I would like
to express my gratitude to Prof. Haruzo Hida for his generous insight, support and guidance.
Contents
1. Introduction
Acknowledgment
2. Modular forms
2.1. Classical modular forms and adelic ones
2.2. Algebro-geometric modular forms
2.3. p-adic modular forms
2.4. Elliptic curves with complex multiplication
2.5. Tower of modular curves
3. Rankin–Selberg L-functions and their special values
4. Hecke relations among CM points on the tower of modular curves
4.1. Outline of the construction of CM points
4.2. Construction of CM points when N = c = 1
4.3. Case when ` = l̄l is split in M
4.4. Case when ` is inert in M
4.5. Construction of CM points for general N and c
4.6. Summary of the construction of CM points
4.7. Isogeny action on modular forms
4.8. Differential operators
5. Hida’s explicit Waldspurger-type of formula
6. Zariski density of CM points on tower of modular curves
7. Non-vanishing of L-values modulo p
References
1
3
3
3
3
4
5
6
7
8
9
9
11
12
13
14
16
17
17
21
22
28
2. Modular forms
2.1. Classical modular forms and adelic ones. We denote by Sk (Γ0 (N ), ψ) the space of holomorphic
cusp forms
f of level Γ0 (N ) and nebentypus ψ with f (γ(z)) = ψ(γ)f (z)j(γ, z)k for γ ∈ Γ0 (N ), where
a
b
j( c d , z) = cz + d for z ∈ H = {z | Im(z) > 0} and ac db ∈ G(R). Here a Dirichlet character ψ
b 0 (N ) via a b 7→ ψ(d). By virtue of the strong approximation theorem
is regarded as a character of Γ
c d
b 0 (N )GL+ (R), where GL+ (R) = {g ∈ G(R) | det(g) > 0}, allowing us to lift f to f :
G(A) = G(Q)Γ
2
2
b 0 (N ) and g∞ ∈ GL+ (R). Note
G(Q)\G(A) → C by f (αug∞ ) = f (g∞ (i))ψ(u)j(g∞ , i)−k for α ∈ G(Q), u ∈ Γ
2
b 0 (N ), and that f is so-called arithmetic lift of f – the often
that f (αgu) = ψ(u)f (g) for α ∈ G(Q) and u ∈ Γ
b 0 (N ), ψ) the space
used automorphic lift involves a determinant factor which we omitted. Denoting by Sk (Γ
b 0 (N ), ψ)
of adelic cusp forms f obtained from f ∈ Sk (Γ0 (N ), ψ) in this way, we have Sk (Γ0 (N ), ψ) ∼
= Sk (Γ
−k
via f ↔ f . Note that the center Z(A) acts on Sk (Γ0 (N ), ψ) via f |ζ(g) = f (ζg) and that f |ζ∞ = ζ∞
f for
ζ ∈ Z(A). Thus Sk (Γ0 (N ), ψ) decomposes into a direct sum of eigenspaces for this action and on each
b 0 (N ) ∩ Z(A) is ψ and which sends ζ∞ to
eigenspace Z(A) acts by a Hecke character whose restriction to Γ
−k
ζ∞
. If we lift ψ to A× in the standard way and set ψ := ψ| · |−k
A , denote the ψ-eigenspace by Sk (N, ψ),
then Sk (Γ0 (N ), ψ) ∼
S
(N,
ψ)
via
f
↔
f
.
= k
2.2. Algebro-geometric modular forms. Let N be a positive integer, B a fixed base Z[ N1 ]-algebra and
S a B-scheme. An elliptic curve E over S is a proper smooth morphism π : E → S whose geometric fibers
are connected curves of genus 1, together with a section 0 : S → E. By a level Γ1 (N )-structure, we refer to
an embedding of finite flat group schemes iN : µN ,→ E[N ], where E[N ] is a scheme-theoretic kernel of the
multiplication by N map – it is a finite flat abelian group scheme over S of rank N 2 .
3
The modular curve M(Γ1 (N )) of level Γ1 (N ) classifies pairs (E, iN )/S , for a B-scheme S. In other words,
M(Γ1 (N )) is a coarse moduli scheme of the following functor from the category of B-schemes to the category
SETS
P(S) = [(E, iN )/S ]/∼
=,
where [ ]/∼
= denotes the set of isomorphism classes of the objects inside the brackets. When N > 3, it is a
fine moduli scheme of this functor.
Let ω denote a basis of π∗ (ΩE/S ), that is a nowhere vanishing section of ΩE/S . Fix a positive integer k
and a continuous character ψ : (Z/N Z)× → B × . Denote by ζN the canonical generator of µN . A B-integral
holomorphic modular form of weight k, level Γ0 (N ) and nebentypus ψ is a function of isomorphism classes
of (E, iN , ω)/A , defined over a B-algebra A, satisfying the following conditions:
(G0) f ((E, iN , ω)/A ) ∈ A if (E, iN , ω) is defined over A;
(G1) If % : A → A0 is a morphism of B-algebras, then f ((E, iN , ω)/A ⊗B A0 ) = %(f ((E, iN , ω)/A ));
(G2) f ((E, iN , aω)/A ) = a−k f ((E, iN , ω)/A ) for a ∈ A× = Gm (A);
(G3) f ((E, iN ◦ b, ω)/A ) = ψ(b)f ((E, iN , ω)/A ) for b ∈ (Z/N Z)× , where b acts on iN by the canonical
action of Z/N Z on the finite flat group scheme µN ;
(G4) For the Tate curve T ate(q N ) over B ⊗Z Z((q)) viewed as algebraization of the formal quotient
b m , the canonical level Γ1 (N )-structure
b m /q N Z , its canonical differential ω can deduced from dt on G
G
T ate
t
can
b m , and all α ∈ Aut(T ate(q N )[N ]) ∼
iT ate,N coming from the canonical image of the point ζN from G
=
G(Z/N Z), we have
can
f ((T ate(q N ), α ◦ ican
T ate,N , ωT ate )) ∈ B ⊗Z Z[[q]] .
The space of B-integral holomorphic modular forms of weight k, level Γ0 (N ) and nebentypus ψ is a B-module
of finite type and we denote it by Gk (N, ψ; B).
As it is well known, over C, the category of test objects (E, iN , ω) is equivalent to the category of the
pairs (L, i), where L is a Z-lattice in C and i : Z/N Z ,→ N1 L/L. The differential ω can be recovered by
pulling back du to E(C) = C/L, for the standard variable u on C. Conversely,
Z
LE =
ω ∈ C γ ∈ H1 (E(C), Z)
γ
is a lattice in C. Then an algebro-geometric modular form f integral over C gives rise to a classical modular
form, whence also to an adelic one, via
f (z) := f (Lz , i, 2πidu) ,
where Lz = Z + Zz and i : 1 7→ 1/N .
2.3. p-adic modular forms. Fix a prime number p that does not divide N . Let B be a p-adic algebra,
that is, an algebra complete and separated in its p-adic topology. For an elliptic curve E/S we consider the
Barsotti–Tate group E[p∞ ] = limn E[pn ], for finite flat group schemes E[pn ] equipped with closed immersions
−→
E[pn ] ,→ E[pm ] for m > n, and the multiplication [pm−n ] : E[pm ] → E[pn ] which is an epimorphism in the
category of finite flat group schemes. We consider a morphism of ind-group schemes ip : µp∞ ,→ E[p∞ ]
bm ∼
b called trivialization of E. Here E
b is the formal
which induces isomorphism of formal groups îp : G
=E
completion of E along its zero-section.
A holomorphic p-adic modular form over B is a function of isomorphism classes of (E, iN , ip )/A , defined
over a p-adic B-algebra A, satisfying the following conditions:
(P0) f ((E, iN , ip )/A ) ∈ A if (E, iN , ip ) is defined over A;
(P1) If % : A → A0 is a p-adically continuous morphism of B-algebras, then f ((E, iN , ip )/A ⊗B A0 ) =
%(f ((E, iN , ip )/A ));
\ which is a p-adic completion of B((q)), the canonical
(P2) For the Tate curve T ate(q N ) over B((q)),
∞
can
×
p -structure iT ate,p , the canonical level Γ1 (N )-structure ican
T ate,N , all p-adic units z ∈ Zp and all
N
∼
α ∈ Aut(T ate(q )[N ]) = G(Z/N Z), we have
can
f ((T ate(q N ), α ◦ ican
T ate,N , z ◦ iT ate,p )) ∈ B[[q]] .
4
We denote the space of p-adic holomorphic modular forms over B by V (N ; B). The fundamental q-expansion
principle holds for both algebro-geometric and p-adic modular forms (see Section 3.2 of [Br11] for the
statement under our notation).
b
Using the trivialization îp , we can push forward the canonical differential dt
t on Gm to obtain an invariant
dt
b
differential ωp := îp,∗ ( t ) on E which then extends to an invariant differential on E. Thus for f ∈ Gk (N, ψ; B)
we can define
f ((E, iN , ip )) := f ((E, iN , ωp )) ,
and we may regard an algebro-geometric holomorphic modular form as a p-adic one. By virtue of the
q-expansion principle, Gk (N, ψ; B) ,→ V (N ; B) is an injection preserving the q-expansions.
2.4. Elliptic curves with complex multiplication. It is well known that each Z-lattice a in M is actually
a proper ideal of the Z-order R(a) := {α ∈ R | αa ⊂ a} of M . On the other hand, every Z-order O of M
is of the form O = Z + cR for a rational integer c called conductor, and the following are equivalent (see
Proposition 4.11 and (5.4.2) in [IAT] and Theorem 11.3 of [CRT])
(1) a is O-projective fractional ideal
(2) a is locally principal, i.e. the localization at each prime is principal
(3) a is a proper O-ideal, i.e. O = R(a).
Thus, one can define the class group Cl−
M (O) := Pic(O) to be the group of O-projective fractional ideals
modulo the globally principal ideals. It is a finite group called the ring class group of conductor c, where c
denotes the conductor of O.
In this paper we are concerned with orders Rc`n := Z + c`n R and their ring class groups Cl−
n := Pic(Rc`n )
when n ≥ 0, where c is a fixed choice of an integer prime to ` that will always be clear from the context.
n
By the class field theory, Cl−
n is the Galois group Gal(Hc`n /M ) of the ring class field Hc`n of conductor c` .
−
The adelic interpretation of Cln is given by
/
.
(∞)
×
b× n
Cl−
(MA )× (A(∞) )× R
n =M
c`
b We define the anticyclotomic class group modulo c`∞ , Cl− := lim Cl− for the
bc`n = Rc`n ⊗Z Z.
where R
∞
←−n n
−
−
n
is
isomorphic
to
the
Galois group
taking
a
to
aR
.
Then
the
group
Cl
→
Cl
projection πm+n,n : Cl−
c`
∞
n
m+n
S
of the maximal ring class field Hc`∞ = n Hc`n of conductor c`∞ of M .
If a Z-lattice a ⊂ M has p-adic completion ap = a ⊗Z Zp identical to R ⊗Z Zp , we consider a complex
torus X(a)(C) = C/a. By the main theorem of complex multiplication ([ACM] 18.6), this complex torus
is algebraizable to an elliptic curve having complex multiplication by M and defined over a number field.
Then applying the Serre–Tate’s criterion of good reduction ([SeTa]) we can conclude that X(a) is actually
defined over the field of fractions K of W and extends to an elliptic curve over W still denoted by X(a)/W .
All endomorphisms of X(a)/W are defined over W and its special fiber X(a)/F̄p = X(a)/W ⊗ F̄p is ordinary
by our assumption that p = pp̄ splits in M .
b
b
Let T (X(a)) = limN X(a)[N ](Q̄) be the Tate module of X(a). Choice of a Z-basis
(w1 , w2 ) of b
a = a ⊗Z Z
←−
xw1 +yw2
2 ∼
gives rise to a level N -structure ηN (a) : (Z/N Z) = X(a)[N ] given by ηN (a)(x, y) =
∈ X(a)[N ].
N
After taking their inverse limit and tensoring with A(∞) , we get a level structure
η(a) = lim ηN (a) : (A(∞) )2 ∼
= T (X(a)) ⊗Zb A(∞) =: V (X(a)) .
←−
N
We can remove the p-part of η(a) and define a level structure η (p) (a) that conveys information about all
prime-to-p torsion in X(a):
η (p) (a) : (A(p∞) )2 ∼
= T (X(a)) ⊗Zb A(p∞) =: V (p) (X(a)) .
Prime-to-p torsion in X(a)/W is unramified at p, and X(a)[N ] for p - N is étale whence constant over W, so
the level structure η (p) (a) is still defined over W ([ACM] 21.1 and [SeTa]).
Since X(a)/W has ordinary reduction over W, identifying µp∞ with the connected component X(a)[p∞ ]◦ ∼
=
∞
ord
∞
∞
X(a)[p ], we obtain the ordinary part of level structure at p, namely ηp (a) : µp ,→ X(a)[p ]. The Cartier
5
duality then yields the étale part of level structure at p, namely ηpét (a) : Qp /Zp ∼
= X(a)[p∞ ]ét ∼
= X(a)[p̄∞ ]
over W. In this way we get a triple
X(a), η (p) (a), ηpét (a) × ηpord (a)
.
/W
2.5. Tower of modular curves. For the affine algebraic group G = GL(2)/Q let S = ResC/R Gm and
denote by h0 : S → G/R the homomorphism of real algebraic groups sending a + bi to the matrix ab −b
.
a
The symmetric domain X for G(R) can be identified with conjugacy class of h0 under G(R) and is isomorphic
to the union H t Hc of complex upper and lower half planes via g ◦ h0 7→ g ◦ i. Here the left actions of G(R)
a b
on X and H t Hc are by conjugation and z 7→ az+b
cz+d , for g = c d , respectively. The pair (G, X) satisfies
Deligne’s axioms for having its Shimura curve Sh ([De71] and [De79] 2.1.1), which in this case is nothing
but a tower of classical modular curves. Sh was first constructed by Shimura in [Sh66] but reinterpreted by
Deligne in [De71] 4.16-4.22 as a moduli of elliptic curves up to isogenies. More precisely, Deligne realized
Sh as a quasi-projective smooth Q-scheme representing the moduli functor F Q from the category of abelian
Q-schemes to SETS :
F Q (S) = {(E, η)/S }/≈ ,
where η : (A(∞) )2 ∼
= T (E) ⊗Zb A(∞) =: V (E) is a Z-linear isomorphism and two pairs (E, η)/S and (E 0 , η 0 )/S
0
are isomorphic up to an isogeny, which we write (E, η)/S ≈ (E 0 , η 0 )/S , if there exists an isogeny φ : E/S → E/S
0
such that φ ◦ η = η .
Let S be a Z(p) -scheme. The pairs (X, η (p) )/S consisting of an elliptic curve X over S and a Z-linear
isomorphism η (p) : (A(p∞) )2 ∼
= T (X) ⊗Zb A(p∞) =: V (p) (X), are classified up to isogenies of degree prime to
(p)
p by a Z(p) -scheme Sh/Z(p) which is a projective limit of smooth Z(p) -schemes (the latter being conctructed
in [Ko]). By its construction, Sh(p) represents a moduli functor F (p) from the category of Z(p) -schemes to
SETS
F (p) (S) = {(X, η (p) )/S }/≈ ,
and two pairs (X, η (p) )/S and (X 0 , η 0(p) )/S are isomorphic up to a prime-to-p isogeny, which we write
0
of degree prime to p such that
(X, η (p) )/S ≈ (X 0 , η 0(p) )/S , if there exists an isogeny φ : X/S → X/S
0
φ◦η =η .
The pair x(a) = X(a), η (p) (a) /W constructed as above, for a Z-lattice a with a ⊗Z Zp = R ⊗Z Zp , gives
rise to a W-point on Sh(p) to which we refer as a CM point.
Each adele g ∈ G(A(∞) ) acts on a level structure η (p) by η (p) 7→ η ◦ g (p∞) inducing G(A(∞) )-action on
Sh(p) . A sheaf theoretic coset η̄ (p) = η (p) K, for an open compact subgroup K ⊂ G(A(∞) ) maximal at p (i.e.
(p)
Kp = G(Zp )), is called a level K-structure. The quotient ShK = Sh(p) /K represents the quotient functor
FK (S) = {(E, η̄ (p) )/S }/≈ ,
(p)
and Sh(p) = limK ShK when K = G(Zp ) × K (p) and K (p) running over open compact subgroups of
←−
(p)
G(A(p∞) ). Let VK/Q̄ denote the geometrically irreducible component of ShK ⊗Q Q̄ containing a geometric
point x(a) = x(a) ⊗Q Q̄. The field of definition of VK in the sense of Weil is contained in the field of fractions
(p)
K of W, and we can think of the schematic closure VK/W in ShK/W that is smooth over W if K (p) is
sufficiently small. The special fiber VK ⊗W F̄p remains irreducible (see [Hi04] Section 2.2 for details). We
define
V (p) := lim VK/W ,
←−
K
where K ranges over all open compact subgroups of G(A(∞) ) maximal at p. In conclusion, the scheme V (p)
is smooth over W, and its generic and special fibers are geometrically irreducible. In this sense, we refer to
(p)
V/W as a geometrically connected component of Sh(p) over W containing a W-point x(a).
b ) for p - N ([PAF] Section 4.2.1), ShK is isomorphic to a, fine (N > 3) or
When K is chosen to be Γ(N
coarse, moduli scheme M(Γ(N ))/Z[1/N ] of level Γ(N ) (the principal congruence subgroup) representing the
6
following functor from the category of Z[1/N ]-schemes to the category SETS
∼ E[N ])/S ]/∼
PΓ(N ) (S) = [(E, φN : (Z/N Z) =
=,
and consequently
(p)
M(Γ(N ))/W .
Sh/W ∼
= lim
←−
p-N
Clearly,
M(Γ(N ))/Z[1/N,µN ] =
G
M(Γ(N ), ζ) ,
ζ
where the union is taken over all primitive N -th roots of unity ζ and M(Γ(N ), ζ) is a modular curve of level
Γ(N ) representing the following functor from the category of Z[1/N, µN ]-schemes to the category SETS
PΓ(N ),ζ (S) = [(E, φN : (Z/N Z) ∼
= E[N ])/S | hφN (1, 0), φN (0, 1)i = ζ]/∼
=,
with h·, ·i denoting the Weil pairing, and with M(Γ(N ), ζ)(C) = Γ(N )\H. Thus, for some generator ζ of µN ,
(p)
M(Γ(N ), ζ)/W .
V/W ∼
= lim
←−
p-N
The complex points of the modular curve Sh have the following expression
/
.
Sh(C) = G(Q)
X × G(A(∞) )
Z(Q) ,
where Z denotes the center of G and the action is given by γ(z, g)u = (γ(z), γgu) for γ ∈ G(Q) and u ∈ Z(Q)
([De79] Proposition 2.1.10 and [Mi] page 324 and Lemma 10.1).
Write G(R)+ for the identity connected component of G(R), and R×
+ for the identity connected component
(p)
×
of the multiplicative group R . The automorphisms of V/F̄p are given by
(2.1)
(p)
Aut(V/F̄p ) =
×
{x ∈ G(A(p) × Zp ) | det(x) ∈ Z×
(p) R+ }
G(Zp )G(R)+ Z(Z(p) )
(this is (2.4) of [Hi04]; see also proof of Proposition 2.8 of [Hi04] and Theorem 4.17 of [PAF]).
b = η −1 (T (X)) ⊂ (A(∞) )2 and the level structure η
To each point (X, η) ∈ Sh we can associate a lattice L
b In the view of a basis w, the G(A(∞) )-action
b
is determined by a choice of a basis w = (w1 , w2 ) of L over Z.
on Sh given by (X, η) 7→ (X, η ◦ g) is a matrix multiplication w> 7→ g −1 w> because (η ◦ g)−1 (T (X)) =
b where > stands for the transpose. We warn the reader about the following
g −1 η −1 (T (X)) = g −1 L,
Remark 2.1. Insisting on modular point of view, the action of matrix g −1 records change of the basis vectors
themselves, rather than coordinates with respect to the basis. Having this on mind and desiring to view
modular forms in adelic, algebro-geometric and p-adic phrasing in a coherent way, it becomes more convenient
for us to use the identifications R ⊗Z Zp = Rp̄ ⊕ Rp and X(R)[pn ] = X(R)[p̄n ] ⊕ X(R)[pn ] = Z/pn Z ⊕ µpn ,
n ≥ 1, in constructing level structures for our CM points due to the definition of nebentypus.
3. Rankin–Selberg L-functions and their special values
Let f be a normalized Hecke eigen-cusp form of level Γ0 (N ), N ≥ 1, weight k ≥ 1, and nebentypus ψ
and let f be its corresponding adelic form on G(Q)\G(A) with the central character ψ. The unitarization
f u (g) := f |ψ(det(g))|−1/2 of f generates a unitary automorphic representation πf . Let M be an imaginary
quadratic field of discriminant d(M ). Let χ be an arithmetic Hecke character of M × \MA× and set χ− :=
(χ ◦ c)/|χ| for its unitary projection. We denote by L(s, πf ⊗ πχ− ) the Rankin–Selberg L-function associated
to πf and the automorphic unitary representation πχ− of G attached to χ− by theta series (see [JaLa] and
[Ja] Section 19 for definitions). It is initially defined as a product of Euler factors over all places of Q and
has a meromorphic continuation to C satisfying the functional equation
L(s, πf ⊗ πχ− ) = (s, πf ⊗ πχ− )L(1 − s, π̃f ⊗ π(χ− )−1 ) ,
where π̃f denotes the contragredient of πf and (s, πf ⊗ πχ− ) is a certain -factor. Under the key condition
χ|A× = ψ −1
7
we know that L(s, πf ⊗ πχ− ) is entire and equal to L(1 − s, π̃f ⊗ π(χ− )−1 ), whence the functional equation
becomes
L(s, πf ⊗ πχ− ) = (s, πf ⊗ πχ− )L(1 − s, πf ⊗ πχ− ) .
Thus the parity of order of vanishing of L(s, πf ⊗ πχ− ) at the central critical point s = 1/2 is determined by
the value of the sign
(πf ⊗ πχ− ) := (1/2, πf ⊗ πχ− ) ∈ {±1} .
The order of vanishing is expected to be minimal for most characters χ, in other words, either L(1/2, πf ⊗πχ− )
or L0 (1/2, πf ⊗ πχ− ) should be nonzero, depending whether the sign is +1 or −1, respectively.
The global sign (πf ⊗ πχ− ) is a product over all places v of Q of the local signs (πf ,v ⊗ πχ− ,v ) which are
attached to the local components of πf and πχ− , normalized as in [Gr]. If η is the quadratic Hecke character
of Q attached to M and we set
S(χ) := {v | (πf ,v ⊗ πχ− ,v ) 6= ηv ψ v (−1)} ,
Q
then the product formula ηψ(−1) = 1 = v ηv ψ v (−1) implies
(πf ⊗ πχ− ) = (−1)#S(χ) .
In this paper we always work with Hecke characters χ of conductor not only supported at N ` but rather
.
sufficiently deep there, as well as having the infinity type (k +2m, 0) for some m ≥ 0, so that χ∞ (a) = ak+2m
∞
These conditions leave the set S(χ) empty. Indeed, the local -factors at places v outside N `d(M ) are equal
to 1, as both πf ,v and πχ− ,v are unramified principal series, and consequently these places do not belong to
S(χ). At places v | N `, following Section 1.1 of [CoVa], we use a combination of Proposition 3.8 of [JaLa]
and Theorem 20.6 of [Ja], so that once we impose that χ is sufficiently ramified at these v, none of them
belongs to S(χ). The only finite places that remain are such that v | d(M ) but v - N . Note that here πf ,v
is unramified principal series and ψ v is unramified, so in this situation we may use local calculations (3.1.1)
and (3.1.2) in [Zh] to conclude that these places do not belong to S(χ). Finally, as an infinite order character
χ has infinity type (k + 2m, 0), we have (πf ,∞ ⊗ πχ− ,∞ ) = (−1)k essentially by the Tate’s thesis ([Ta]) and
the archimedean place does not belong to S(χ). In conclusion, the global -factor is 1.
Remark. The proof of the main Theorem 1.1 is completely independent of this discussion. All the assumptions on the conductor of considered Hecke characters are imposed by the Waldspurger-type of formula we
are using (see the statement of the main Theorem 4.1 in [Hi10b]) – that the global -factor is 1 is essentially
proven in the course of computing the formula itself.
We may take the base-change lift π̂f of πf to ResM/Q G and consider the modular L-function L(s, π̂f ⊗ χ− )
due to the automorphic induction identity L(s, π̂f ⊗ χ− ) = L(s, πf ⊗ πχ− ).
The rationality of the central critical L-values L(1/2, π̂f ⊗ χ− ), after being divided by suitable periods,
is proved by Shimura in [Sh76], and is consistent with the conjectures of Deligne. The nature of a period
depends on the ∞-type of χ, or more precisely, its comparison to the weight k of f . When χ is of ∞-type
(κ, 0) and κ ≤ k − 2, the period depends on f but is independent of χ, and can be expressed in terms of
the Shimura periods u± (f ). On the other hand, if κ ≥ k, the period is independent of f and depends on M
only – it is a power of a CM period attached to M . The central critical L-values studied in this paper are
of the latter kind.
4. Hecke relations among CM points on the tower of modular curves
We fix once and for all a positive integer N such that (N, p`d(M )) = 1 as well as a normalized Hecke
newform f0 ∈ Sk (Γ0 (N ), ψ) of conductor N and nebentypus ψ. Let f0 ∈ Sk (N, ψ) be the corresponding adelic
form with the central character ψ, and πf0 the automorphic representation generated by the unitarization
f0u . We
Q work under the
Q assumption (N `, d(M )) = 1 throughout the paper. For the prime factorization
N = l lν(l) , let c := l|N lν̃(l) where ν̃(l) ≥ ν(l) is an arbitrary fixed integer. Throughout the paper we
−
work with Cl−
n := Pic(Rc`n ), n ≥ 0. We write a for a proper Rc`n -ideal, and [a] ∈ Cln for its proper ideal
class.
In this section we associate to each proper Rc`n -ideal a prime to p, n ≥ 0, two triples
x(a) = X(a), η (p) (a), ηpét (a) × ηpord (a)
and x̃(a) = X(a), η̃ (p) (a), η̃pét (a) × η̃pord (a)
/W
/W
8
giving rise to points x(a) and x̃(a) on Sh, as well as the corresponding points in Sh(p) (W) after removing
the p-part of the level structure. We refer to these points by CM points on Sh.
4.1. Outline of the construction of CM points. For the sake of clarity, we first give an overview of
the key steps in our construction, and provide the reasoning underlying them. The construction of the CM
points x(a) and x̃(a) goes in parallel, in the following sense. We start from fixing once and for all a single
triple x(R) = x̃(R). We will choose once and for all a complete set of representatives {A1 , . . . , AH − } for Cl−
0,
0
and we assume A1 = Rc . Assume for the moment that the construction is done for n = 0, i.e. assume that
for any proper Rc -ideal A the corresponding points x(A) and x̃(A) are already constructed. When n ≥ 1,
for a proper Rc`n -ideal a prime to p, we can write aRc = βAj for some β ∈ M × and 1 ≤ j ≤ H0− . Then we
×
(∞)
choose a “global” g(a) ∈ G(Q) such that det(g(a)) ∈ Z×
) for the
(p) R+ and, writing g(a)` ∈ G(Q` ) ⊂ G(A
“local” part of g(a) concentrated at `, we set
x(a) = x(βAj ) ◦ g(a)−1
and x̃(a) = x̃(βAj ) ◦ g(a)−1 .
`
That being said, for n = 0, when constructing the points x(A) and x̃(A) corresponding to a proper Rc -ideal A,
we will also use such “local” and “global” choices, respectively, regarding the primes l | c. Here g(a) is chosen
independently of the ideals in the proper Rc`n -ideal class of a. In practice, this means that when we get a
point x(a) on Sh corresponding to a certain individual proper Rc`n -ideal a, we can retrieve g(a)` ∈ G(Q` )
from there, and then use it to define points corresponding to every single other proper Rc`n -ideal belonging
to its proper ideal class [a] ∈ Cl−
n.
Two most important ingredients in the proof of the Main theorem 1.1 are the Hecke relation among the
CM points on modular curves and the Zariski-density argument of Hida, and we use the CM points x(a)
for former, while CM points x̃(a) for later purpose. To utilize the Hecke relations among the CM points in
question, it is eventually more convenient to work with points x(a) that are related by the Hecke actions
by the “local” matrices (as the computation of the relevant q-expansions in the proof of the Main Theorem
1.1 will be simpler). However, these Hecke actions by the “local” matrices typically do not preserve the
connected component of Sh containing a fixed CM point. Combined with the fact that the Zariski-density
argument of Hida is dealing with the full tower Sh as it uses a full-scale Serre–Tate deformation theory,
this is precisely the reason we need to construct x̃(a) by bringing into play a “global” g(a) ∈ G(Q) such
×
that det(g(a)) ∈ Z×
(p) R+ . By (2.1), the action of such “global” g(a) will preserve a geometrically connected
component of Sh containing a fixed CM point. Two points are worth emphasizing here:
• Even though the CM points x(a) and x̃(a) are initially constructed on Sh (and Sh(p) ), hence initially
will be indexed by proper Rc`n -ideals a, when working with a modular form of level N , we ultimately
b 0 (N `n ). On Shb
descend these points to ShΓb0 (N `n ) := Sh/Γ
Γ0 (N `n ) the points will depend on the
−
proper ideal classes [a] ∈ Cln only, and will consequently be denoted by x([a]) and x̃([a]). (See
Corollary 4.2 below.)
n
b
• We will always choose g(a) ∈ G(A(∞) ) such that g(a)(`) := g(a)g(a)−1
` ∈ Γ0 (N ` ) so that we actually
have x([a]) = x̃([a]) on ShΓb0 (N `n ) . (See Corollary 4.1 below.)
Finally, we will also separately treat cases when ` = l̄l is split in M and when it is inert, as the CM points
that (implicitly) appear in Hida’s Waldspurger-type of formula are different depending on these two cases
and we need a precise match with our CM points constructed here.
4.2. Construction of CM points when N = c = 1. For the sake of clarity, we first explain our construction of CM points when N = c = 1, as all the key ideas and steps are already present in this case. The CM
points associated to proper R`n -ideals a prime to p are constructed in three stages.
To briefly outline the three stages, first, we equip X(R)/W with appropriate level structures and a fixed
choice of an invariant differential. Second, for a fractional R-ideal A such that Ap = R ⊗Z Zp , we then
induce from X(R)/W the corresponding level structures and an invariant differential on X(A)/W . Third,
when ` = l̄l is split in M , we consider the geometric quotients of X(A) by suitable rank `n subgroup schemes
C ⊂ X(A)[`n ] étale locally isomorphic to Z/`n Z ([ABV] Section 12), and explain how these give rise to
desired CM points associated to certain proper R`n -ideals a prime to p. When ` is not split in M the third
stage is similar, however we start by inducing points X(R`n )/W from X(R)/W , n ≥ 1, and then examine the
9
geometric quotients of X(R`n ) by suitable rank ` subgroup schemes C ⊂ X(R`n )[`] étale locally isomorphic
to Z/`Z.
b
b Outside of `, we take this
b = R ⊗Z Z.
A choice of z1 ∈ R such that R = Z + Zz1 induces a Z-basis
of R
b
b To specify our choice of basis w` at ` we distinguish
basis to be our choice of a Z-basis
w = (w1 , w2 ) of R.
cases when ` = l̄l is split or inert in M :
Case when ` = l̄l is split in M : We choose w` as a standard
√ basis ((1, 0), (0, 1)) of R` = Rl̄ ⊕ Rl .
Case √
when ` is inert in M : We can write R ⊗Z Z` = Z` [ d] with d ∈ Z√
` , and set w` =
√ (w1,` , w2,` ) =
(1, d). Moreover, we choose a basis of R`n ⊗Z Z` given by 10 `0n ` (1, d)> = (1, `n d) .
The basis w = (w1 , w2 ) gives rise to a level structure η(R) : (A(∞) )2 ∼
= V (X(R)) defined over W and
a point x(R) = (X(R), η(R)) on Sh as explained above. In the case when ` is inert we also get points
x(R`n ) = (X(R`n ), η(R`n )) on Sh, for all n ≥ 1, such that
x(R`n ) = x(R) ◦
1 0 −1
0 `n `
.
We start from a single x(R) = x̃(R). In the case when ` is inert, replacing the “local” matrices by their
“global” counterparts we define
−1
x̃(R`n ) = x̃(R) ◦ 10 `0n
.
Since X(R)/W has ordinary reduction over W, identifying µp∞ with the connected component X(R)[p∞ ]◦ ∼
=
X(R)[p∞ ], we obtain the ordinary part of level structure at p, namely ηpord (R) : µp∞ ,→ X(R)[p∞ ]. The
Cartier duality then yields the étale part of level structure at p, namely ηpét (R) : Qp /Zp ∼
= X(R)[p∞ ]ét ∼
=
∞
∞
X(R)[p̄ ] over W. More precisely, X(R)[p ] is multiplicative (étale locally) and X(R)[p̄∞ ] is étale over
W and we can identify X(R)[p∞ ] ∼
= X(R)[p∞ ]ét . We fix once and for all an
= X(R)[p∞ ]◦ and X(R)[p̄∞ ] ∼
0
invariant differential ω(R) on X(R)/W such that H (X(R), ΩX(R)/W ) = Wω(R).
Let A be a fractional R-ideal whose p-adic completion Ap = A ⊗Z Zp is identical to R ⊗Z Zp . We already
specified a level structure η (p) (R) : (A(p∞) )2 ∼
= V (p) (X(R)). We choose once and for all a complete set
FH0− × b× ×
×
b j = Aj R,
b
=
M Aj R M∞ . We write A
of representatives {A1 , . . . , A − } ⊂ M × of Cl−
0 so that M
H0
A
A
j=1
so that {A1 , . . . , AH − } is a complete representative set for Cl−
0 , and we assume A1 = R. Then we have
0
−
b = αAj R
b for some α ∈ M × and 1 ≤ j ≤ H , and we can define η(A) = α−1 A−1 η(R) so that we have a
A
0
j
commutative diagram
αAj
(4.1)
T (p) (X(A)) ←−−−− T (p) (X(R))
x
x


∼
∼
=η (p) (A)
=η (p) (R)
b (p)
A
αAj
←−−−−
.
b(p)
R
Since Ap = R ⊗Z Zp , X(R ∩ A) is an étale covering of both X(A) and X(R), we get ηpord (A) : µp∞ ∼
=
∞ ét
ord
ét
X(A)[p∞ ]◦ and ηpét (A) : Qp /Zp ∼
X(A)[p
]
first
by
pulling
back
η
(R)
and
η
(R)
from
X(R)
to
X(R∩A)
=
p
p
and then by push-forward from X(R ∩ A) to X(A). In this way we create a CM point
x(A) = X(A), η (p) (A), ηpét (A) × ηpord (A)
/W
on Sh associated to a fractional R-ideal A. Furthermore, ω(R) induces a differential ω(A) on X(A) first by
pulling back ω(R) from X(R) to X(R ∩ A) and then by the pull-back inverse from X(R ∩ A) to X(A). The
projection π1 : X(R ∩ A) X(A) is étale so the pull-back inverse (π1∗ )−1 : ΩX(R∩A)/W → ΩX(A)/W is an
isomorphism, whence H 0 (X(A), ΩX(A)/W ) = Wω(A).
In this setting of N = c = 1, for a proper R-ideal A the CM points x̃(A) happen to coincide with x(A), as
we start our constructions from a single point x(R) = x̃(R), so the level structure η̃(R) coincides with η(R).
In the setting of nontrivial N and c, the points x(Rc ) and x̃(Rc ), as well as x(A) and x̃(A) for a proper
Rc -ideal A, will not necessearily coincide, as we will already use “local” and “global” choices at primes l | c.
10
4.3. Case when ` = l̄l is split in M . For the moment, assume that M is neither Q(i) nor Q(exp(2πi/6))
−
n−1
which contain non-rational units. Note that then for n ≥ 1 the projection πn,0 : Cl−
(`−1)–
n → Cl0 is a “`
n
n
to–1” map. Let C ⊂ X(R)[` ] be a rank ` subgroup scheme of a finite flat group scheme X(R)[`n ] that is
étale locally isomorphic to Z/`n Z and such that C ∩ X(R)[ln ] = {0}, but also C is different from X(R)[l̄n ].
Note that X(R)[`n ] = X(R)[l̄n ]⊕X(R)[ln ] = Z/`n Z⊕µ`n over W. If ζ`n and γ`n are the canonical generators
of µ`n and Z/`n Z, respectively, then we actually consider C to be one of the `n−1 (` − 1) rank `n finite flat
n
n
subgroup schemes Cu = hζ`−u
and gcd(u, `) = 1. We shall examine the
n γ`n i of X(R)[` ], for 1 ≤ u ≤ `
geometric quotients of X(R) by such finite flat subgroup schemes C ([ABV] Section 12). If a is a lattice so
that X(R)/C = X(a), then a/R = C and a is a Z-lattice of M , because C is Z-submodule. Since `n C = 0
we have `n Ra ⊂ a, which means that a is R`n -ideal. Moreover a is not R`n−1 -submodule so we conclude that
a is a proper R`n -ideal. The quotient map π : X(R) X(R)/C is étale over W so we obtain level structures
η (p) (a) = π∗ η (p) (R) = π ◦ η (p) (R), ηpord (a) = π∗ ηpord (R) = π ◦ ηpord (R) and ηpét (a) = π∗ ηpét (R) = π ◦ ηpét (R), as
well as an invariant differential ω(a) = (π ∗ )−1 ω(R) on X(R)/C. Note that H 0 (X(a), ΩX(a)/W ) = Wω(a) as
π : X(R) X(a) is étale and consequently (π ∗ )−1 : ΩX(R)/W → ΩX(a)/W is an isomorphism. In this way
we created `n−1 (` − 1) CM points
x(a) = X(a), η (p) (a), ηpét (a) × ηpord (a)
/W
on Sh, equipped with an invariant differential ω(a) on X(a), where these a’s are certain representatives of
−n
R] ∈ Cl−
the exactly `n−1 (` − 1) proper R`n -ideal classes in Cl−
0.
n that project to the ideal class of [l̄
In the Section 6 of [Br11] we showed that the adelic interpretation of the above construction is given as
n
n
×
follows. If Cu = hζ`−u
n γ`n i ⊂ X(R)[` ], for u ∈ (Z/` Z) , we have
n
x(a) = x(R) ◦ `0 u1 ` if x(a) = x(R)/Cu .
Moreover, we have the following instances of Shimura’s reciprocity law (([ACM] 26.8 and [Hi10a] Section
3.2)):
• If C = X(R)[l̄n ], we have the Verschiebung map
n
x(R)/C = X(l̄−n R) = x(R) ◦ `0 10 ` ,
• If C = X(R)[ln ], we have the Frobenius map
x(R)/C = x(l−n R) = x(R) ◦
1 0
0 `n `
.
Repeating this procedure for each element of the representative set {A1 , . . . , AH − } of Cl−
0 fixed above,
0
and considering geometric quotients X(Aj )/Cu of X(Aj )/W by rank `n étale finite flat subgroup schemes
n
n
Cu = hζ`−u
and gcd(u, `) = 1, we create CM points
n γ`n i ⊂ X(Aj )[` ], where 1 ≤ u ≤ `
n
(4.2)
x(aj,u ) = X(aj,u ), η (p) (aj,u ), ηpét (aj,u ) × ηpord (aj,u )
= x(Aj ) ◦ `0 u1 `
/W
on Sh, equipped with an invariant differential ω(aj,u ) on X(aj,u ), where for every fixed 1 ≤ j ≤ H0− , these
aj,u are certain representatives of the exactly `n−1 (` − 1) proper R`n -ideal classes in Cl−
n that project to the
proper ideal class [l̄−n Aj ] ∈ Cl−
.
0
Proceeding as explained in Section 4.1 we also define
n
(4.3)
x̃(aj,u ) = X(aj,u ), η̃ (p) (aj,u ), η̃pét (aj,u ) × η̃pord (aj,u )
= x̃(Aj ) ◦ `0 u1
/W
`n u
0 1
for a “global”
and aj,u as above. Here we view u as an element of A× being nontrivial at ` and having
zeros outside `.
Finally, if b is any proper R`n -ideal prime to p then after finding its proper ideal class representative aj,u
in Cl−
there is β ∈ M × such
n , for a unique j and u as above, we proceedas outlined in Section 4.1. Namely,
n
n
`
u
`
u
that bR = βAj and we set x(b) := x(βAj ) ◦ 0 1 ` and x̃(b) := x̃(βAj ) ◦ 0 1 .
Assume now that M is either Q(i) or Q(exp(2πi/6)). As c = 1 and Cl−
0 is the usual class group,
−
−
1 n−1
1 n−1
(` − 1)–to–1” map when M = Q(i), or “ 3 `
(` − 1)–to–1” map when M =
πn,0 : Cl0 → Cln is “ 2 `
Q(exp(2πi/6)). This is due to existence of non-rational roots of unity and does not happen when c > 1.
11
As ` is a split prime ` = l̄l, note that `−1 R/R is a two-dimensional vector space over Z/`Z, and among
` + 1 one-dimensional vector subspaces of `−1 R/R, only ` − 1 come from proper R` -ideals (precisely those
which are not equal to l̄ and l). For R`2 and on, p−1 R` /R` is a two-dimensional vector space over Z/`Z and
among its ` + 1 one-dimensional vector subspaces, ` of them come from proper ideals of R`2 , because R` has
only one prime l1 above ` (so precisely those not equal to l−1
1 ).
That being said, Shimura’s reciprocity law assures us that in the above construction (4.2) each proper
R`n -ideal class is represented twice among aj,u ’s when M = Q(i), and three times when M = Q(exp(2πi/6)).
Considering (4.2) first in the setting n = 1, we extract from aj,u ’s and fix a complete set of representatives
1
×
of Cl−
1 and their corresponding set of indices U1 ⊂ (Z/`Z) in (4.2), so that |U1 | = 2 (` − 1) when M = Q(i),
1
and |U1 | = 3 (` − 1) when M = Q(exp(2πi/6)). Then in the general setting n ≥ 1 we work only with points
x(aj,u ) for which u ∈ (Z/`n Z)× projected down to (Z/`Z)× land in U1 .
4.4. Case when ` is inert in M . Assume from now that n ≥ 1. Note that the projection πn+m,n :
−
m
Cl−
n+m → Cln is “` –to–1” map. Let C ⊂ X(R`n )[`] be a rank ` subgroup scheme of the finite flat group
scheme X(R`n )[`] that is étale locally isomorphic to Z/`Z. We consider the geometric quotients of X(R`n )
by such finite flat subgroup schemes C ([ABV] Section 12). The quotient map π : X(R`n ) X(R`n )/C
is étale over W so we transfer to X(R`n )/C the level structure π∗ η (p) (R`n ) = π ◦ η (p) (R`n ), as well as the
invariant differential (π ∗ )−1 ω(R`n ).
Note that the ideal `n = `Z + `n R = `R`n−1 is a prime ideal of R`n but not a proper one – it is a proper
ideal of R`n−1 , and we have X(R`n )[`n ] = Rc`n−1 /R`n , so X(R`n )/X(R`n )[`n ] ∼
= X(Rc`n−1 ). Assume now
that C ⊂ X(Rc`n )[`] is different from X(Rc`n )[`n ]. Note that there are precisely ` such group subschemes.
If a is a lattice such that X(R`n )/C = X(a), then a/R`n = C, and a must be not only a Z-lattice of
M but rather a proper R`n+1 -ideal. Since C generates over R`n all `-torsion points of X(R`n ), we have
−
aR`n = `−1 R`n , so the proper ideal class [a] ∈ Cl−
n+1 projects down to the (identity) class [R`n ] ∈ Cln . In
conclusion, this procedure gives rise to ` CM points x(a) on Sh associated to certain representatives a of the
precisely ` proper ideal classes [a] ∈ Cl−
down to the given (identity) class [R`n ] ∈ Cl−
n+1 that project
n.
√
√
>
n
1 0
n
Since the basis of R` ⊗Z Z` is given by 0 `n ` (1, d) = (1, ` d), we find that
−1
−1
(4.4)
x(R) ◦ 10 `0n ` = x(R`n ), as well as x(R`n−1 ) ◦ 10 0` ` = x(R`n ) .
We view u ∈ Z as an element of A× being nontrivial at ` and having zeros outside `. Then, for a suitable
u ∈ Z we have
−1
u −1
(4.5)
x(a) ◦ 0` 0` ` = x(R`n+1 ) ◦ 01 1` ` if x(a) = x(R`n )/C for C as above,
because the basis of `a` is given by
√ 1 + `n√
u d
=
`n+1 d
1 u
`
0 1
1 0
` 0 `n+1 `
√1
d
.
By replacing
the “local” matrices in (4.4) and (4.5) with their “global” counterparts, and noting that
` 0 ∈ Z(Q) acts trivially on Sh, we define
0 `
−1 1 u −1 n −1
0
`
(4.6)
x̃(a) = x(R) ◦ 10 `n+1
= x(R) ◦ 10 `u`
.
n+1
0 1
We consider x(a) in ShK := Sh/K for an open subgroup K ⊂ G(A(∞) ) maximal at p and containing
b
Z(Z). Note that
` 0
` 0
` 0 −1
` 0 (`) ∈ Z(Z)
b ⊂K
0 ` ∈ Z(Q) and 0 ` ◦ 0 ` ` = 0 `
imply
−1
−1
(4.7)
x(a) = x(a) ◦ 0` 0`
= x(a) ◦ 0` 0` ` on ShK .
Combining (4.5) and (4.7) we conclude
(4.8)
x(a) = x(R`n+1 ) ◦
1 −u
`
0 1
`
on ShK , if x(a) = x(R`n )/C for C as above.
By repeating (4.5), if x(a) = x(R`n )/C for C ∼
= Z/`m Z with C ∩ X(R`n )[`n ] = {0}, then a is a proper
R`n+m -ideal, and this gives us `m CM points x(a) on Sh associated to certain representatives a of the precisely
−
`m proper ideal classes [a] ∈ Cl−
n+m that project down to the given (identity) class [R`n ] ∈ Cln . Rather
12
than working with A1 = R in (4.4), by repeating (4.4) and (4.5) for any Aj from the chosen complete set of
representatives of Cl−
0 , and then proceeding as outlined in Section 4.1, we can construct the CM points x(b)
and x̃(b) on Sh corresponding to any proper R`n -ideal b prime to p.
4.5. Construction of CM points for general N and c. We decompose c = ci cs into “inert” and “split”
part. For each l | ci we repeat the procedure performed for the prime ` when ` is inert in Section 4.4, whereas
for each l | cs we repeat the procedure performed for the prime ` when ` was split in Section 4.3.
b
b
b = R ⊗Z Z.
A choice of z1 ∈ R such that R = Z + Zz1 we made in Section 4.2 has induced a Z-basis
of R
b
b
Outside of primes dividing N `, we take this basis to be our choice of Z-basis w = (w1 , w2 ) of R. We already
chose a basis w` at ` in Section 4.2, and in the same fashion we specify our choice of basis wl at l | N ,
distinguishing cases when l = l̄l is split or inert in M :
Case when l | N is split in M : We choose wl as a standard
√basis ((1, 0), (0, 1)) of Rl = Rl̄ ⊕ Rl .
Case √
when l | N is inert in M : We can write R⊗Z Zl = Zl [ d] with d ∈ Z√
l , and set wl = (w
√ 1,l , w2,l ) =
1 0
>
ν̃(l)
(1, d). Moreover, we choose a basis of Rlν̃(l) ⊗Z Zl given by 0 lν̃(l) l (1, d) = (1, l
d).
The basis w = (w1 , w2 ) gives rise to a level structure η(R) : (A(∞) )2 ∼
= V (X(R)) defined over W and a point
x(R) = (X(R), η(R)) on Sh as explained above. We have
Y
−1
1 0
x(Rci ) = x(R) ◦
,
0 lν̃(l) l
l|ci
and replacing the “local” matrices by their “global” counterparts we define
Y
−1
−1
1 0
x̃(Rci ) = x(R) ◦
.
= x(R) ◦ 10 c0i
0 lν̃(l)
l|ci
We choose a representative set {B1 , . . . , BHc− } of Pic(Rci ) and first induce from x(Rci ) the points x(B)
i
(and their corresponding x̃(B)) for every proper Rci -ideal B, as we did in Section 4.2 Q
when discussing (4.1).
We repeat the procedure behind (4.2) for each l | cs and a tuple u = (u(l))l|cs ∈ l|cs (Z/lν̃(l) Z)× , thus
creating CM points
Y ν̃(l)
l
u(l)
(4.9)
x(bj,u ) = X(bj,u ), η (p) (bj,u ), ηpét (bj,u ) × ηpord (bj,u )
= x(Bj ) ◦
l
0
1
/W
l|cs
on Sh, equipped with an invariant differential ω(bj,u ) on X(bj,u ), where for every fixed 1 ≤ j ≤ Hc−i , these
bj,u are certain representatives of the exactly ϕQ (cs ) proper Rc -ideal classes in Cl−
0 that project to the proper
Q
−ν̃(l)
ideal class [ l|cs l̄
Bj ] ∈ Pic(Rci ). Then proceeding as in (4.3) we also define
(4.10)
x̃(bj,u ) = X(bj,u ), η̃ (p) (bj,u ), η̃pét (bj,u ) × η̃pord (bj,u )
/W
= x̃(Bj ) ◦
Y
lν̃(l) u(l)
0
1
l|cs
working with “global” matrices and bj,u as above. Here we view u(l) as an element of A× being nontrivial
at l and having zeros outside l.
To take care of `, when ` is split we simply repeat the argument behind (4.2). In the case when ` is inert
we first get points x(Rc`n ) = (X(Rc`n ), η(Rc`n )), for n ≥ 1, with
x(Rc`n ) = x(Rc ) ◦
1 0 −1
0 `n `
,
and replacing the “local” matrices by their “global” counterparts we define
x̃(Rc`n ) = x̃(Rc ) ◦
Then we repeat the argument behind (4.5).
13
1 0 −1
0 `n
.
4.6. Summary of the construction of CM points. In this section we summarize the facts about CM
points we outlined in Section 4.1. We start with the following
Corollary 4.1. For every proper Rc`n -ideal a, we have x(a) = x̃(a) on ShΓb0 (N `n ) .
Proof. In case when ` is split (and the very same reasoning applies for split l | N ), the assertion follows from
(`)
n
b 0 (N `n ). In case when ` is inert (and the very same
comparing (4.2) and (4.3) and noting that `0 u1
∈Γ
reasoning applies for inert l | N ), the assertion follows from comparing (4.4) and (4.5) on one hand side, and
(`)
n (`)
b 0 (N `n ).
(4.6) on the other hand side, and noting that 10 `u`
, 0` 0`
∈Γ
n+1
Note that to any x = (Ex , ηx ) ∈ Sh one can associate a representation ρx : MA× ,→ G(A(∞) ) by
αηx = ηx ◦ ρx (α) ,
0
and when x = x ◦ g on Sh for some g ∈ G(A(∞) ), we have ρx0 = g −1 ρx g.
Recall that we fixed a choice of z1 ∈ R such that R = Z + Zz1 . We can define ρ : M ,→ M2 (Q) by a
regular representation
z
αz1
(4.11)
ρ(α) 1 =
,
1
α
and, after tensoring with A(∞) , we get ρ : MA× ,→ G(A(∞) ) associated to [z1 , 1] ∈ Sh(C). We introduce a
representation ρx(R) : MA× ,→ G(A(∞) ) by
αη(R) = η(R) ◦ ρx(R) (α) .
(4.12)
For all split l | N ` we can identify Ml = Ml̄ × Ml = Ql × Ql , and write ιl (α) = α and c ◦ ιl (α) = ᾱ, where ιl
and c ◦ ιl are the projections of Ml to Ml and Ml̄ , respectively. Note that ρx(R) = h−1
1 ρh1 for a matrix h1
that realizes
ρx(R) (α) = ᾱ0 α0 at split l | N `
(for example, setting h1,l = z̄11 z11 at the desired l will do). Moreover, at inert l | N ` we realized R ⊗Z Zl =
√
Zl [ d] and consequently
√
x y
ρx(R) (x + y d) = dy x .
We can also introduce representations ρn = ρx(Rc`n ) : MA× ,→ G(A(∞) ), for all n ≥ 0, by
αη(Rc`n ) = η(Rc`n ) ◦ ρn (α) .
(4.13)
Note that ρ = ρn on
(4.14)
(N `)×
MA
and ρn = g −1
ρg n where
n
(
n
h1,` `0 11 `
g n,` =
1 0 −1
0 `n `
if ` is split,
if ` is inert,
and
(4.15)
g n,l =
(
h1,l
lν̃(l) 1
0 1 l
−1
1 0
ν̃(l)
0l
l
for split l | N,
for inert l | N.
Here in specifying gn,` when ` is split, we assume without loss of generality that in (4.2), the proper Rc`n
n
`n u0
ideal a1,1 from x(a1,1 ) = x(Rc ) ◦ `0 11 ` , represents the identity class [Rc`n ] ∈ Cl−
n . Working with
0 1 `
instead, for some fixed u0 ∈ (Z/`n Z)× , does
not
affect
our
argument
in
Corollary
4.2
and
Proposition
5.3
n
below (for the latter we would set `0 u10 ` in (5.2)). The same could also be achieved by a suitable choice
of embedding ι` . We make the same convention at split l | N .
Recall that we fixed once and for all a complete set of representatives {A1 , . . . , AH − } ⊂ MA× of Cl−
0 so
0
FH0− × b× ×
×
that MA = j=1 M Aj Rc M∞ . For the representation ρ0 from (4.13) and for any proper Rc -ideal A, after
b = αAj R
bc for some α ∈ M × , we have
writing A
x(A) = x(Rc ) ◦ ρ0 (α−1 A−1
j ).
Let n ≥ 1. In the case when ` is split in M we examined the quotients x(Aj )/C for carefully chosen rank `n
finite flat subgroup schemes Cu and we obtained points x(aj,u ) for a certain complete set of representatives
14
aj,u of Cl−
n . Similarly, in the case when ` is inert in M , we first obtained points x(a) for a certain set of
representatives a of Cl−
n and then used this knowledge to define a CM point corresponding to any proper
Rc`n -ideal. Denote these complete set of representatives of Cl−
} ⊂ MA× , so that MA× =
n by {a1 , . . . , ah−
n
−
Fhn
×
×
b×
j=1 M aj Rc`n M∞ . For the representation ρn from (4.13) and any proper Rc`n -ideal a, n ≥ 1, after writing
bc`n for some α ∈ M × , we have
b
a = αaj R
x(a) = x(Rc`n ) ◦ ρn (α−1 a−1
j )
by our construction.
Corollary 4.2. For every proper Rc`n -ideal a, the point x(a) = x̃(a) on ShΓb0 (N `n ) depends on the proper
ideal class [a] ∈ Cl−
b0 (N `n ) by x([a]) accordingly.
n only. We denote these points on ShΓ
b× n ) ⊂ Γ
b 0 (N `n ). Note that when ` is split in M (the very same
Proof. This follows from the fact that ρn (R
c`
×
b n = {α ∈ R
b× | α ≡ ᾱ mod `n } and
reasoning applies when l | N is split) we have R
`
ρn (α) =
`n 1 −1 ρ
`n 1
x(R) (α) 0 1 `
0 1 `
=
n `n 1 −1 ᾱ 0
` 1
0 α
0 1 `
0 1 `
ᾱ−α
`n
ᾱ
0
=
α
b 0 (`n ) for α ∈ R
b×n .
∈Γ
`
Note that when ` is inert√(the very same reasoning applies when l | N is inert), using the above realization
of ρx(R) at `, for α = x + y d ∈ (R`n ⊗Z Z` )× we have `−n y ∈ Z` and
√
ρn (x + y d) =
=
√
−1
ρx(R) (x + y d) 10 `0n `
y 1 0 −1
x `−n y
b 0 (`n ) .
∈Γ
x
0 `n ` = `n dy x
−1
1 0 −1
0 `n `
x
1 0
0 `n ` dy
−
Corollary 4.3. Let n ≥ 1. For any fixed [b0 ] ∈ Cl−
n and [a0 ] ∈ Cln+1 , such that πn+1,n ([a0 ]) = [b0 ], we have
{x([a]) | [a] ∈ Cl−
n+1 and πn+1,n ([a]) = [b0 ]} = {x([a0 ]) ◦
1 u
`
0 1
`
| u ∈ Z/`Z} on ShΓb0 (N `n+1 ) .
−
More generally, for any s ≥ 1, and fixed [b0 ] ∈ Cl−
n , [a0 ] ∈ Cln+s such that πn+s,n ([a0 ]) = [b0 ], we have
(4.16)
{x([a]) | [a] ∈ Cl−
n+s and πn+s,n ([a]) = [b0 ]} = {x([a0 ]) ◦
1 `us
0 1
`
| u ∈ Z/`s Z} on ShΓb0 (N `n+s ) .
Proof. The assertion follows from (4.2) and (4.8). Note that the same was also verified in Section 3.1 of
[Hi04].
Corollary 4.4. Let m, n ≥ 1. If a is a proper Rc`m -ideal prime to p and b a proper Rc`n -ideal prime
to p such that aR and bR belong to the same ideal class in Pic(R), then x̃(a) and x̃(b) stay in the same
(p)
geometrically connected component V/F̄p containing the starting x(R) = x̃(R).
×
Proof. Note that each α ∈ R(p)
induces an isomorphism
(X(R), η (p) (R)) ∼
= (X(αR), η (p) (αR))
and the central endomorphism ρx(R) (α) then fixes x(R), whence preserves its geometrically connected com(p)
×
ponent V/F̄p . Let aR = αbR for some α ∈ R(p)
. The assertion essentially follows from (2.1) combined with
(4.3) and (4.6). Depending whether ` is split or inert in M we used (4.3) or (4.6), respectively, and at the
same time at the split primes l | N we used an analogue of (4.3), whereas at the inert primes l | N we used
an analogue of (4.6). In any case, we used the “global” matrices having determinant in Z(p) R×
+.
15
4.7. Isogeny action on modular forms. Let N and c be the integers fixed at the beginning Section 4
and let q be a prime outside N p`. Let X(a) = (x(a), ω(a)) = (X(a), η (p) (a), ηpét (a) × ηpord (a), ω(a)) be a
b 0 (q) induces
quadruple associated to a proper Rc`n -ideal a. Taking the sheaf theoretic coset η (p) (a) mod Γ
a finite group subscheme C of X(a) defined over W and isomorphic to Z/qZ étale locally. Thus, we can
construct canonically the image of X(a) under q-isogeny
[q](X(a), η (p) (a), ηpét (a) × ηpord (a), ω(a)) = (X(a)/C, π∗ η (p) , π∗ (ηpét (a) × ηpord (a)), (π ∗ )−1 ω(a))
for the projection π : X(a) X(a)/C. If q = qq̄ splits in M , then we can choose ηq to be induced by
X(a)[q ∞ ] ∼
= X(a)[q̄∞ ] ⊕ X(a)[q∞ ] ∼
= Mq̄ /Rq̄ ⊕ Mq /Rq ∼
= Qq /Zq ⊕ µq∞ ,
so that we have a level Γ0 (q)-structure C = X(a)[q] on X(a) which depends on the choice of a factor q. Then
[q](X(a)) = X(q−1 a)
(4.17)
and
[q](x(a)) = x(q−1 a) = x(a) ◦
(4.18)
10
0 q q
on Sh .
When a prime r = r2 ramifies in M (hence outside N p`), X(a) has a subgroup C = X(a)[rn ] isomorphic to
−1
), and (4.17) and (4.18)
Z/rZ for rn = r ∩ Rc`n . Thus, we can still define [r](X(a)) = X(ar−1
n ) = X(ar
continue to hold for a ramified prime.
Let N1 be a positive integer prime to p. Note that a point x = (E, η (p) ) ∈ Sh(p) (S), for a W-scheme S,
b 1 (N1 ) and the coset is taken as a
projects down to x = (E, iN1 ) ∈ M(Γ1 (N1 ))(S), where iN1 = η (p) mod Γ
(p) ét
ord
sheaf theoretic one. Thus, for a test object X = (X, η , ηp × ηp , ω), an algebro-geometric modular form
f ∈ Gk (Γ0 (N1 ), ψ1 ; W) can be evaluated at this test object via
f (X) := f (X, iN1 , ω) ,
and the same holds for a p-adic modular form f ∈ V (N1 ; W ) via
f (X) := f (X, iN1 , ηpord ) .
If a prime q - N1 we have a linear operator [q] : V (N1 ; W ) → V (qN1 ; W ) by f |[q](X) = f ([q](X)).
For a test object (L, C, i) in the lattice viewpoint, this is tantamount to choosing a lattice LC ⊃ L with
LC /L ∼
= Z/qZ and f |[q](L, C, i) = f (LC , i). Over C, from the classical point of view this is nothing but
f |[q](z) = ψ(q)q k f (qz).
For the discriminant d(M ) ∈ Z (d(M ) < 0) of M we set d0 (M ) = |d(M )|/4 if 4|d(M ), while d0 (M ) =
|d(M )| otherwise. Now we specialize to the case of our interest fixing N1 := N d0 (M ), for N fixed once and for
all at the beginning of Section 4, such that (N, p`d(M )) = 1. At the same time we fixed f0 ∈ Sk (Γ0 (N ), ψ),
and recall that f0 ∈ Sk (N, ψ) denotes the corresponding adelic form with the central character ψ, and πf0
the automorphic representation generated by the unitarization f0u . In [Hi10b], when expressing the L-value
L( 12 , π̂f0 ⊗ χ− ) as the square of a finite sum of the values of a nearly holomorphic cusp form in πf0 , the author
effectively works with a suitable normalized Hecke eigen-cusp form f (that will be explicitly made out of the
starting f0 at the end of our Section 5), such that its arithmetic lift f is inside the automorphic representation
πf0 . The Hecke eigen-cusp form f will be of level N1 := N d0 (M ), but having the same nebentypus ψ of
conductor dividing N as the starting f0 , or in the notation of Section 2.1, f ∈ Sk (N1 , ψ).
For a split prime q - N1 , by (4.18) we have f |[q] = f | 10 0q q , where the latter is defined by
f |g(X) := f (X ◦ g)
(4.19)
(∞)
for g ∈ G(A ) and f ∈ V (N1 ; W ).
For a ramified prime r | d0 (M ) such that r = r2 , we can reverse the order of steps and define f |[r] :=
f | 10 0r r . Here the assumption (N `, d(M )) = 1 becomes crucial, as it guarantees r - c`n so our construction
(p)
of CM points allows us to additionally choose ηr (without interfering with the choices we made at primes
b 0 (r) induces a finite group subscheme C = X(a)[rn ]
l | N `) so that taking a sheaf theoretic coset η (p) (a) mod Γ
n
of X(a) isomorphic to Z/rZ, where rn = r ∩ Rc` . In other words, we choose a Zr -basis (w1,r , w2,r ) of
16
(p)
(p)
R ⊗Z Zr so that C = ηr (a)(Zr w1,r + r−1 Zr w2,r )/ηr (a)(Zr ) amounts to X(a)[rn ]. Then for X(a) =
(X(a), η (p) (a), ηpét (a) × ηpord (a), ω(a)) associated to a proper Rc`n -ideal a we have
f |[r](X(a)) = f ([r](X(a))) = f (X(r−1 a)) .
We end this section by emphasizing
Remark 4.5. Note that the Corollary 4.1, Corollary 4.2 and Corollary 4.3 remain valid if we descend points
b 0 (N1 `n ) and Γ
b 0 (N `n ) have the same l-parts for l | N `,
x(a) to ShΓb0 (N1 `n ) instead of ShΓb0 (N `n ) . Since Γ
b 0 (r) at primes r | d0 (M ) and this
for the Corollary 4.2 we only need to check that ρx(R) ((R ⊗Z Zr )× ) ⊂ Γ
(p)
follows from the above choice of ηr (R). For the Corollary 4.3, note that all we worked with in (4.7) is an
b The fact that f is of level N1 := N d0 (M ),
open subgroup K ⊂ G(A(∞) ) maximal at p containing Z(Z).
but having the same nebentypus ψ of conductor dividing N as the starting f0 ∈ Sk (Γ0 (N ), ψ), makes f
b 0 (r) at primes r | d0 (M ); when being evaluated
insensitive of the Γ0 (r)-structure induced by η (p) (a) mod Γ
b 1 (r).
f only takes into account the Γ1 (r)-structure induced by η (p) (a) mod Γ
4.8. Differential operators. Recall the definition of the Maass–Shimura differential operators on H indexed
by k ∈ Z:
∂
k
1
+
and δkm = δk+2m−2 . . . δk
δk =
2πi ∂z
z − z̄
for a non-negative integer m. They preserve rationality of the value at a CM point ([AAF] III and [Sh75])
when applied to modular forms. Let a be a proper Rc`n -ideal prime to p. Note that the complex uniformization X(a)(C) = C/a induces a canonical invariant differential ω∞ (a) in ΩX(a)/C by pulling back du, where u
is the standard variable on C. Then one can define a period Ω∞ ∈ C× by ω(a) = Ω∞ ω∞ (a) ([Ka78] Lemma
5.1.45). Note that Ω∞ does not depend on a since ω(a) is induced by ω(R) on X(R) by construction. Then
for f ∈ Gk (N1 , ψ; W), for any N1 ≥ 1, we have
δkm f (x(a), ω∞ (a))
= δkm f (x(a), ω(a)) ∈ W
Ωk+2m
∞
([Ka78] Theorem 2.4.5).
Katz introduced a purely algebro-geometric definition of the Maass–Shimura differential operator ([Ka78]
Chapter II) by interpreting it in terms of the Gauss–Manin connection of the universal elliptic curve over
the modular curve M(Γ1 (N1 )). In this way he extended the operator δ∗ to algebro-geometric and p-adic
modular forms; we denote the latter extension of δ∗m by dm : V (N1 ; W ) → V (N1 ; W ). The ordinary part
[ for the p-adic formal
bm ∼
of level structure at p, ηpord (a) : µp∞ ∼
= X(a)[p∞ ] induces a trivialization G
= X(a)
[
[
completion X(a)
of X(a) along its zero-section. We obtain an invariant differential ωp (a) on X(a)
by
/W
dt
t on
/W
b m , which then extends to an invariant differential on X(a)/W also denoted by ωp (a).
pushing forward
G
Then one can define a period Ωp ∈ W × , independent of a, by ω(a) = Ωp ωp (a) ([Ka78] Lemma 5.1.47). The
fact that will be of instrumental use for us is
(dm f )(x(a), ωp (a))
= (dm f )(x(a), ω(a)) = (δkm f )(x(a), ω(a)) ∈ W
(4.20)
Ωpk+2m
([Ka78] Theorem 2.6.7). The effect of dm on the q-expansion of a modular form is given by
X
X
(4.21)
dm
a(n, f )q n =
nm a(n, f )q n
n≥0
n≥0
([Ka78] (2.6.27)).
5. Hida’s explicit Waldspurger-type of formula
Recall that for the discriminant d(M ) ∈ Z (d(M ) < 0) of M we set d0 (M ) = |d(M )|/4 if 4|d(M ), while
d0 (M ) = |d(M )| otherwise. Let N such that (N, p`d(M )) = 1 and f0 ∈ Sk (Γ0 (N ), ψ) be the ones fixed at
the beginning of Section 4 for the rest of the paper. Recall we denoted by f0 ∈ Sk (N, ψ) the corresponding
adelic form with the central character ψ. Let f be a suitable normalized Hecke eigen-cusp form that will
be explicitly made out of f0 later in this section, such that its arithmetic lift f is inside the automorphic
17
representation πf0 generated by the unitarization f0u . Since πf0 = πf , we will eventually write πf for the
automorphic representation. The Hecke eigen-cusp form f will be of level N1 := N d0 (M ), but having the
same nebentypus ψ of conductor dividing N as the starting f0 , or in the notation of Section 2.1 we have
f ∈ Sk (N1 , ψ).
m
We adelize the Maass–Shimura
m-th derivative δk f , m ≥ 0, to a function fm on G(A) as in Section 3.1 of
1 1 i
[Hi10b]. We regard X = 2 i −1 ∈ sl2 (C) – the Lie algebra of SL2 (C), as an invariant differential operator
Xg∞ on SL2 (C) for a variable matrix g∞ ∈ G(R) (here identifying G(R) with SL2 (R) × R× by the natural
isogeny). We set
m
fm (g) = (−4π)−m |det(g)|−m
A Xg∞ f (g) ,
where g∞ is the infinite part of g ∈ G(A). Then fm (g∞ ) = (δkm f )(g∞ (i))j(g∞ , i)−k−2m , and when det(g∞ ) =
1 we have
m
fm (g) = |det(g (∞) )|−m
A δk f (g) ,
(5.1)
where δkm f : G(Q)\G(A) → C is the arithmetic lift of δkm f as in Section 2.1, given by δkm f (αug∞ ) =
b 0 (N1 ) and g∞ ∈ GL+ (R) ([Hi10b] Definition 3.3 and
δkm f (g∞ (i))ψ(u)j(g∞ , i)−k−2m for α ∈ G(Q), u ∈ Γ
2
(∞)
Lemma 3.1). Here g
is the finite part of g ∈ G(A). The central character of fm is given by ψ m (x) =
b 0 (N1 ).
ψ(x)|x|−2m
and fm (gu) = ψ m (u)fm (g) when u ∈ Γ
A
×
×
Let ρ : M \MA ,→ G(Q)\G(A) be the regular representation defined by (4.11). We fix g ∈ G(A) such
that g∞ (i) = z1 and det(g∞ ) = 1, while the finite places of g will be specified shortly. We recall Lemma 3.7
of [Hi10b].
k+2m
.
Lemma 5.1. Let χm : M × \MA× → C× be a Hecke character
with χm |A× = ψ −1
m and χm (a∞ ) = a∞
−
×
×
(∞) ×
×
Then a 7→ fm (ρ(a)g)χm (a) factors through IM := M
MA (A ) M∞ (the anticyclotomic idele class
group).
×
×
so that
be the idele class group and choose a Haar measure d× a on MA× /M∞
Let IM := M × MA× M∞
R
×
×
×
of
I
,
we
get a
d
a
=
1
as
in
Section
2.1
of
[Hi10b].
Taking
a
fundamental
domain
Φ
⊂
M
/M
M
b×
∞
A
R
×
measure on IM still denoted by d a. Set
Z
fm (ρ(a)g)χm (a)d× a .
Lχm (fm ) :=
IM
In [Hi10b] Hida related Lχm (fm ) to the central critical L-value L( 21 , π̂f0 ⊗ χ−
m ) for all arithmetic Hecke
,
computing
explicitly
all
local
Euler-like
factors
without any ambiguity.
characters χm with χm |A× = ψ −1
m
Moreover this is done under optimal assumptions on the conductor of χm , one of them being a sufficient
depth at the prime Q
divisors of the conductor N of πf0 .
Recall that N = l lν(l) denotes the prime factorization and we work under the assumptions (N, p`d(M )) =
1 and (`, d(M )) = 1. In this section we assume that χm is unramified outside N and `. Moreover, we assume
that the conductor of χm satisfies the following:
• at primes l | N is equal to lν̃(l) , where ν̃(l) ≥ ν(l) is an arbitrary integer, and
• at ` is equal to `n , where n ≥ 1 is an arbitrary integer.
Q
In other words, if we denote by C the conductor of χm and set c := l|N lν̃(l) , we can write the above
assumptions as CN = c and C` = `n .
We briefly explain Hida’s recipe for a choice of g = gn at finite places (see Section 4 of [Hi10b]) adjusted to
our need. In the Section 4 of [Hi10b] one divides the set of prime factors of N (C)N d0 (M ) into disjoint union
AtC as follows (that being said, the primes dividing d0 (M ) play a role in choosing g = gn only if they divide
N (C)N as well, but this is not the case under our assumptions). Under our assumptions (N `, d(M )) = 1
and ` - N , it suffices for our purpose to set A = {l : l | N ` is split in M }. Set C = C0 t C1 , where C1 is
the set of prime factors of d0 (M ) and C0 = Ci t Cs t Cr , so that Ci = {l : l | N ` is inert in M }, Cr = {2}
if ord2 (d(M )) = 2 with ν(2) > 2 and Cr = ∅ otherwise. Then Cs = ∅ in our setting. Thus, there are two
possibilities for `: if it splits in M it is placed in the set A, otherwise it is placed in Ci .
If l | N ` splits in M we choose a prime l̄ over l in M ; we set A = {l̄ | l ∈ A}. Recall that in Section 4.6
for all l ∈ A we identified Ml = Ml̄ × Ml = Ql × Ql , and wrote ιl (α) = α and c ◦ ιl (α) = ᾱ, where ιl and
c ◦ ιl are the projections of Ml to Ml and Ml̄ , respectively. Note that these identifications follow reversed
2
18
notation from the ones in [Hi10b] due to a reason explained in Remark 2.1 at the prime p – we proceed
similarly at other split primes to keep our notation uniform. Then for l ∈ A, we specify gn,l and gn,l by
first
ᾱ 0 , and
invoking from the Section 4.6 the choice of h1,` ∈ G(Z` ) and h1,l ∈ G(Zl ) so that h−1
ρ(α)h
=
1,l
0 α
1,l
then setting:
(
n
h1,` `0 11 ` if ` is split, i.e. ` ∈ A ,
(5.2)
gn,` =
`n 0
if ` is inert, i.e. ` ∈ Ci ,
0 1 `
and
(
(5.3)
gn,l =
lν̃(l) 0
0 1 l
lν̃(l) 0
0 1 l
h1,l
for split l | N, i.e. l ∈ A\{`} ,
for inert l | N, i.e. l ∈ Ci \{`} .
Here we exclude the possibility that 2 | N ` when 2 is inert in M , since if exceptionally this is the case, Hida
gives a different choice of gn,2 for in Lemma 2.5 of [Hi10b] not suitable for our purpose. We already chose
gn,∞ ∈ G(R) (independently of n) so that gn,∞ (i) = z1 and det(gn,∞ ) = 1. We set gn,l to be the identity
matrix in G(Zl ) for l 6∈ A t C t {∞} (see the proof of Proposition 2.2 in [Hi10b]).
The above Lemma 5.1 is appropriately improved under these new assumptions:
Lemma 5.2. Let χm : M × \MA× → C× be a Hecke character such that its conductor C satisfies CN ` =
. Then a 7→ fm (ρ(a)gn )χm (a) factors through Cl−
with χm (a∞ ) = ak+2m
c`n and χm |A× = ψ −1
n =
∞
× (∞) × ×m ×
×
b
M
MA (A ) Rc`n M∞ .
Proof. Note that fm (ρ(a)gn ) = fm (gn (gn−1 ρ(a)gn )), so the assertion follows from combining the assumption
n
−1
b×
b
χm |A× = ψ −1
m with gn ρ(Rc`n )gn ∈ Γ0 (N1 ` ), which at primes l | N ` is checked in the same way as in
Corollary 4.2, whereas the reasoning for r | d0 (M ) is explained in Remark 4.5.
Thus, for any complete set of representatives a of the proper ideal classes [a] ∈ Cl−
n and their corresponding
bc`n , we immediately conclude
a ∈ MA× such that b
a = aR
(5.4)
Lχm (fm ) =
vol(IM ) X
ϕQ (c`n ) X
χm (a−1 )fm (ρ(a−1 )gn ) =
χm (a−1 )fm (ρ(a−1 )gn ) .
−
n)
2ϕ
(c`
|Cln |
M
−
−
[a]∈Cln
[a]∈Cln
Cl−
n
×
We formally write the summation over proper ideal classes [a]R ∈
as each summand is independent of the
choice of a proper ideal class representative. Here vol(IM ) = IM d a = h(M )/|R× |, where h(M ) is the class
R
number of M , because under the chosen normalization of the Haar measure we have Rb× /R× d× a = 1/|R× |.
Additionally,
2h(M )ϕM (c`n )
−
h−
:=
|Cl
|
=
n
n
|R× |ϕQ (c`n )
can be read off from the exact sequence
0 −→ ClQ (c`n ) −→ ClM (c`n ) −→ Cl−
n −→ 0 ,
where ClM (c`n ) and ClQ (c`n ) denote the ray class groups modulo c`n of M and Q, respectively.
In the Section 6 of [Br11], we studied the G(A(∞) )-action by gn on Sh and we examined in great detail
how the local components gn,l affect the conductor of the lattice
associated
to a CM point on Sh in the sense
of Deligne’s treatment of Sh. Recall that Sh(C) = G(Q) X × G(A(∞) ) Z(Q) and write [z, g] ∈ Sh(C)
for the image of (z, g) ∈ X × G(A(∞) ).
Let {a1 , . . . , ah−
} be a complete set of representatives of proper ideal classes in Cl−
n , such that for their
n
×
b
corresponding aj ∈ MA with b
aj = aj Rc`n one has aj,l = 1 for a finite set of primes l | N1 `. We can always
(∞)
choose such a complete set of representatives. Then ρ(a−1
j ) commutes with gn , and under the above
(∞)
assumption on aj each point [z1 , ρ(a−1
j )gn
] ∈ Sh(C) once descended to ShΓb0 (N1 `n ) will depend only on
b 0 (N1 `n ). The same holds for CM points x(aj ) (see
the proper ideal class [aj ] ∈
since
⊂Γ
Corollary 4.2 and Remark 4.5). Having this on mind we state
Cl−
n,
b× n )gn
gn−1 ρ(R
c`
19
Proposition 5.3. For any choice of a complete set of representatives {a1 , . . . , ah−
} of the proper ideal classes
n
(∞)
−
b
in Cln with b
aj = aj Rc`n and aj,l = 1 for a finite set of primes l | N1 `, the set of points {[z1 , ρ(a−1 )gn ] ∈
j
−
Sh(C) | j = 1, . . . , h−
b0 (N1 `n ) to the set {x(a) | [a] ∈ Cln }. More precisely, there exists
n } is identical on ShΓ
b × ) such that:
a fixed zn ∈ Z(Z
(∞)
−
−
{[z1 , ρ(a−1
b1 (N1 `n ) .
j )gn ] ∈ Sh(C) | j = 1, . . . , hn } = {x(aj ) ◦ zn | j = 1, . . . , hn } on ShΓ
b × ) due to (5.2) and (4.14),
= 1 ∈ Z(Q)Z(Z
Proof. It is a matter of a straightforward check that gn,` g −1
n,`
b × ) for l | N due to (5.3) and (4.15).
whereas gn,l g −1
∈ Z(Q)Z(Z
n,l
bc`n ),
Working with arbitrary representatives a of all classes [a] ∈ Cl−
a = aR
n (no assumption on a from b
and the particular {a1 , . . . , ah−
} from Proposition 5.3, using Proposition 5.3 and Corollary 4.2 we get
n
−1
ϕQ (c`n )
(5.4) X
L
(f
)
=
χm (a−1 )fm (ρ(a−1 )gn )
χ
m
m
2ϕM (c`n )
−
[a]∈Cln
−
(5.4)
=
hn
X
−1
χm (a−1
j )fm (ρ(aj )gn )
j=1
−
(5.1)
=
|det(gn(∞) )|−m
A
hn
X
−1
−m m
−1
χm (a−1
j )|det(ρ(aj ))|A δk f (ρ(aj )gn )
j=1
−
(Prop. 5.3)
=
−k−2m
j(gn,∞ , i)
|det(gn(∞) )|−m
A ψ m (zn )
hn
X
−1 −m m
χm (a−1
j ) |aj |MA (δk f )(x(aj ), ω∞ (aj ))
j=1
X
= j(gn,∞ , i)−k−2m |det(gn(∞) )|−m
A ψ(zn )
m
χm (a−1 )|a|m
MA (δk f )(x(a), ω∞ (a)) .
[a]∈Cl−
n
Using the Katz–Shimura rationality result (4.20), we conclude
X
m
Lχm (fm )
=
χm (a−1 )|a|m
(5.5)
C̃ k+2m
MA (d f )(x(a), ω(a)) ∈ W ,
Ω∞
[a]∈Cl−
n
where
(5.6)
−1
C̃ := j(gn,∞ , i)k+2m |det(gn(∞) )|m
A ψ(zn )
ϕQ (c`n )
2ϕM (c`n )
−1
.
We are now ready to invoke the main Theorem 4.1 of [Hi10b] that relates square of Lχm (fm ) to the central
critical value L(1/2, π̂f0 ⊗χ−
m ). For the starting Hecke newform f0 ∈ Sk (Γ0 (N ), ψ), let f0 |T (l) = a(l, f0 )f0 for
all primes l. As usual, we define the Satake parameters αl , βl ∈ C by the equations αl + βl = a(l, f0 )/l(k−1)/2
and αl βl = ψ(l) when l - N , while we set αl = a(l, f0 )/l(k−1)/2 and βl = 0 when l | N . Then the primitive
L-function is the product
Y
L(s, π̂f ⊗ χ− ) =
El (s)
l
of Euler factors given by
h
α χ− (l̄)
β χ− (l)
α χ− (l)

)(1 − l lm
)(1 − l lm
)(1 −
 (1 − l lm
s
s
s
−1
El (s) = 2/e −
2/e −
χm (l)
β
χm (l)

 (1 − αl 2s/e
)(1 − l l2s/e
)
l
i−1
αl χ−
m (l̄)
)
ls
if l = ll̄ splits in M,
if l = le is non-split,
−
where χ−
m (l) = 0 if l divides the conductor C̄ of χm .
Now we explain how starting from f0 , we choose a suitable f such that its arithmetic lift f is inside the
automorphic representation πf0 generated by the unitarization f0u (see Section 4.1 of [Hi10b]). The form f
20
is a normalized Hecke eigen-cusp form in πf0 with f |T (n) = a(n, f )f and a(l, f ) = a(l, f0 ) for all primes l
outside N `d0 (M ). It is possible to choose f inside πf0 such that for primes l | N `d0 (M ) we have
(
a(l, f0 )
if l | N,
a(l, f ) =
αl l(k−1)/2 if l - N.
Note that πf0 = πf , and we write πf from now. Our character χm is of the form χm := λ · χ · | · |m
MA , where
λ is a fixed choice of a Hecke character of ∞-type (k + m, −m) such that λ|A× = ψ −1 and of conductor c,
and χ is any finite order anticyclotomic character of conductor `n – so it factors through Cl−
n . Then our χm
has conductor C = c`n and the condition (F) of the main Theorem 4.1 of [Hi10b] is satisfied, so by invoking
the theorem we have
1
Γ(k + m)Γ(m + 1)
(5.7)
Lχm (fm )2 = c1
E(1/2)L(N `d) ( , π̂f ⊗ (λχ)− ) .
(2πi)k+2m+1
2
In the following, we set ν̃(`) := n. The constant c1 = c2 · G · v with
p
2πi
) d(M )(2i)−k−2m (c`n )k+2m
c2 = exp(− Q
ν̃(l)
l∈A l
is given by
v=Q
1
l2ν̃(l) (1 + 1l )2 (1 − 1l )
Y
Y
ν̃(l) −1
G=
χ−
)
G(χ−
),
m,l (l
m,l̄
l∈A
(5.8)
lν̃(l) (1
1 3
l)
−
Q
l∈A
l|N `
χ−
m,l
,
χ−
m |Q×
l
χ−
m,l̄
where
=
Euler factor is given by
(5.9)
=
χ−
m |M × ,
l̄
,
l∈Ci
)
G(χ−
m,l̄
= χ−
is the Gauss sum of χ−
m |M × , and the modification
m,l̄
l̄
−1 −1
Y χ−
χ−
m (l)αl
m (l)βl
E(1/2) =
1−
1−
.
N (l)1/2
N (l)1/2
l|d(M )
Then (5.5) and (5.7) suggest that we normalize the L-value as follows
(5.10)
L(N `d) ( 12 , π̂f ⊗ (λχ)− )
1
Γ(k + m)Γ(m + 1)
Lalg ( , π̂f ⊗ (λχ)− ) := G
E(1/2)
.
2(k+2m)
2
π k+2m+1
Ω∞
As we are going to study Lalg ( 21 , π̂f ⊗ (λχ)− ) modulo p by utilizing (5.5), we are clearly going to exclude p
from the primes that divide the fudge factors c1 and v above, as well as C̃ given in (5.6). Thus, let S(N, `)
be a finite set of prime divisors of elements of
(5.11)
{N, ` − 1} ∪ {l − 1 : prime l | N ` is split in M } ∪ {l − 1, l + 1 : prime l | N ` is inert in M } .
6. Zariski density of CM points on tower of modular curves
Each integral R-ideal A prime to c` induces a unique proper Rc`n -ideal An = A ∩ Rc`n , and after taking
alg
the inverse limit [A] := limn An , we obtain an element of Cl−
of
∞ . We are able to embed the group Cl
←−
−
fractional ideals of R prime to c` as a subgroup of Cl∞ by extending this procedure to fractional ideals in
the obvious way.
(p)
(p)
Let V/F̄p be a fixed irreducible component of Sh/F̄p containing the CM point x(R) = x̃(R). In other words,
(p)
(p)
V/F̄p is the special fiber of the geometrically connected component V/W of Sh(p) introduced in the Section
(p)
2.5. Hereafter, for simplicity we write V for V/F̄p . Let n = {nj }∞
1 be any infinite sequence of integers and
−
n
denote by πnj ,r : Cl−
nj Clr the projection, 0 ≤ r ≤ n1 . We emphasize that for all proper Rc` j -ideals a
whose proper Rc`nj -ideal class [a] lies in the kernel Ker(πnj ,r ), all the reductions x̃(a)/F̄p indeed belong to
the same fixed component V by Corollary 4.4, so we can define
Ξ = Ξn := {x̃(a) ∈ V (F̄p ) | [a] ∈ Ker(πnj ,r ), j = 1, 2, . . . , for an integer r with 0 ≤ r ≤ n1 } .
21
Elliptic curves sitting over points in Ξ are non-isomorphic by a result of Deuring ([Deu]) which assures that
the isomorphism class of an elliptic curve over F̄p is determined by the action of the relative Frobenius map
on its `-adic Tate module. Thus, the set Ξ is infinite and we record the following obvious
Fact 6.1. Since dimV/F̄p = 1, the set Ξ is Zariski dense in V (F̄p ).
alg
Let Q ⊂ Cl−
and, without loss of generality, such that
∞ be a finite subset independent modulo Cl
the elements δ ∈ Q project down to the identity class in Pic(R). Denote by δ(a) a proper ideal class
representative of the product of π∞,nj (δ) and [a] in Cl−
nj . Note that the x̃(δ(a)) here stay in the same fixed
component V by Corollary 4.4, since all δ(a), δ ∈ Q, project down to the identity class in Pic(R) and we
can set
ΞQ := {(x̃(δ(a)))δ∈Q ∈ V Q | x̃(a) ∈ Ξ} ,
Q
where V Q = δ∈Q V . We recall the Proposition 2.8 of [Hi04] since it is of instrumental use for us.
alg
Theorem 6.2. If Q is finite and injects into Cl−
, the subset ΞQ is Zariski dense in V/Q
.
∞ /Cl
F̄p
A detailed proof is given in [Hi04] and we just give a brief sketch noting that its key ingredient is an
instance of the Chai’s Hecke orbit principle (seeQ[Ch] Section 8) stated below. Let Q = {δ1 , . . . , δh }. The
×
h
via ρx(R) , for ρx(R) defined by (4.12). Let Z1 be the Zariski
torus T = R(p)
/Z×
(p) acts diagonally on V
Q
h
closure of Ξ in V . The group T1 = {α ∈ T | α ≡ 1 (mod `n1 )} leaves Z1 stable, as it just permutes ΞQ by
ρx(R) (α)(x̃(a)) = x̃(αa). If Z0 denotes the irreducible component of Z1 containing (x̃(Rc`n1 ), . . . , x̃(Rc`n1 )),
then the stabilizer T0 of Z0 is of finite index in T1 and its p-adic closure is open in T . Thus the following
theorem applies.
Theorem. ([Hi10a] Corollary 3.19) Let Z0 be an irreducible subvariety of V m , m ≥ 1, containing a fixed
point of T . If there exists a subgroup T0 ⊂ T that stabilizes Z0 and whose p-adic closure is open in Rp× /Z×
p,
then Z0 is a Shimura subvariety of V m .
As Z0 is a nontrivial Shimura subvariety of V h (i.e. not of the form V h−1 × {x} for a fixed CM point
x), this leaves two possibilities: either Z0 = V h or, after a permutation of factors of V h , Z0 is a Shimura
subvariety of V h−2 × ∆β,β 0 , where for some β, β 0 ∈ R(p) we define the diagonal ∆β,β 0 by
∆β,β 0 = {(x ◦ ρx(R) (β), x ◦ ρx(R) (β 0 )) | x ∈ V } = {(x, x ◦ ρx(R) (β −1 β 0 )) | x ∈ V } ⊂ V 2 .
However, the second possibility imposes δh−1 /δh = β −1 β 0 ∈ M × , whence δh−1 Clalg = δh Clalg and Theorem
6.2 follows.
Let CΞ denote the space of functions on Ξ with values in P1 (F̄p ) = F̄p t {∞}. The anticyclotomic class
group Cl−
∞ acts on CΞ by left translation. By virtue of the Zariski density of Ξ in V , we can embed the
function field of V into CΞ . Then Theorem 6.2 has the following
Corollary 6.3. ([Hi04] Corollary 2.9) Let L be a line bundle over V/F̄p . Then for a finite set Q ⊂ Cl−
∞
alg
that injects into Cl−
, and a set {fδ ∈ L | δ ∈ Q} of non-constant global sections fδ of L finite at Ξ,
∞ /Cl
the functions fδ ◦ δ, δ ∈ Q, are linearly independent in CΞ .
7. Non-vanishing of L-values modulo p
In this section we prove Theorem 1.1 following closely Section 3.4 of [Hi04], particularly the proof of
Theorem 3.2 there. That being said, we avoid constructing the anticyclotomic measure employed there, in
order not to impose the condition a(`, f ) 6= 0. Before proceeding to the proof, we need a technical lemma.
To this end, for α ∈ GL+
2 (R) and a classical modular form f ∈ Sk (Γ0 (N1 ), ψ), for any N1 ≥ 1, we define
f kk α = det(α)k/2 f (α(z))j(α, z)−k .
Note that the operator kk depends on the weight of the form, but since the weight will always be clear
from the context, we shall write it as k. We use a non-standard notation to distinguish it from the isogeny
action defined by (4.19) in Section 4.7. In the following lemma we investigate the effect of the latter on the
q-expansions of Katz’s p-adic derivatives of modular forms. If r = r1 · . . . · rn is a square-free product of
primes, we write
f |[r] = f |[r1 ]| . . . |[rn ] .
22
Let N and the starting Hecke newform f0 ∈ Sk (Γ0 (N ), ψ) be the ones fixed at the beginning of Section 4
for the rest of the paper. Let from now on N1 := N d0 (M ) and let f ∈ Sk (Γ0 (N1 ), ψ) be the Hecke eigen-cusp
form explicitly made out of f0 in Section 5, so that its arithmetic lift f is inside the unitary automorphic
representation πf0 . In particular, ψ is unramified at r | d0 (M ).
In this section, from now on we use q to denote the variable in the q-expansion of a modular form in
order to avoid abuse of notation in the proof of the main theorem. Then we have the following
Lemma 7.1. For a square-free product r = r1 · . . . · rn of primes split or ramified in M and prime to N , we
have
−1 .
(dm f )|[r] = ψ(r)rk/2+m dm f k 10 0r
The q-expansion of (dm f )|[r] is given by
ψ(r)rk+2m
X
nm a(n, f )qnr .
n≥0
Proof. Let X = (X, η (p) (x), η ét (x) × η ord (x), ω(x)) denote a general test object that gives rise to a point
be
x = (X, η (p) (x), η ét (x) × η ord (x)) in the ordinary locus of Sh, and let ω∞ (x) ∈ ΩX/C and ωp (x) ∈ ΩX/W
b
the differentials induced from ω(x) as in the Section 4.8.
We first prove the assertion when r is a prime. If we set the global α = 10 0r ∈ G(Q) ⊂ G(A), then
[r]x = x ◦ αr on Sh by (4.18), and the crux of the proof is the following identity that holds for all m ≥ 0:
(7.1)
−1
δkm f ((x, ω∞ (x)) ◦ αr ) = ψ(r)rk/2+m δkm (f kα∞
)(x, ω∞ (x)) .
To verify it, recall that if x = [z, g (∞) ] ∈ Sh(C), for some z ∈ X and g (∞) ∈ G(A(∞) ), then by definition
δkm f (x, ω∞ (x)) = δkm f ([z, g (∞) ], ω∞ (x)) = δkm f (g)j(g∞ , i)k+2m ,
where g∞ ∈ G(R) is such that g∞ (i) = z and g = g (∞) g∞ . Note that we need to check the identity over
ShΓb1 (N1 ) only, so without loss of generality we may assume that g (∞) = 1, i.e. x = [z, 1]. Indeed,
δkm f ([r]x, ω∞ ([r]x)) = δkm f ([z, 1] ◦ αr , ω∞ (x ◦ αr ))
= δkm f (αr g∞ )j(g∞ , i)k+2m
−1
= δkm f α(α(r∞) )−1 α∞
g∞ j(g∞ , i)k+2m
−1
= ψ(α(r∞) )−1 δkm f α∞
g∞ j(g∞ , i)k+2m
−1
−1
= ψ(r)δkm f (α∞
g∞ (i))j(α∞
g∞ , i)−k−2m j(g∞ , i)k+2m
−1
−1
= ψ(r)δkm f (α∞
(z))j(α∞
, z)−k−2m
−1
= ψ(r)rk/2+m (δkm f )kα∞
(z)
−1
= ψ(r)rk/2+m δkm (f kα∞
)([z, 1], ω∞ (x))
−1
= ψ(r)rk/2+m δkm (f kα∞
)(x, ω∞ (x))
−1
−1
b 0 (N1 ) for this
as desired. Here we used a general fact (δkm f )kα∞
= δkm (f kα∞
) and that (α(r∞) ) ∈ Γ
particular α. By the Katz–Shimura rationality result (4.20) we have
dm f ([r]x, ωp ([r]x))
δ m f ([r]x, ω∞ ([r]x))
= k
k+2m
Ωp
Ωk+2m
∞
−1
δ m (f kα∞
)(x, ω∞ (x))
= ψ(r)rk/2+m k
k+2m
Ω∞
m
−1
d
(f
kα
∞ )(x, ωp (x))
= ψ(r)rk/2+m
,
Ωk+2m
p
which yields
−1
(dm f )|[r] = ψ(r)rk/2+m dm (f kα∞
) for all m ≥ 0 .
m
The assertion about the q-expansion of (d f )|[r] now easily follows from (4.21).
23
To prove the lemma for an arbitrary square-free integer r, by induction it suffices to verify it when r = r1 r2
is a product of two primes. Indeed,
−1 k/2+m m
dm f |[r] = dm f |[r1 ]|[r2 ] = ψ(r1 )r1
f k 10 r01
|[r2 ]
d
−1 1 0 −1 k/2+m
k/2+m m
f k 10 r01
= ψ(r1 )r1
ψ(r2 )r2
d
0 r2
−1
= ψ(r)rk/2+m dm f k 10 0r
and the claimed effect of the operator [r] on the q-expansion of dm f again easily follows from (4.21).
Now we are ready to prove Theorem 1.1.
Proof. Classical modular forms are defined over a number field and we may assume that f is defined over a
localization V of the integer ring in a number field E. We take a finite extension of W generated by V and
the values of the arithmetic Hecke character λ and, abusing the symbol, we keep denoting it W . We write
P for the prime ideal of W corresponding to ιp . The considered values dm f (x(a), ω(a)) are algebraic and
P-integral over V by results of Shimura and Katz ([Sh75] and [Ka78]).
We fix a decomposition Cl−
∞ = ∆ × Γ, where Γ is a torsion free subgroup topologically isomorphic to
Z` and ∆ is a finite group. Denote by Fp [f, λ, µ`|∆| ] the finite subfield of F̄p generated by the values of
×
λ mod P and all `|∆|-th roots of unity over V/P ∩ V. Similarly, if χ : Cl−
n → F̄p is a character, denote by
Fp [f, λ, µ`|∆| ](χ) the finite extension of Fp [f, λ, µ`|∆| ] generated by the values of χ.
As ∆ is a finite group, it suffices to fix a branch character ν : ∆ → F̄×
p and, aiming for contradiction,
suppose that
1
Lalg ( , π̂f ⊗ (λχj )− ) ≡ 0 (mod P)
2
−
∞
for infinitely many characters χj : Clnj → F̄×
p such that χj |∆ = ν, where {nj }j=1 is an infinite sequence of
integers. We assume that nj is the smallest integer such that χj factors through Cl−
nj . Suppose that the
prime p > 2 lies outside the finite set S(N, `) given by (5.11). Then by plugging χj,m := λ · χj · | · |m
MA in
(5.5) and (5.7), we have
X
λχj (a−1 )dm f (x(a), ω(a)) = 0 in F̄p .
[a]∈Cl−
n
j
An important point here is that each summand in the above sum is independent of the choice of a proper
ideal class representative in Cl−
nj . This follows from Lemma 5.2 and the Katz–Shimura rationality result
(4.20). Alternatively, we can directly quote the statement (G30 ) on page 762 of [Hi04]:
dm f (x(αa), ω(αa)) = α−k−m(1−c) dm f (x(a), ω(a)) for α ∈ R(p) .
Since the Hecke character λ is of ∞-type (k + m, −m) we can introduce the modified value dm f ([a]) by
dm f ([a]) := λ(a−1 )dm f (x(a), ω(a))
which is independent of the choice of a in its proper ideal class. Then the above identity becomes
X
(7.2)
χj ([a]−1 )dm f ([a]) = 0 in F̄p .
[a]∈Cl−
n
j
Moreover, for each σ ∈ Gal(F̄p /Fp [f, λ, µ`|∆| ]) we have
X
(7.3)
χσj ([a]−1 )dm f ([a]) = 0 .
[a]∈Cl−
n
j
Indeed, by Shimura’s reciprocity law ([ACM] 26.8 and [PAF] 2.1.4), we have
Φ(dm f (x([a]))) = dm f (x([p−1 a]))
24
for the Frobenius map Φ(x) = xp for x ∈ F̄p . Thus, if σ = Φn for a positive integer n, we have


X
 X

Φn 
χj ([a]−1 )dm f ([a]) = (λχj )σ (p−n )
χσj ([a]−1 )dm f ([a])
[a]∈Cl−
n
[a]∈Cl−
n
j
j
and hence (7.2) implies (7.3).
Consider the trace map from the field Fp [f, λ, µ`|∆| ](χj ) to Fp [f, λ, µ`|∆| ], given by
X
TrFp [f,λ,µ`|∆| ](χj )/Fp [f,λ,µ`|∆| ] (ξ) =
σ(ξ)
σ∈Gal(Fp [f,λ,µ`|∆| ](χj )/Fp [f,λ,µ`|∆| ])
for ξ ∈ Fp [f, λ, µ`|∆| ](χj ), and note that if Fp [f, λ, µ`|∆| ]× ∩ µ`∞ = µ`s we have
(
`ñj χj (x) if χj (x) ∈ Fp [f, λ, µ`|∆| ], for some ñj ≥ nj − s,
TrFp [f,λ,µ`|∆| ](χj )/Fp [f,λ,µ`|∆| ] (χj (x)) =
0
otherwise.
Combining (7.3) with the above trace identity, we conclude
X
χj ([a]−1 )dm f ([a]) = 0 .
(7.4)
−1
[a]∈Cl−
n :[a]∈χj (µ`s )
j
Moreover, if Γj denotes the image of Γ in Cl−
nj , as the summands are independent of the choice of a proper
ideal class representative in Cl−
,
we
conclude
from (7.4) that for every y ∈ Γj we have
nj
X
χj ([a]−1 )dm f ([a]) = 0 .
(7.5)
−1
[a]∈Cl−
n :[a]∈yχj (µ`s )
j
alg
alg
The group ∆
= ∆ ∩ Cl
is generated by prime ideals of M non-split over Q and we can choose a
complete representative set for ∆alg consisting of product of prime ideals of M outside N , p and `. As in
[Hi04], we choose this set as {r−1 | r ∈ R}, where R is made of square-free products of rational primes
outside N and ` that are ramified in M , and r is a unique ideal in M such that r2 = r. Thus, {r | r ∈ R} is
a complete representative set for 2-torsion elements in ClM , the class group of M .
alg
We also choose a complete set of representatives Q for Cl−
consisting of prime ideals q of M that
∞ /Γ∆
are split over Q and prime to p and `. We denote by hqi the projection of [q] ∈ Cl−
∞ to Γ, where [q] is the
image of q under the embedding of Clalg into Cl−
defined
at
the
beginning
of
Section
6. Easy to check,
∞
alg
−
but nevertheless a very subtle point here is that even though [q] ∈ Cl ,→ Cl∞ , the projections hqi are
alg
independent modulo
inside Cl−
∞.
F Cl
−
−1 −1
Then Cl∞ = q,r [q r ]Γ and since χj |∆ = ν we can rewrite (7.5) as
XX
X
ν(rq)
χj ([a−1 hqi])dm f ([q−1 r−1 a]) = 0 .
q∈Q r∈R
s
[a]∈yχ−1
j (µ` )
If we set
(dm f )ν :=
(7.6)
X
λ(r)ν(r)(dm f )|[r] ,
r∈R
then the above identity becomes
X
(7.7)
ν(q)
q∈Q
X
χj ([a]−1 )(dm f )ν ([q−1 hqia]) = 0 .
s
[a]∈yχ−1
j (µ` )
Fix q ∈ Q. From (4.16) (see also Remark 4.5) we know that on ShΓb0 (N1 ) the set {x([a]) | [a] ∈ yχ−1
j (µ`s )}
1 `us
−1
s
is given by {x([a0 ]) ◦ 0 1 ` | u ∈ Z/` Z}, where [a0 ] ∈ yχj (µ`s ) is any fixed member. In particular, we
−1
s
s
can identify yχ−1
j (µ`s ) with Z/` Z via [a] 7→ u mod ` . We formally write the element [a] of yχj (µ`s )
u
−1
corresponding to u ∈ Z/`s Z as [ 01 `1s ` a0 ]. Note that, after tacitly assuming that nj ≥ 2s (which could
be achieved by passing to a suitable subsequence if necessary), the above identification is tantamount to
25
nj −s
nj
identifying the multiplicative group Γ`
/Γ` with the additive one Z/`s Z by 1 + `nj −s u 7→ u. If we
1 `us −1
choose a primitive `s -th root of unity ζ = exp(2πi/`s ) and [ay ] ∈ yχ−1
ay ]) =
j (µ`s ), we can write χj ([ 0 1
χj (u) = ζ −uvj for some vj ∈ (Z/`s Z)× , independent of y ∈ Γj , that initially depends on χj . However, we
may assume that vj ’s are constant v by resorting to a suitable subsequence of {χj }∞
j=1 . Then, using the
description of the operators [r] and [q] in Section 4.7, the inner sum of (7.7) is equal to
X
u ζ uv (dm f )ν |[q]| 10 `1s ` ([hqiay ]) .
u mod `s
Note here that
1 `us
0 1
`
10
0 q q
and
P
commute. Setting gq := u mod `s ζ uv (dm f )ν |[q]|
X
ν(q)gq ([hqia]) = 0 .
1 `us
0 1
`
, (7.7) becomes
q∈Q
alg
Note that the elements of Q = {hqi | q ∈ Q} are all distinct in Cl−
by our choice, since hq1 ihq2 i−1 ∈
∞ /Cl
alg
−1
alg
Q0
Cl would imply q1 q2 ∈ Γ∆ . Thus, the set Ξ defined in Section 6, for Ξ = Ξn and n = {nj }∞
1 , is
Q0
Zariski dense in V/F̄p by Theorem 6.2. Note that here Corollary 4.1 and Corollary 4.2 allow us to climb to
the tower Sh and safely use CM points x̃(a) instead of x(a) in the context of Section 6. Then Corollary 6.3
furnishes gq = 0.
Note that for a classical modular form f1 of arbitrary level and weight, one has the identity
u u −1
(dm f1 )| 10 `1s ` = dm f1 k 10 `1s
0
which can be verified by literally the same argument as in the proof of Lemma 7.1 – the appropriate
incarnation of the identity (7.1) has no scalar factors due to the fact that the matrix here is unipotent. Then
fixing q0 ∈ Q and using Lemma 7.1, we have
X
u gq0 =
ζ uv (dm f )ν |[q0 ]| 01 `1s `
u mod `s
!
=
X
ζ uv
u mod `s
=
X
X
λ(r)ν(r)(dm f )|[r] |[q0 ]|
1 `us
0 1
`
r∈R
λ(r)ν(r)ψ(rq0 )(rq0 )
X
k/2+m m
d
ζ
uv
fk
1 0 −1 1 − `us
0 rq0
0 1
!
.
u mod `s
r∈R
Thus, using (4.21) we conclude that the q-expansion of gq0 is given by


!
X
X
X
gq0 (q) =
λ(r)ν(r)ψ(rq0 )(rq0 )k+m dm 
ζ u(v−nrq0 ) a(n, f )qnrq0 
n≥0
r∈R
u mod `s
!
=
XX
k+2m
λ(r)ν(r)ψ(rq0 )(rq0 )
X
ζ
u(v−nrq0 )
nm a(n, f )qnrq0 .
u mod `s
n≥0 r∈R
The Fourier coefficient a(n, gq0 ) is given by
( P
`s r∈R λ(r)ν(r)ψ(rq0 )(rq0 )k+m nm a( rqn0 , f ) if n ≡ v (mod `s ),
a(n, gq0 ) =
0
otherwise,
where we accept the usual convention that a(n, f ) = 0 when n is not an integer.
Thus, if we fix a prime r0 ∈ R, in order to contradict gq = 0 it suffices to find a prime l outside N p` and
R such that l ≡ v(r0 q0 )−1 (mod `s ) and a(l, f ) 6≡ 0 (mod p). Indeed, then lr0 q0 ≡ v (mod `s ) and
a(lr0 q0 , gq0 ) = `s λν(r0 )ψ(r0 q0 )(r0 q0 )k+2m lm a(l, f ) 6≡ 0 (mod p) .
Let P be a prime ideal of E above p. To do the outlined, recall that the celebrated constructions of
Shimura (k = 2), Deligne (k > 2) and Deligne and Serre (k = 1) attach to the starting Hecke newform f0 a
Galois representation
ρ : Gal(Q̄/Q) → GL2 (EP )
26
such that for all primes l - N p, ρ is unramified at l and for a Frobenius element Frl one has
Trρ(Frl ) = a(l, f0 ) and detρ(Fl ) = ψ(l)lk−1 .
Recall that a(l, f ) = a(l, f0 ) for all primes l - N `d0 (M ) by the choice of f in Section 5. Let ρ̄ be the reduction
of ρ modulo P. Note that the fields K = Q̄Kerρ̄ and Q(µ`s ) are linearly disjoint. Indeed, ` - N p is unramified
in the former while totally ramified in the latter. Then one can choose
σ ∈ Gal(K(µ`s )/Q) = Gal(K/Q) × Gal(Q(µ`s )/Q)
such that
σ|K = idK
and σ|Q(µ`s ) = v(r0 q0 )−1 ∈ (Z/`s Z)× ∼
= Gal(Q(µ`s )/Q) .
Then the Čebotarev density theorem furnishes a prime l outside N `d0 (M ) such that Frl |K(µ`s ) = σ, and
we clearly have a(l, f ) = a(l, f0 ) = Trρ(Frl ) = Tr(idK ) = 2 6≡ 0 (mod p) and l ≡ v(r0 q0 )−1 (mod `s ) as
desired.
Note that we used the assumption ` - N only to conclude that K = Q̄Kerρ̄ and Q(µ`s ) are linearly disjoint.
That being said, if we allow possibility of ` | N , notation in the statement of the main Theorem 1.1 as well
as in the Sections 4 and 5 would have to be slightly adapted. Namely, in place of N and c = ci cs one would
(`) (`)
use their outside-of-` parts N (`) and c(`) = ci cs , respectively. However the argument remains valid. Thus,
the Theorem 1.1 is valid under the following assumption milder than ` - N :
Assumption 7.2. K = Q̄Kerρ̄ and Q(µ`∞ ) are linearly disjoint.
In particular, if ` | N but the modulo P reduction ρ̄ happens to be unramified at `, this assumption is
clearly satisfied. Assume for simplicity `kN . If the component πf ,` = π(µ1 , µ2 ) is a principal series, where
×
µ1 and µ2 are characters of Q×
` with values in E , then one of these characters is unramified and the other
×
has conductor `. Write µ̂1 and µ̂2 for the characters of the decomposition group D` ∼
= Gal(Q̄` /Q` ) → EP
corresponding to µ1 and µ2 , respectively, by the local class field theory. By a well known result of Carayol
([Ca]), we know that the semisimplification ρ|D` is isomorphic to µ̂1 ⊕ µ̂2 . Supposing that the ramified one
among µ1 and µ2 has unramified reduction modulo P in order to obey the Assumption 7.2, would impose
p | ` − 1. This in turn places p inside the prohibited set S(N, `) and consequently we would not get the
desired result.
We hope to treat this question in a future paper and here we limit ourselves to the following
Proposition 7.3. Suppose that `kN and that πf ,` is a special representation. Then Theorem 1.1 holds.
×
Proof. Let πf ,` = sp(α| · |1/2 , α| · |−1/2 ) for an unramified character α : Q×
` → E . It is a well known result
of Langlands ([La73]) that ρ|D` is isomorphic to
cy
α̂ ⊗ χ0p ∗1 ,
×
where α̂ denotes the character of D` ∼
corresponding to α by the local class field theory
= Gal(Q̄` /Q` ) → EP
cy
ab
and χp is the p-th cyclotomic character. Let Q` denote the maximal abelian extension of Q` and µ(`),∞
the set of all roots of unity in Q̄` of order not divisible by `. Then for the Artin local reciprocity map
×
×
ab
n
( · , Qab
` /Q` ) : Q` → Gal(Q` /Q` ) and x = ` u (n ∈ Z, u ∈ Z` ), we have
(
Frn` on Q` (µ(`),∞ ),
ab
(x, Q` /Q` ) =
u−1 on Q` (µ`∞ ).
Going back to the proof of the Theorem 1.1, it suffices to choose σ ∈ K(µ`s ) by first lifting (v −1 r0 q0 , Qab
` /Q` ) ∈
∼
Gal(Qab
` /Q` ) to D` = Gal(Q̄` /Q` ) and then projecting to K(µ`s ). Again, by virtue of the Čebotarev density
theorem we get a prime l such that Frl |K(µ`s ) = σ, and since both α̂ and χcy
p are unramified at `, we conclude
a(l, f ) = Trρ(Frl ) = Tr(σ) = 2 6≡ 0 (mod p) and l ≡ v(r0 q0 )−1 (mod `s ) as desired.
27
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Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West,
Montreal, QC, Canada H3A 0B9
E-mail address: miljan.brakocevic@mcgill.ca
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