ON A p-ADIC INTERPOLATION OF THE KOHNEN-ZAGIER FORMULA AND ITS APPLICATION TO THE GOLDFELD CONJECTURE
KENJI MAKIYAMA
1. Introduction
Let p be an odd prime, D a fundamental discriminant and χD :=
(D)
the Kronecker character attached to the quadratic field Q(√D). The Kohnen-Zagier formula introduced in [KZ81] states that the equation
(
the central L-value attached to the χD-twist of a primitive form f
) =
(
the |D|-th coefficient of the cusp form corresponding to f under the D-th Shimura correspondence
)2
holds, up to explicit factors in ¯Q. This type formula has been generalized in several cases (cf. [Sak08]). The left hand side above is essentially equal to the |D|-th coefficient of the D-th Shintani lifting of f. Thus, a p-adic interpolation of the|D|-th Shintani liftings implies a p-adic interpolation of the central L-values of χD-twists of primitive forms.
Stevens established a p-adic interpolation of original Shintani liftings introduced in [Shi75] for p-adic families with slope 0 (Hida families) in [Ste94]. Park generalized Stevens’s work to p-adic families with finite slope not necessarily 0 (Coleman families) in [Par10]. Following their methods, we will obtain a p-adic interpolation of D-th Shintani liftings defined in [KT04] for Coleman families (see Theorem 3.2).
In the last section, we will obtain an application to a conjecture of Goldfeld (see Theorem 4.1).
2. Preliminaries
Throughout the paper, we fix an odd prime p and a positive integer N satisfying p∤ N. We denote by ¯Q and ¯Qp an algebraic closure of the rational number field Q, and the p-adic number field Qp,
respectively. Let C be the complex number field. We take the p-adic completion Cp of ¯Qp. Then, we
fix two embeddings i∞ : ¯Q ,→ C and ip : ¯Q ,→ ¯Qp, and an isomorphism Cp −→ C which commutes∼
with i∞ and ip. Let ordp be the normalized p-adic additive valuation onCp so that ordp(p) = 1 and
| · |p the absolute value given by ordp. Then, we denote byO the subring of Cp consisting of elements
s with |s|p ≤ 1.
Let k and M be positive integers and χ a Dirichlet character modulo M . We denote by Sk(M, χ)
the space of holomorphic cusp forms of weight k with character χ on Γ0(M ). We denote by Sknew(M, χ)
the subspace of newforms at level M . For the usual Hecke operator Tp at p and a Hecke eigenform
f ∈ Sk(M, χ), we refer to ordp(ap(f )) as the Tp-slope of f . Let α be a non-negative rational number.
We denote by Sknew(M, χ)α the Tp-slope α subspace. Assume that M is odd. Let χD :=
(D)
be the
Date: March 31, 2014.
2 KENJI MAKIYAMA
Kronecker character attached to a quadratic field with discriminant D. Put ˜χ := χχ(−1)χ. We denote by Sk+1/2+ (4M, ˜χ) the Kohnen plus space, i.e.,
Sk+1/2+ (4M, ˜χ) :={g ∈ S2k+1(4M, ˜χ)| bn(g) = 0 if χ(−1)(−1)k+1n≡ 2, 3 (mod 4)},
(2-0-1)
where S2k+1(4M, ˜χ) is the space of cusp forms of half-integral weight k + 1/2 with level 4M and a character ˜χ modulo 4M in the sense of Shimura [Shi73, P.447]. Let D be a fundamental discriminant with χDχ(−1)(−1)k+1= 1 and (D, M ) = 1. We define the D-th Shimura lift Sk,χ,DM by
Sk,χ,DM : Sk+3/2+ (4M, ˜χ)→ S2k+2(M, χ2) ; g7→ ∑ n≥1 ∑ 0<d|n χD(d)χ(d)dkbn2|D|/d2(g) qn. (2-0-2)
Then the|D|-th Shintani lifting θk,χ,D(M ) := Sk,χ,DM,∗ is defined as the adjoint mapping with respect to the Petersson inner product. To ensure that θk,χ,D(M ) takes the image in the space of cusp forms, we always agree that one of the following is fulfilled:
(i) k≥ 1.
(ii) M is square-free.
(iii) M is cubic-free and χ = 1M,
where let 1M denote the trivial character modulo M .
3. Main results
Let f ∈ Sknew0 (N, ε)αbe a primitive form with k0−1 > α ̸= (k0−1)/2. We take the root αp(f ) of the
polynomial X2−ap(f )X + ε(p)pk0−1∈ O[X] satisfying ordp(αp(f )) = α. We define the p-stabilization
of f by f∗ := f−ε(p)p k0−1 αp(f ) · f|Vp , (3-0-1)
which is a Hecke eigenform in Sk0(N p, ε)α with the same eigenvalues as f outside p and the Tp
-eigenvalue ap(f∗) = αp(f ). Then we have the following theorem:
Theorem 3.1. (cf. [Col97, Corollary B5.7.1] and [Yam12, Corollary 2.3]) There exist a formal power series f = ∑n≥1anqn ∈ O⟨Xp−km00⟩[[q]] with a non-negative integer m0 and a primitive form fk ∈
Sknew(N, ε)α for each k ∈ K := {k ∈ Z | k ≡ k0 (mod (p− 1)pm0) , k− 1 > α} except at most one k′
different from k0 satisfying the following properties:
(i) fk0 = f .
(ii) f (k) := ∑n≥1an(k)qn = fk∗ and f (k′) is a primitive form in Sknew(N p, ε)α if k′ ∈ K is an
exceptional element.
We refer to {f(k)}k∈K as a Coleman family (Hida family when α = 0) passing through f∗ at k0.
In this section, we keep the notation in the theorem above.
From now on, we assume that N is odd, that k0 = 2˜k0+ 2 with some non-negative integer ˜k0 and
that ε = χ2 for a Dirichlet character χ. Let D be a fundamental discriminant with (D, N p) = 1 and
χDχ(−1)(−1) ˜
p-ADIC INTERPOLATION OF THE KOHNEN-ZAGIER FORMULA 3
Theorem 3.2. There exist a formal power series Θ =∑n≥1bnqn∈ eA[[q]] with a non-negative integer
˜
m0 and there exists an error termEk∈ C×p for each ˜k in
e K := {˜k ∈ Z | ˜k ≡ ˜k0 (mod (p− 1)pm0+ ˜m0/2ϵ) , 2˜k + 1 > α} with ϵ := { 1 if p≥ 5 0 if p = 3 (3-0-2)
satisfying the following properties: (i) Ek0 = 1.
(ii) Θ(˜k) :=∑n≥1bn(˜k)qn=E˜k· θ(N p)˜k,χ,D(f (2˜k + 2)),
where let eA denote the metaplectic algebra (cf. [Par10, Definition 4.14]) of the affinoid algebra A(B) associated with the affinoid disk B of radius p−(m0+ ˜m0) around k
0 defined overCp.
Hereafter, we assume that the conductor of χ is equal to N . Then we obtain the following: Proposition 3.3. Suppose that f (2˜k + 2) is p-old at ˜k∈ eK, that is, f(2˜k + 2) = f∗
2˜k+2 with a primitive
form f2˜k+2 ∈ Sknew(N, ε)α. Then we have
b|D|(θk,χ,DN p (f2k+2∗ )) = ( 2 + χD(p)χ(p)p−k−1(p− 1) + 2p−1 ) ( 1−χD(p)χ(p)p k αp(f2k+2) ) (3-0-3) × b|D|(θNk,χ,D(f2k+2)) = ( 2 + χD(p)χ(p)p−k−1(p− 1) + 2p−1 ) ( 1−χD(p)χ(p)p k αp(f2k+2) ) × |D|k+1/2N2k+1k!L (k + 1, f2k+2⊗ χDχ) πk+1 ,
where the last equation is a consequence of [KT04, (3-18) and (4-21)]. Finally, we obtain the following corollary:
Corollary 3.4. We have a p-adic L-function b|D| as the |D|-th coefficient of the formal power series Θ as in Theorem 3.2 which satisfies the following interpolation property:
b|D|(˜k) E2k+2 = ( 2 + χD(p)χ(p)p−k−1(p− 1) + 2p−1 ) ( 1−χD(p)χ(p)p k αp(f2k+2) ) (3-0-4) × |D|k+1/2N2k+1k!L (k + 1, f2k+2⊗ χDχ) πk+1 for each ˜k∈ eK. 4. Application
Let f ∈ S2knew(N, χ) be a primitive form. For X > 0, let F (X) denote the set of fundamental discriminants D satisfying|D| < X. A conjecture of Goldfeld, which was originally posed for ranks of elliptic curves in [Gol79], states that
Mf(X) := ♯{D ∈ F (X) | L(k, f ⊗ χD)̸= 0} ≫ X
(4-0-1)
(see [Ono04, Conjecture 9.10] for a more precise statement), that is, there exists a positive constant c such that, for sufficiently large X > 0, we have Mf(X)≥ cX. Vatsal proved that Mf(X)≫ X for
4 KENJI MAKIYAMA
good ordinary reduction at 3 (cf. [Vat99, Theorem 0.3]). By combining the result of Vatsal with main results, we obtain the following:
Theorem 4.1. For each non-negative integer k satisfying k ≡ 0 (mod 2 · 3m) with some non-negative integer m, there exists a primitive form f ∈ Snew
2k+2(N, 1N)0 satisfying Mf(X)≫ X.
Acknowledgement. The author is very grateful to the organizers of the summer school, in particular, Professor Harashita for the opportunities. He is also grateful to Professor Atsushi Yamagami for valuable guidance and kind help.
References
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Department of Mathematics, Kyoto Sangyo University, Kyoto, 603-8555, Japan E-mail address: kenji [email protected]
