The Jacquet-Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2

0.22 MB·

by  Kimball Martin

The Jacquet–Langlands correspondence, Eisenstein
L
congruences, and integral -values in weight 2
6
1 Kimball Martin∗
0
2
March 11, 2016
r
a
M

0 Abstract
1
We usethe Jacquet–Langlandscorrespondencetogeneralizewell-knowncongruence

resultsofMazuronFouriercoefficientsandL-valuesofellipticmodularformsforprime
]
T level in weight 2 both to nonsquare level and to Hilbert modular forms.
N
A celebrated result of Mazur says that, for N a prime and p an odd prime dividing (the
.
h numerator of) N−1, there exists a cusp form f S (N) congruent to the Eisenstein series
t 12 ∈ 2
a E of weight 2 and level N mod p [Maz77, II(5.12)]. Furthermore, one has a congruence
2,N
m
for the algebraic part of the central L-value L(1,f ) = L(1,f)L(1,f η ), where η
K K K
[ ⊗
is the quadratic character associated to a quadratic field K/Q [Maz79]. For instance, if

2 N = 11 and K is not split at 11, there is one cusp form f S (N) and one gets that
2
v Lalg(1,f ) 0 mod 5 if and only if 5 ∤ h . (If K splits at 11∈, the root number is 1 so
4 K 6≡ K −
8 L(1,fK) = 0). A form of Mazur’s L-value result was reproved by Gross [Gro87] for K/Q
2 imaginary quadratic using quaternion algebras and the height pairing, whereas Mazur used
3
modular symbols. Ramakrishnan pointed out to me that one can also deduce this from
0
. his average L-value formula with Michel [MR12]. (Note Gross’s argument also involves an
1
averaging type procedure, so these two arguments are not entirely different in spirit.)
0
6 In this article, we use the Jacquet–Langlands correspondence and an explicit L-value
1
formula to extend these results of Mazur both to more general levels and to parallel weight
:
v 2 Hilbert modular forms over a totally real field F with K/F a quadratic CM extension.
i
X For simplicity, we only state our results precisely for F = Q in this introduction.
r We first discuss the Hecke eigenvalue congruence result and a nonvanishing L-value
a
result, and will state the more precise result on L-value congruences below in Theorem B.
Theorem A. Let N be a nonsquare, and write N = N N where (N ,N ) = 1 and N
1 2 1 2 1
has an odd number of prime factors, all of which occur to odd exponents. Let p be a prime
dividing (the numerator of) 1 ϕ(N )N (1+q−1) and p a prime of Q above p. Then
12 1 2 q|N2
there exists f S (N) which is an eigenform for all T with (n,N) = 1 and is congruent
2 n
∈ Q
to the Eisenstein series E mod p away from N.
2,N
Suppose moreover N = N with N squarefree and that p ϕ(N1). Then we can take f
1 1 | 24
to be a newform such that f E mod p (as q-expansions). If K/Q is an imaginary
2,N
≡
quadratic field not split at any prime dividing N such that p ∤h , then there exists such an
K
f with L(1,f ) = 0.
K
6
∗Department of Mathematics, University of Oklahoma, Norman, OK 73019
1
Here E is the Eisenstein series of weight 2 and level N defined by E (z) =
2,N 2,N
µ(d)dE (dz), where µ is the M¨obius function and E is the weight 2 Eisenstein series
d|N 2 2
for SL(2,Z) normalized so that the Fourier coefficient of q is 1.
P
The statement about congruence away from N means that f and E have the same
2,N
Hecke eigenvalues mod p for T for any prime ℓ ∤ N. If N = N is squarefree, then E has
ℓ 1 2,N
ϕ(N) ϕ(N) ϕ(N)
constant term , and the only reason we need to assume p rather than p in
24 | 24 | 12
the second part is so the constant term of E will be 0 mod p. E.g., if N = 73 there is a
2,N
rational newform f S (73) all of whose Hecke eigenvalues match with those of E mod
2 2,73
∈
2, but the constant term of E is 1 mod 2 so they are not congruent as q-expansions.
2,73
Note E2,N = E2,N′ where N′ = p|N p is the “powerfree part” of N so E2,N really has
level N′, and E is an eigenfunction of the Hecke operators T on M (N) for all primes
2,N Q ℓ 2
ℓ having eigenvalue ℓ + 1 when ℓ ∤ N and eigenvalue 1 when ℓ N. One might prefer to
|
replace E with a form whose T eigenvalue is 0 when ℓ2 N and look for a congruence
2,N ℓ
|
mod ℓ as well. However, since we do not know the exact level of f in general, this does not
make a difference for the statement of the theorem. Still, we give non-squarefree examples
below where one can replace E with a different Eisenstein series to get a congruence of
2,N
all Fourier coefficients.
The Hecke eigenvalue congruence in the N = N squarefree case of Theorem A was
1
also proved by Ribet (unpublished, announced in 2010) under the additional assumptions
p ∤ N and p 5. In fact, Ribet also obtained a converse (there are no other such p ∤ N
≥
with p 5) and more generally addressed finding newforms congruent to E away from
2,N
≥
N with specified p -th Fourier coefficient a 1 for each p N. This was studied
i pi ∈ {± } i|
further by Yoo in [Yooa], which also explains Ribet’s work in detail. (When N = N is
1
squarefree, our Theorem A only concerns finding newforms with a = 1 for each p N.)
pi i|
Their arguments involve studying Eisenstein ideals using Jacobian calculations and then
appealing to the Jacquet–Langlands correspondence. See also [Yoob] for related results
when N is not squarefree.
In contrast, our proof follows from using a (generalized) Eichler mass formula to con-
structaquaternioniccuspformφ“of level N”onthedefinitequaternionalgebraB ramified
at each prime dividing N such that φ 1 mod p. (Here φ is just a function on a certain
1
≡
finite set of ideal classes with the cuspidality criterion amounting to a linear relation among
the values of φ.) In addition to our method being more straightforward, we have no trouble
dealing with N not squarefree and arbitrary p, or extending this to Hilbert modular forms
as in Theorem 2.1 (though in the Hilbert modular case we make a simplyfing assumption
that our totally real field F satisfies h = h+—see Remark 2.2). We do not get a converse
F F
or consider a = 1, though see Remark 2.4 regarding the converse.
pi −
Here are some simple consequences of Theorem A.
Corollary 1. If p > 2, then there exists f S2(p3) such that f and E2,p3 = E2,p are
∈
congruent mod p away from p, for p a prime above p in Q.
For example, when p = 3, there is a unique normalized (new)form f S (27), which
2
∈
has Fourier expansion
a qn = q 2q4 q7+5q13+4q16 7q19 5q25 +2q28 4q31+11q37 +8q43 +
n
− − − − − ···
X
2
and it satisfies a ℓ +1 mod 3 for all primes ℓ = 3. In fact we have f(z) E (z)
ℓ 2,3
≡ 6 ≡ −
E (3z) mod 3. (Note the constant term of the right hand side is 0.)
2,3
We remark that the analogous statement of the corollary is not true in level p2—e.g.,
for N = N = 25, then prime p = 5 divides the numerator of 1 φ(N ), but S (25) = 0 .
1 12 1 2 { }
In the case of p = 2, the theorem says that there exists f S (32) which is congruent
2
∈
to E mod 2 away from 2. Here S (32) is 1 dimensional, and spanned by the form
2,2 2
a qn = q 2q5 3q9+6q13+2q17 q25 10q29 2q37+10q41+6q45 7q49 + .
n
− − − − − − ···
X
In fact, f(z) E (z) E (2z) mod 2.
2,2 2,2
≡ −
Corollary 2. Let p,q be distinct primes with p > 3. Then there exists f S (qp2) such
2
∈
that f and E are congruent mod p away from pq, for p a prime of Q above p.
2,q
For instance, S (50) = S (50)new has dimension 2, and one eigenform f S (50) has
2 2 2
∈
Fourier expansion
q+q2 q3+q4 q6 2q7+q8 2q9 3q11 q12+4q13 2q14+q16+3q17+
− − − − − − − ···
In this case it happens that f and E are congruent mod 5 everywhere away from 5, and
2,50
f(z) E (z) E (5z) mod 5.
2,10 2,10
≡ −
The L-value result at the end of Theorem A is a direct generalization of [Gro87, Cor
11.8], and follows from our proof of Theorem 2.1 and Waldspurger’s formula [Wal85]—see
Proposition 3.1. The p = 2 case (not treated by Mazur) has consequences in the spirit of
Goldfeld’s conjecture:
Example 3. Let C be the unique-up-to-isogeny elliptic curve over Q of conductor N = 17.
Then among negative prime discriminants d, the central value L(1,C ) for the quadratic
−d
−
twist C is nonzero exactly 50% of the time.
−d
In fact, we can say precisely when L(1,C ) = 0 for d < 0 a prime discriminant.
−d
6 −
We know C is associated to the unique normalized form f S (17). We take p = 2 in
2
∈
our theorem and conclude L(1,f ) = 0 when 17 is inert in K = Q(√ d) and 2 ∤ h .
K K
6 −
However, for d prime, Gauss’ genus theory implies 2 ∤ h . Thus, if 17 is inert in K,
K
L(1,f ) = L(1,C)L(1,C ) = 0. On the other hand, if 17 is split in Q(√ d), then
K −d
6 −
L(1,C ) = 0 because the root number is 1. In particular, at least 50% of these prime
−d
−
quadratic twists C have finitely many rational points.
−d
Now we discuss congruences of L-values. Let K/Q be a quadratic extension and χ an
idealclass character ofK. LetL(s,f,χ)denotethetwistL(s,f χ)ofthebasechangef
K K
⊗
of f to K (a degree 2 L-function over K), or equivalently L(s,f θ ) the degree 4 Rankin–
χ
×
Selberg L-function over Q of f times the GL(2) theta series θ attached to χ. When χ = 1
χ
is trivial, this L-function is just L(s,f,1) = L(s,f )= L(s,f)L(s,f η ).
K K
⊗
Given a congruence of modular forms, say as in Theorem A, we expect a congruence
of algebraic parts of the special L-values L(1,f,χ). In the case where one of the modular
formsisanEisensteinseries,thiswasstudiedbyVatsalandHeumann[Vat99],[HV14](after
Mazur [Maz79] and Stevens [Ste82]) using modular symbols.
3
The Jacquet–Langlands correspondence, together with an explicit L-value formula of
earlier joint work with Whitehouse [MW09], also yields a congruence of central L-values
when one starts with a congruence of quaternionic modular forms. We just consider the
case where one of these quaternionic modular forms is constant (i.e., corresponds to E )
2,N
but the same approach can also be used when one starts with two quaternionic cusp forms
which are congruent mod p.
Theorem B. Suppose N is squarefree product of an odd number of primes and f S (N)
2
∈
corresponds via Jacquet–Langlands to a quaternionic cusp form φ of level N such that
φ 1 mod pr for a prime p of Q above p. Then we may normalize Lalg(1,f,χ) so that (i)
≡
it is an algebraic integer, and (ii) there exist an algebraic integer c invertible mod p such
φ
that, for any imaginary quadratic field K inert at each prime dividing N and any ideal class
character χ of K, we have
c 2h2 mod pr χ = 1,
Lalg(1,f,χ) | φ| K
≡ (0 mod p χ = 1.
6
Given f, the normalization of Lalg(1,f,χ) depends on K and φ in a simple way, and
does not depend on χ—see (3.5). The integer c can also be determined in a simple way
φ
from φ.
Vatsal and Heumann [Vat99], [HV14] show that if f is congruent to E mod pr, p > 2
2,N
and p ∤ N, then there is a congruence of special values as above. In fact they treat higher
weights, other Eisenstein series and non-central special values (when k 4). However their
≥
normalization of algebraic parts of special values makes use of canonical periods, which are
onlydefineduptop-adic units,whereasournormalization determinesLalg(1,f,χ)uniquely.
On the other hand, we do not require any restriction on p, and we can easily treat Hilbert
modular forms also.
Our method is similar in spirit to Gross’s approach [Gro87], and also an approach by
Quattrini [Qua11] using half-integral weight modular forms. (These works also make use of
definite quaternion algebras.) In fact we get a period congruence, which is stronger than
the L-value congruence as the period also accounts for the sign of the “square root” of
Lalg(1,f ).
K
Themaindeficiencyinourresultisthatwerequireacongruenceofquaternionic modular
forms, which is a priori stronger than a congruence of elliptic (or Hilbert) modular forms.
However, we expect that these two notions of congruence are equivalent (see Remark 2.4
and (3.3)). This issue is also present in Quattrini’s approach, who used work of Emerton
[Eme02] to show these notions are equivalent when r = 1 and N is prime [Qua11, Thm
3.6]. Hence our Theorem B covers the case addressed in [Maz79]. We can also show that,
for r = 1, under the hypotheses of Theorem A, if there is a unique cuspidal newform
f E mod p, then it corresponds to a quaternionic form φ 1 mod p. The difficulty
2,N
≡ ≡
in general is separating quaternionic eigenforms with congruent eigenvalues.
Now we outline the contents and briefly remark on other related literature.
In Section 1, we explain some preliminaries on weight 2 quaternionic modular forms
for totally definite quaternion algebras B over totally real number fields F. These will be
functions on the finite set of ideal classes Cl( ), for some order of B.
O O
4
In Section 2, we use the Eichler mass formula to show the existence of a quaternionic
cusp form φ congruent to the constant function 1 modulo suitable primes. Then we apply
the Jacquet–Langlands correspondence to get a Hilbert cusp form f corresponding to φ
whose Hecke eigenvalues are congruent to those of a Hilbert Eisenstein series associated
to the constant function on Cl( ). This (Theorem 2.1) is the first main result, which
O
specializes to the Hecke eigenvalue congruence statements in Theorem A.
Wenoterecent workofBerger, KlosinandKramer[BKK14]gives analgebraic approach
to counting congruences of Hecke eigenvalues in more general settings, which yields a re-
finement of Mazur’s congruence of Fourier coefficients result for S (N) with N is prime. In
2
the case of Hilbert modular forms, some results are already known about finding primes of
congruence between two cusp forms, e.g., the work of Ghate [Gha02] generalizing a result
of Hida [Hid81] for elliptic cusp forms, but we are not aware of results along the lines of
Theorem2.1guaranteeingtheexistenceofHilbertcuspformscongruenttoEisensteinseries.
Then in Section 3, we use Waldspurger’s formula [Wal85] relating central L-values to
periodsonquaternionalgebras. Thisimmediately gives thenonvanishingL-valuestatement
inTheorem A.TogettheprecisecongruenceinTheorem B,weuseamoreexplicitversionof
Waldspurger’s formula from [MW09]. That L-value formula applies to arbitrary quadratic
extensions K/F (F totally real or not), but we need to restrict to K/F CM here in order
for the relevant period to lie on the definite quaternion algebra B. Again, we are not aware
of such L-value congruences in the Hilbert modular case.
On the other hand, there are some existence results of a different nature about even
weight cusp forms with Lalg(1,f,χ) 0 mod p using average value formulas, e.g., the
6≡
aforementioned work of Michel and Ramakrishnan [MR12] for elliptic cusp forms and joint
work of File and Pitale with the present author [FMP] for Hilbert cusp forms (though
excluding parallel weight 2 for simplicity). These results are of just the form that for a
given pandany suitably largelevel N,thereexists somef S (N)such thatLalg(1,f,χ)
k
∈ 6≡
0 mod p, but the bound on the level depends on K. I am not aware of any other results on
thevanishingof Lalg(1,f,χ) modpfor χnontrivial whenthesign of thefunctionalequation
is +1.
This project grew out of several discussions with Dinakar Ramakrishnan about nonvan-
ishing L-values, and I am grateful to him for leading me to think about these things. This
work was done in part while visiting Osaka City University with the support of a JSPS
Invitation Fellowship (Long Term, L14518), and in part with support from a Simons Col-
laboration Grant. I also thank Nicolas Billerey, Masataka Chida, Lassina Demb´el´e, Ariel
Pacetti and Hwajong Yoo for helpful comments and directions to related literature. I am
especially grateful to thereferee for a careful readingand findinga gap in an earlier version.
1 Quaternionic modular forms
Throughout, we fix the following notation. Let F be a totally real number field of degree
d, and o its ring of integers. Let N = N N be a nonzero integral ideal in o such that
F 1 2 F
N = pe1 per and N = qf1 qfs where the p ’s and q ’s are distinct prime ideals of o ,
1 1 ··· r 2 1 ··· s i j F
e ,f N, each e is odd, and r d mod 2. Note that N can be any nonzero ideal when
i j i
∈ ≡
d is even, and N can be any ideal which is not a square when d is odd. The letter ℓ will
5
denote a finite prime of F.
Denote by B the unique (up to isomorphism) totally definite quaternion algebra over F
ramified at each p and no other finite prime. Let be an order of level N in B such that
i
is conjugate to R (qfi), the subring of M (o O) with lower left entry 0 mod qfi, for
Oqi 0 i 2 F,qi i
1 i s. Note this is an Eichler order if N is squarefree—otherwise it is the intersection
1
≤ ≤
of an Eichler order of level p N with an order, which is unique up to isomorphism, of
i 2
·
level N .
1
Q
We want to work with a certain space of automorphic forms on B×. Adelically, we will
be looking at functions on
Cl( )= B× Bˆ×/ˆ× = B× B×(A )/( ˆ× B×). (1.1)
O \ O \ F O × ∞
Here Cl( ) can naturally be viewed as the (finite) set of invertible (locally principal) right
O
ideal classes of . We denote the class number #Cl( ) by h( ). Let , , be a set
1 h
O O O I ··· I
of representatives for the right ideal classes of , and let x ,...,x be a corresponding set
1 h
of representatives for the double coset classes BO× Bˆ×/ˆ×.
\ O
We consider the space of (weight 2, or weight 0, depending on convention) quaternionic
modular forms on defined by
O
M( ) = φ :Cl( ) C ,
O { O → }
a complex vector space of dimension h( ). To work adelically, we will typically view
O
φ M( ) as a function on Bˆ× which is left invariant by B× and right invariant by ˆ×.
∈ O O
For simplicity, we will work with the subspace of forms with trivial central character:
= M( ,1) = φ M( ) : φ(z) = φ(1) for z Fˆ× . (1.2)
M O { ∈ O ∈ }
Note that M( ,1) = M( ) if F has class number 1, because in this case Fˆ× = F׈o×
O O F ⊂
B× ˆ×. In general is a complex vector space of dimension h( ) (m 1) where m is
O M O − −
size the image of the map Cl(o ) Cl( ) induced from inclusion A× B×(A ), so we
F → O F → F
always have h( ) h +1 dim h( ). When h is odd, composingwith the reduced
F F
O − ≤ M≤ O
norm shows the map Cl(o ) Cl( ) is injective so dim = h( ) h +1.
F F
→ O M O −
We make into an inner product space by defining
M
(φ,φ′) = φ(g)φ′(g)dg = (4π2)d φ(g)φ′(g)dg,
ZA×FB×\B×(AF) ZFˆ×B×\Bˆ×
wheredg denotestheHaar measureontherelevant quotient inducedby theproductof local
Tamagawa measures on B× and F× and the counting measure on B×. The inner product
v v
converges by compactness.
We can write Fˆ×B×x ˆ× = Bˆ×zx ˆ×, where z Fˆ×/(Fˆ× B×x ˆ×x−1). The
iO z iO ∈ ∩ iO i
latter setis finiteof sizeat mosth sinceF׈o× Fˆ× B×x ˆ×x−1. Denote its cardinality
FF F ⊂ ∩ iO i
by c .
i
Since φ and φ′ are right invariant by ˆ×, we may write
O
(φ,φ′)= ω−1φ( )φ′( ), φ,φ′ M( ),
i Ii Ii ∈ O
i
X
6
where φ( ) = φ(x ) and
i i
I
ω = (4π2)−dvol(Fˆ×B× Fˆ×B×x ˆ×)−1c .
i i i
\ O
Note
c
vol(Fˆ×B× Fˆ×B×x ˆ×)= vol(Fˆ×B× Fˆ×B×x ˆ×x−1)= vol(ˆo×B× B×xˆ ×x−1) i .
\ iO \ iO i F \ iO i h
F
Since ( )= x ˆ×x−1 B×, we have
Oℓ Ii iO i ∩
[ ( )× :o×] h
ω = Oℓ Ii F F . (1.3)
i vol(ˆ×/ˆo×) (4π2)d
O F
One can obtain the following Tamagawa volume computations from [Vig80].
1
First, vol(o× ) = ∆−2 for v < .
Fv Fv ∞
For finite v ∤ N, we have
vol( ×/o× )= (1 q−2)vol(o× )3 = L(2,1 )−1vol(o× )3.
Ov Fv − v Fv Fv Fv
For v = p N , we have
i 1
|
vol( ×/o× ) = (1 q−1)−1q−eiL(2,1 )−1vol(o× )3.
Ov Fv − v v Fv Fv
For v = q N , we have that
j 2
|
[GL (o ) :R (qfj)×] = qfj(1+q−1),
2 F,qj 0 j v v
and therefore
vol( ×/o× )= q−fj(1+q−1)−1L(2,1 )−1vol(o× )3.
Ov Fv v v Fv Fv
Putting together the local measures gives
−3/2
∆ 1 1
vol(ˆ×/ˆo×) = F . (1.4)
O F N(N)ζF(2) 1 qv−1 1+qv−1
vY|N1 − vY|N2
Note that computing (φ,φ) for the constant function φ = 1 gives
0
ω−1 = (φ ,φ )= (2π)2dvol(Fˆ×B× Bˆ×) = vol(B×A× B×(A )) = 2.
i 0 0 \ F\ F
If we pXut w = [ ( )× : o×] = (2π)2dvol(Oˆ×/ˆo×F)ω and define the normalized paring
i Oℓ Ii F hF i
h
[φ,φ′] = F (φ,φ′)= w−1φ( )φ′( ), φ,φ′ M( ), (1.5)
(2π)2dvol(ˆ×/ˆo×) i Ii Ii ∈ O
O F X
then we recover a generalized form of the usual Eichler mass formula,
m( ) = w−1 = [φ ,φ ] = 21−2dπ−2d ∆ 3/2h ζ (2)N(N) (1 q−1) (1+q−1)
O i 0 0 | F| F F − v v
X vY|N1 vY|N2
= 21−dh ζ ( 1)N(N) (1 q−1) (1+q−1). (1.6)
F| F − | − v v
vY|N1 vY|N2
7
Here N(N) is the level (reduced norm) of N. The rational number m( ) is called the mass
O
of . When F = Q, N = (N ) and N = (N ), this simplifies to
1 2 2
O
ϕ(N )
m( )= w−1 = 1 N (1+p−1). (1.7)
O i 12 2
X pY|N2
Now we want to define the Eisenstein and cuspidal subspaces of . The Eisenstein
M
space will be generated by the one-dimensional representations of Bˆ×, which all factor
through the reduced norm map N : Bˆ× Fˆ+. The reduced norm induces a surjective map
→
N : Cl( ) Cl+(o ) to the narrow class group of F. Define the Eisenstein subspace of
F
O → E
to be the subspace of all φ which factor through N, and the cuspidal space to
M ∈ M S
be the orthogonal complement of in . We can describe in terms of our normalized
E M S
pairing (1.5) by
= φ : [φ,ψ N]= 0 for all characters ψ :Cl+(o )/Cl(o ) C .
F F
S { ∈ M ◦ → }
One has in the usual way Hecke operators T for each prime ℓ ∤N, which commute with
ℓ
each other and are self-adjoint with respect to the inner product. One can also describe
the action on (or M( )) in terms of (generalized) Brandt matrices. Hence , and
M O M
also , has a basis consisting of eigenforms for each such T . Via the Jacquet–Langlands
ℓ
S
correspondence, each eigenform φ transfers to an eigenform f S (N), the space of
2
∈ S ∈
parallel weight 2 Hilbert modular forms of level N, with the same Hecke eigenvalues away
from N.
Suppose now ℓ N but ℓ2 ∤ N . Then one can define a Hecke operator by (T φ)(x) =
1 1 ℓ
|
φ(x̟ )where̟ isauniformizerforB . Thiscorrespondstothedoublecoset ̟ =
Bℓ Bℓ ℓ Oℓ BℓOℓ
̟ . Now one has bases for and which are eigenforms for all T with ℓ ∤ N and
BℓOℓ M S ℓ 2
ℓ2 ∤ N , and the Jacquet–Langlands lift of such an eigenform φ will be a modular form
1
f S (N) which, at all such ℓ, is new and is an eigenform for T with the same eigenvalue
2 ℓ
∈
as φ.
We refer the reader to [Hid06, Chap 2] or [DV13] for details.
2 Congruence with Eisenstein series
We keep the notation of the previous section.
For f M (N) (resp. φ ) which is an eigenfunction of the Hecke operator T , let
2 ℓ
∈ ∈ M
λ (f) (resp. λ (φ)) denote the corresponding eigenvalue.
ℓ ℓ
Let E be the normalized Eisenstein series on M (N) with Hecke eigenvalues λ (E) =
2 ℓ
N(ℓ)+1 when ℓ ∤ N and λ (E) = 1 when ℓ N. When F = Q, this is the E explicitly
ℓ 2,N
|
constructed in the introduction. Recall φ = 1 . Then φ is an eigenform for the Hecke
0 0
∈E
operators T because the Brandt matrices have constant row sums, and in fact λ (E) =
ℓ ℓ
λ (φ ) when ℓ ∤ N and ℓ2 ∤ N .
ℓ 0 2 1
The field of rationality of an eigenform φ is the field Q(φ) generated by its Hecke
∈ M
eigenvalues. We may normalize φ so that all values of φ are integers in Q(φ), in which case
we say φ is integral. Let Q(N) be the compositum of the fields of rationality Q(φ) of all
eigenforms in φ . Note this is a subfield of the compositum of all Q(f) where f ranges
∈ M
over eigenforms in S (N).
2
8
Ifφ,φ′ areintegral andpis anideal of asuitablerationality field, wewriteφ φ′ mod p
≡
if φ( ) φ′( ) mod p for all Cl( ). If φ,φ′ are eigenforms, then φ( ) φ′( ) mod p
I ≡ I I ∈ O I ≡ I
implies their Hecke eigenvalues are also congruent mod p because the Hecke operators act
by integral Brandt matrices. We expect the converse to generally be true, though do not
know how to show it. See Remark 2.4 and (3.3) below.
Denote by h+ the narrow class number of F.
F
Theorem 2.1. Assume h+ = h . Let num be the numerator of m( ). Suppose p num
F F O |
and p is a prime above p in Q(N). Then there exists an eigenform f S (N) such that
2
∈
λ (f) λ (E) mod p for all ℓ with ℓ ∤ N and ℓ2 ∤ N . If N = N is squarefree, we may
ℓ ℓ 2 1 1
≡
take f to be a newform.
Proof. First we show there exists an integral φ such that φ 1 mod p. Consider
∈ S ≡
φ M( ) integral with φ 1 mod p so, for each Cl( ), φ( ) = 1+pa for some
i i i
∈ O ≡ I ∈ O I
a Z. Order our ideals so that , , represent each of the ideal classes in the image
i 1 m
∈ I ··· I
of the map Cl(o ) Cl( ). Recall m h . Then, by (1.2), φ just means: (i)
F F
→ O ≤ ∈ M
a = = a . Moreover, φ if in addition [φ,φ ]= 0, i.e., φ if and only if (i) holds
1 m 0
and w·e··have (ii) m( )= p∈ Sh w−1a . ∈ S
O − i=1 i i
We claim there exist a Z such that these conditions hold. Let w = h w and
i ∈P i=1 i
w∗ = w. We can rewrite (ii) as w∗ = p w∗a . To also account for (i), put w′ =
i wi i − i i Q 1
w∗ + + w∗ , w′ = w∗ , b = a , and b = a for 2 i k := h m.
1 ··· m i i+m−1 1P 1 Pi i+m−1 ≤ ≤ −
Note = means h > m, i.e., k 2. Then (i) and (ii) are achievable if and only if
k Mw′ =6 Ep k b w′ is solvable fo≥r b Z. For this, it suffices to show gcd(w′, ,w′)
i=1 i − i=1 i i i ∈ 1 ··· k
divides p−1 k w′, which is obvious at primes away from p. Say pj gcd(w′, ,w′).
P iP=1 i k 1 ··· k
Then p num implies wp w′ = wm( ), and therefore pj+1 w′. This proves the claim
| P | i O | i
and gives us our desired φ. Note this argument also shows that p num implies = .
P P | M6 E
Now let Φ be the set of integral φ which are congruent to a nonzero multiple of
∈ S
φ mod p. Fix a basis of eigenforms φ , ,φ of . Let r be minimal such that, after a
0 1 s
··· S
possible reordering of the φ ’s, there exists φ Φ with φ = c φ + +c φ , c Q(N).
i 1 1 r r i
∈ ··· ∈
Say φ cφ mod p. Then, for ℓ such that ℓ ∤ N and ℓ2 ∤ N ,
0 2 1
≡
[T λ (φ )]φ (λ (φ ) λ (φ ))cφ mod p,
ℓ ℓ j ℓ 0 ℓ j 0
− ≡ −
and thus [T λ (φ )]φ Φ unless λ (φ ) λ (φ ) mod p. (Note [T λ (φ )]φ Φ is also
ℓ ℓ j ℓ 0 ℓ j ℓ ℓ j
− ∈ ≡ − ∈
integral because T φ is, so it makes sense to consider this mod p.) However, [T λ (φ )]φ
ℓ ℓ ℓ j
−
is a linear combination of φ ,...,φ ,φ ,...,φ , which would contradict the minimality
1 j−1 j+1 r
of r if [T λ (φ )] Φ. Hence λ (φ ) λ (φ ) mod p for all ℓ as above and all 1 j r.
ℓ ℓ j ℓ 0 ℓ j
− ∈ ≡ ≤ ≤
Since f has the same Hecke eigenvalues as φ for T with ℓ as above, this yields the
ℓ
theorem.
Remark 2.2. (a) The reason we assume h+ = h is to guarantee the existence of φ
F F ∈ S
such that φ 1 mod p. If h+ = h , one gets an additional linear constraint [φ,ψ N]= 0
≡ F 6 F ◦
on the a ’s in the proof for each nontrivial character ψ of Cl+(o )/Cl(o ). Then, at least
i F F
a priori, one needs to place some conditions on the w ’s, m( ) and p to guarantee the
i
O
existence of a Z solving this system of Q-linear equations.
i
∈
(b) When h+ > 1, there are other Eisenstein series in M( ), and one might ask for
F O
congruences of these Eisenstein series as well. However, such Eisenstein series which are
9
also eigenforms can be obtained by twisting φ by narrow ideal class characters of F, and
0
analogous congruences are just obtained by twisting both E and f.
We observe that there are often multiple choices for B for a given N. For instance, if
F = Q and N = 11 13, we get num = 5 7 if we take N = 11, N = 13 and num = 12 if
1 2
· ·
we take N = 13, N = 11. Thus one gets congruences of cusp forms in level 11 13 with
1 2
·
Eisenstein series modulo the primes p = 2,3,5,7.
Some examples with F = Q were given in the introduction. Here is a simple example
with F = Q.
6
Example 2.3. Let F = Q(√5), which has narrow class number 1. Suppose N = N is one
2
of the two prime ideals with N(N) = 31. Then ζ ( 1) = 1 and m( ) = 32 = 8 , so we
F − 30 O 60 15
take p = 2. Here S (N) is one dimensional and has rationality field Q. Then for nonzero
2
f S (N) our theorem says that λ (f) N(ℓ)+1 mod 2 for ℓ = N. It is also true that
2 ℓ
∈ ≡ 6
λ (f) 1 mod 2. Indeed, the first few nonzero Hecke eigenvalues λ (f) satisfy the values
N ℓ
≡
listed in the following table:
N(ℓ) 4 5 9 11 11 19 19 29 29 31 31 41 41
λ (f) 3 2 2 4 4 4 4 2 2 1 8 6 6
ℓ
− − − − − − − − −
Remark 2.4. Suppose N = N is squarefree. If f S (N) such that f E mod p, we
1 2
∈ ≡
expect (cf. (3.3) below) that f corresponds to a quaternionic form φ such that, after
∈ S
normalization, φ 1 mod p. Let p be the rational prime below p. For such a cusp form
≡
φ to exist, we need w[φ,φ ] w[φ ,φ ] 0 mod p where as before w = w . Thus if
0 0 0 i
≡ ≡
(p,w) = 1, a congruence f E mod p should only exist if p num as in the theorem. If
≡ | Q
F = Q, then no prime > 3 divides w, and this agrees with the converse obtained by Ribet.
We also note that when p num, if there is a unique newform f S (N) such that
2
| ∈
f E mod p, the above proof implies f corresponds to a quaternionic form φ with
≡ ∈ S
φ 1 mod p, as we must have r = 1 in the final paragraph. When r = 1, F = Q and
≡
N = N is prime, this is also true by [Qua11], [Eme02].
3 Quadratic twist L-values
We keep our previous notation, but now assume is a maximal order of B, i.e., N = N
1
O
and N is squarefree. For consistency and clarity, we will denote complete L-functions by
L∗(s, ) and incomplete (i.e., the finite part of) L-functions just by L(s, ). However, we
− −
follow the usual convention that automorphic L-functions are normalized so that s = 1/2 is
the central point, whereas L-functions for f S (N) are normalized classically with s = 1
2
∈
the central point.
Let K/F be a CM quadratic extension such that each pN is inert in K. This means
|
K embeds in B and, since we are now assuming is maximal, we may fix an embedding
O
of K into B such that o . Hence, to each t Cl(o )= K× Kˆ×/ˆo× we can associate
K ⊂ O ∈ K \ K
the quaternionic ideal class x(t) Cl( ) via x(t) = B×t ˆ×. Fix a character χ of Cl(o ).
K
∈ O O
Put the product of local Tamagawa measures on A× and A×, and the counting measure on
F K
10