INDIVISIBILITY OF SPECIAL VALUES OF L-FUNCTIONS AND ORDERS OF SELMER GROUPS FOR MODULAR FORMS

MASATAKA CHIDA

Abstract. In this article we consider the propblem of indivisibility of special values of L-functions and orders of Selmer groups for modular forms. Then we give a generalization of a result of Kohnen and Ono [12] and James and Ono [9] to higher weight modular forms. We also discuss about this problem for non-central values of L-functions using congruences of modular forms.

1. Introduction

In this section we give a review of recent results on indivisibility of class numbers of quadratic fields and orders of Selmer groups for quadratic twists of elliptic curves.

For a number field K, let h(K) be the class number of K. For a quadratic field ℚ(\sqrt{D}), we put h(D) = h(ℚ(\sqrt{D})). Let p be a prime number. The research of distribution of class numbers in several families of number fields is a interesting problem. Especially, we recall some results on p-indivisibility of class numbers in the family of imaginary or real quadratic fields. Cohen and Lenstra [5] gave heuristics describing the expected behavior of h(D) and they predicted that the density that p ∤ h(D) is ∏_{k=1}^{∞}(1 − p^{−k}). For this problem, Kohnen and Ono [12] proved that if p > 3 is a prime and ε > 0, then for all sufficiently large X > 0

#\{0 < D ∈ S(X) | p ∤ h(−D)\} ≥ (\frac{2(p − 2)}{\sqrt{3}(p − 1)} − ε) \frac{\sqrt{X}}{log X} ,

where

S(X) := {D ∈ ℤ | |D| < X, D : square-free}.

For real quadratic fields, Byeon [3] proved

#\{0 < D ∈ S(X) | p ∤ h(D)\} ≫ \frac{\sqrt{X}}{log X} .

Also we can consider the same problem for Selmer groups or Tate-Shafarevich groups for the family of quadratic twists of elliptic curves.

Let E be an elliptic curve over ℚ with conductor N which is defined by the Weierstrass equation

E : y^{2} = x^{3} + ax + b,

where a,b are rational integers. Then we define the L-function for E by

L(E, s) = ∑_{n=1}^{∞} \frac{a(n)}{n^{s}} := ∏_{p∤N}(1 − a(p)p^{−s} + p^{1−2s})^{−1} ∏_{p|N}(1 − a(p)p^{−s})^{−1},

where a(p) = 1 + p − #E(𝔽_{p}) for good prime p. The p-Selmer group for E is defined by

Sel_{p}(E/ℚ) := Ker [H^{1}(ℚ, E[p^{∞}]) → ∏_{l≦∞} \frac{H^{1}(ℚ_{l}, E[p^{∞}]))}{E(ℚ_{l}) ⊗ ℚ_{p}/ℤ_{p}} ] ,

where

E[p^{∞}] := \underrightarrow{lim}_{n} E[p^{n}] = \underrightarrow{lim}_{n} (Ker[p^{n} : E(\overline{ℚ}) → E(\overline{ℚ})]).

For a square-free integer D, let E_{D} denote the D-quadratic twist of E that is given by the equation

E_{D} : y^{2} = x^{3} + aD^{2}x + bD^{3}.

Then Kohnen and Ono predicted that

#{D ∈ S(X) | #Sel_{p}(E_{D}/ℚ) = 1} ≫ X

as same as the case of class numbers of quadratic fields. For this problem, Kohnen and Ono [12] and James and Ono [9] proved that if E/ℚ is an elliptic curve, then for every prime p ≫_{E} 1

#\{D ∈ S(X) | #Sel_{p}(E_{D}/ℚ) = 1\} ≫_{E,p} \frac{\sqrt{X}}{log X} .

This result follows from a Kolyvagin’s result on the bounds of Tate-Shafarevich groups of elliptic curves and the indivisibility of central values of L-functions associated to E_{D} using Waldspurger’s formula for Fourier coefficients of modular forms of half integral weight.

2. Statement of result

Here, we consider the generalization of the result of [12] and [9] to the case of higher weight modular forms.

Let f = ∑_{n=1}^{∞} a(n)q^{n} be a normalized newform of weight 2k for Γ_{0}(N). We put K = ℚ({a(n)}), then it is known that K is a finite extension of the rational field ℚ. Let λ be a finite place of K. For a rational integer D, we define the D-quadratic twist of f by

f ⊗ χ_{D} = ∑_{n=1}^{∞} a(n)χ_{D}(n)q^{n},

where χ_{D} is the quadratic character for ℚ(\sqrt{D})/ℚ. Then f ⊗ χ_{D} is a normalized newform of weight 2k with respect to Γ_{0}(D^{2}N). For a newform f, we can define the Selmer group H_{f}^{1}(ℚ, A_{λ}(f ⊗ χ_{D}, k)) as a generalization of the case of elliptic curves. We give the definition of Selmer groups for modular forms in the next section.

The following theorem is the main result of [4].

Theorem 2.1. Let f be a normalized newform of weight 2k for Γ_{0}(N). For almost all finite places λ of K we have

#\{D ∈ S(X) | #H_{f}^{1}(ℚ, A_{λ}(f ⊗ χ_{D}, k)) = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

In this note, we introduce a proof of this theorem for non-CM case in the sense of Ribet [17]. Here if there exists a Hecke character ψ of an imaginary quadratic field F such that L(f, s) = L(ψ, s), we say that f has complex multiplication by F.

3. The Selmer groups for modular forms

We introduce the definition of Selmer groups for modular forms according to Bloch-Kato [2]. Let f = ∑_{n=1}^{∞} a(n)q^{n} be a normailzed newform of weight 2k, level N. We put K = ℚ({a(n)}). Then Deligne constructed the Galois representations associated to modular forms.

Theorem 3.1 (Deligne [6]). Let λ be a finite place of K above p. Then there exists 2-dimensional vector space V_{λ} = V_{λ}(f) over K_{λ} and a continuous homomorphism

ρ_{f,λ} : Gal(\overline{ℚ}/ℚ) → Aut(V_{λ})

satisfying that trace ρ_{f,λ}(Frob_{ℓ}) = a(ℓ) and det ρ_{f,λ}(Frob_{ℓ}) = ℓ^{2k−1} for any prime ℓ not dividing pN.

We fix a Gal(\overline{ℚ}/ℚ)-stable lattice T_{λ} of rank 2 in V_{λ} and put V_{λ}/T_{λ}. We denote the j-th Tate twist of Gal(\overline{ℚ}/ℚ)-module M by M(j). Then Bloch-Kato [2] defined the “finite part” of Galois cohomology groups H^{1}(ℚ_{l}, V_{l}(j)) by

H_{f}^{1}(ℚ_{l}, V_{l}(j)) := \begin{cases}Ker(H^{1}(D_{l}, V_{λ}(j)) → H^{1}(I_{l}, V_{λ}(j))) & l ≠ p, \\ Ker(H^{1}(D_{l}, V_{λ}(j)) → H^{1}(D_{l}, V_{λ}(j)⊗_{ℚ_{l}}B_{cris})) & l = p, \end{cases}

where B_{cris} is a ring defined by Fontaine [7], D_{l} is the decomposition group at l and I_{l} is the inertia group at l. We set

H_{f}^{1}(ℚ, V_{λ}(j)) := Ker [H^{1}(ℚ, V_{λ}(j)) → ∏_{l≦∞} \frac{H^{1}(ℚ_{l}, V_{λ}(j))}{H_{f}^{1}(ℚ_{l}, V_{λ}(j))}]

and

H_{f}^{1}(ℚ_{l}, A_{λ}(j)) := π_{∗}H_{f}^{1}(ℚ_{l}, V_{λ}(j)),

where π is the projection V_{λ}(j) →^{π} A_{λ}(j) Then we define the Selmer group for T_{λ} to be

H_{f}^{1}(ℚ, A_{λ}(j)) := Ker [H^{1}(ℚ, A_{λ}(j)) → ∏_{l≦∞} \frac{H^{1}(ℚ_{l}, A_{λ}(j))}{H_{f}^{1}(ℚ_{l}, A_{λ}(j))}] .

and Tate-Sharevich group for T_{λ} to be

� (T_{λ}, j) := H_{f}^{1}(ℚ, A_{λ}(j))/π_{∗}H_{f}^{1}(ℚ, V_{λ}(j)).

One can show if the residue representation of ρ_{f,λ} is irreducible, T_{λ} is uniquely determined. Then we denote T_{λ}(f, j) := T_{λ}(j),A_{λ}(f, j) := A_{λ}(j) and � _{λ}(f, j) := � (T_{λ}, j).

4. The central values of L-functions and modular forms of half integral weights

Let M_{k}(N, χ) be the space of modular forms of weight k, level N with character χ and S_{k}(N, χ) the space of cusp forms of weight k, level N with character χ. We recall the following result about the relation between central values of twisted modular Lfunctions and Fourier coefficients of half integral weight modular forms which is a useful version of Waldspurger’s formula.

Theorem 4.1 (Ono-Skinner [16]). Let f(z) = ∑_{n=1}^{∞} a(n)q^{n} be a normalized newform of weight 2k, level M with trivial character and let δ ∈ {±1} be the sign of the functional equation of L(f, s) = ∑_{n=1}^{∞} a(n)n^{−s}, then there is a positive integer N with 4M | N, a Dirichlet character χ modulo N, a complex number Ω_{f} ∈ ℂ^{×} and a non-zero eigenform

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+1/2}(N, χ)

with the property that g(z) maps to a twist of f under the Shimura correspondence and if δD > 0, then

b(|D|)^{2} = \begin{cases}ϵ_{D} \frac{L(f⊗χ_{D},k)|D|^{k−1/2}}{Ω_{f}} & if (D, N) = 1, \\ 0 & otherwise, \end{cases}

where ϵ_{D} and b(n) are algebraic numbers in some finite extension of ℚ. Moreover for all but finitely many primes p, if D is a square-free integer for which

1. δD > 0,

2. (D, N) = 1,

then we have |L(f, χ_{D}, k)|D|^{k−1/2}/Ω|_{p} = |b(|D|)^{2}|_{p}, where | · |_{p} is an extension of multiplicative p-adic valuation.

Remark 4.2. In this theorem we can choose the complex number Ω_{f} as the period associated to modular symbols defined by Mazur, Tate and Teitelbaum [15].

5. Indivisibility of Fourier coefficients of half-integral weight modular forms

To get an estimate for the indivisility of Fourier coefficients of half integral weight modular forms, we need the following theorem.

Theorem 5.1 (Sturm [20], Theorem 1). Let

f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ M_{k}(N, χ)

be a half-integral or integral weight modular form for which the coefficients a(m) are algebraic integers contained in a number field K. Let v be a finite place of K and let

ord_{v}(f) = \begin{cases}+∞ & if a(n) ≡ 0 mod v for all n, \\ min\{ n | a(n) ≢ 0 mod v\} & otherwise \end{cases}

Moreover put

λ = \frac{k}{12}[Γ_{0}(1) : Γ_{0}(N)] = \frac{kN}{12} ∏_{p|N} \frac{p + 1}{p}.

Assume that

ord_{v}(f) > λ,

then ord_{v}(f) = +∞.

Remark 5.2 (cf. [9], Proposition 5). In [20], Sturm proved this theorem for integral weight modular forms with trivial character, but the general case follows by taking an appropriate power of f.

Lemma 5.3 (Shimura,[19]). Suppose

f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{k+1/2}(N, χ)

is a half integral weight cusp form and l is a prime. We define (U_{l}f)(z), (V_{l}f)(z) by

(U_{l}f)(z) = ∑_{n=1}^{∞} u_{l}(n)q^{n} = ∑_{n=1}^{∞} a(ln)q^{n},

(V_{l}f)(z) = ∑_{n=1}^{∞} v_{l}(n)q^{n} = ∑_{n=1}^{∞} a(n)q^{ln}.

Then

(U_{l}f)(z), (V_{l}f)(z) ∈ S_{k+1/2}(Nl, χ (\frac{4l}{·})).

The following lemma gives a estimate for indivisibility of Fourier coefficients of modular forms of half integral weight.

Lemma 5.4. Let f(z) = ∑_{n=1}^{∞} a(n)q^{n} be a normalized Hecke eigen newform of weight 2k, level M with trivial character. Put K = ℚ({a(n)|n ≧ 1}) and let

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+1/2}(N, χ)

be the eigenform given in Theorem 4.1. We fix a prime number p greater than 3. Assume that f does not have complex multiplication and for any primes λ in K above p, mod λ Galois representation associated to f

ρ_{f,λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O_{K}/λ)

is surjective. If there exists a negative integer D_{0} such that (D_{0}, N) = 1, ϵ = (\frac{D_{0}}{p} ) ≠

0 and |b(|D_{0}|)|_{λ} = 1, then

#\{D ∈ S(X) | (\frac{D}{p}) = ϵ, |b(D)|_{λ} = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

6 MASATAKA CHIDA

Proof. If we put

b_{0}(n) = \begin{cases}b(n) & if (n, Np) = 1 and (\frac{n}{p}) = ϵ, \\ 0 & otherwise, \end{cases}

then

g_{0}(z) = ∑_{n=1}^{∞} b_{0}(n)q^{n} ∈ S_{k+1/2}(Np^{2}, χ′).

If a(l) ≡ 0 mod λ, by the formula for the action of Hecke operator T (l^{2}) we find that

b(l^{2}n) + χ′(l)l^{k−1} (\frac{(−1)^{k}n}{l}) b(n) + χ′^{2}(l)l^{2k−1}b(n/l^{2}) ≡ 0 mod λ.

Now, for a prime λ and positive integers r, t, we define

T (λ, r, t) := {l : prime | a(l) ≡ 0 mod λ, l ≡ r mod t}.

Since Galois representation

ρ_{f,λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O_{K}/λ)

is surjective, Chebotarev density theorem implies

#T (λ, r, t, X) := #\{l ≤ X| l ∈ T (λ, r, t)\} ≫_{f,λ} \frac{X}{log X}

and for each l ∈ T (λ, r, t)

b(l^{2}n) ≡ −χ′(l)l^{k−1} (\frac{−n}{l}) b(n) − χ′^{2}(l)l^{2k−1}b(n/l^{2}) mod λ. (5.1)

Put κ = (k + 1/2)[Γ_{0}(1) : Γ_{0}(Np^{2})]/12 + 1. Now, we choose (r_{0}, t_{0}) satisfying the following properties.

1. Np^{2}|t_{0}, (r_{0}, t_{0}) = 1, χ′(r_{0}) = 1.

2. If l is a prime with l ≡ r_{0} mod t_{0}, then (\frac{−n}{l}) = −1 for each 1 ≤ n ≤ κ with (n, Np^{2}) = 1.

3. For each prime l ≡ r_{0} mod t_{0} we have (\frac{D_{0}}{l} ) = −1.

4. Each prime l ≡ r_{0} mod t_{0} has the property that 1 + l ≢ 0 mod λ.

If l ∈ T (λ, r_{0}, t_{0}) is a sufficiently large prime, for all 1 ≤ n ≤ κ

u_{l}(ln) = b_{0}(l^{2}n) ≡ −l^{k−1} (\frac{−n}{l}) b_{0}(n)−l^{2k−1}b_{0}(n/l^{2}) = l^{k−1}b_{0}(n) = l^{k−1}v_{l}(ln) mod λ.

By the relation 5.1, we have

v_{l}(l^{3}|D_{0}|) = b_{0}(l^{2}|D_{0}|) ≡ l^{k−1}b_{0}(|D_{0}|) mod λ,

also

u_{l}(l^{3}|D_{0}|) = b_{0}(l^{4}|D_{0}|) ≡ −l^{2k−1}b_{0}(|D_{0}|) mod λ.

Therefore by assumption and the choice of (r_{0}, t_{0}), l^{k−1}v_{l}(l^{3}|D_{0}|) − u_{l}(l^{3}|D_{0}|) ≡ l^{2k−2}(1 + l)b_{0}(|D_{0}|) ≢ 0 mod λ Hence

ord_{λ}(U_{l}g_{0} − l^{k−1}V_{l}g_{0}) < +∞.

By Theorem 5.1 and Lemma 5.3, there exists a integer n_{l} such that

1 ≤ n_{l} ≤ (k + 1/2)[Γ_{0}(1) : Γ_{0}(Np^{2}l)]/12 = κ(l + 1), (n_{l}, l) = 1

and

b_{0}(n_{l}l) = u_{l}(n_{l}) ≢ v_{l}(n_{l}) ≡ 0 mod λ.

Consequently, let D′ be the square-free part of D = n_{l}l, then

|b_{0}(D′)|_{λ} = 1.

For convenience, let l_{i} denote the primes in T (p, r_{0}, t_{0}) in increasing order, and let D_{i} denote the square-free part of l_{i}n_{l_{i}}. If r < s < t and D_{r} = D_{s} = D_{t}, then l_{r}l_{s}l_{t}|D_{r}. However this can only occur for finitely many r, s and t since |D_{i}| < κl_{i}(l_{i} + 1). Therefore, the number of distinct |D_{i}| < X is at least half the number of l ∈ T (p, r_{0}, t_{0}) with l ≤ \sqrt{X/κ}. Therefore the Lemma follows from

#T (r_{0}, t_{0}, X) ≫_{f} X/ log X.

6. Indivisibility of orders of Selmer groups

In this section we give the proof of main theorem for non-CM case. So we assume that f = ∑_{n=1}^{∞} a(n)q^{n} is a normalized newform of weight 2k, level N with trivial character without complex multiplication. We use the following result by Kato which is a generalization of Kolyvagin’s results [13], [14].

Theorem 6.1 (Kato [10], Proposition 14.20., Theorem 14.5. (3)). Assume L(f, k) ≠ 0. Suppose λ is a prime satisfying λ ∤ 2Nk! and

SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f,λ} : Gal(\overline{ℚ}/ℚ(μ_{p^{∞}})) → GL_{O_{λ}}(T_{λ}) ≅ GL_{2}(O_{λ})],

where O_{λ} is the integer ring of K_{λ}.

Then there exists a Gal(\overline{ℚ}/ℚ)-stable O_{λ}-lattice T_{λ} of rank 2 in V_{λ} such that

#H_{f}^{1}(ℚ, A_{λ}(k)) ≦ [O_{λ} : L(f^{∗}, k)/Ω_{f}] · #H^{0}(ℚ, A_{λ}(k))^{2},

where Ω_{f} is the period associated to modular symbols and f^{∗} is the dual cusp form introduced by Kato [10] which is also a normalized newform of weight 2k, level N.

For almost all finite places λ of K, the assumption of Theorem 6.1

SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f,λ} : Gal(\overline{ℚ}/ℚ(μ_{p^{∞}})) → GL_{2}(O_{λ})]

holds.

Since ℚ(μ_{p^{∞}}) does not include the quadratic feild ℚ(\sqrt{D}) with p ∤ D, if

SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f,λ}]

is satisfied, then we have

SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f⊗χ_{D},λ}]

for any sqare-free integer D with p ∤ D.

We note that A_{λ} depends on a choice of a lattice T_{λ}, but if the residue representation of ρ_{f,λ} is irreducible, this lattice is uniquely determined. Also Ω_{f} is uniquely determined up to units of O_{K}.

Now we choose a Gal(\overline{ℚ}/ℚ)-stable lattice T_{λ}⊗χ_{D} in V_{λ}(f ⊗χ_{D}) and put A_{λ}⊗χ_{D} := (V_{λ} ⊗ χ_{D})/(T_{λ} ⊗ χ_{D}).

For the global points H^{0}(ℚ, A_{λ}(k)), it is known that if there is a prime ℓ not dividing N such that ℓ^{k} +ℓ^{k−1} ≢ a(ℓ) mod λ, then we have H^{0}(ℚ, A_{λ}) = 0.(see Kato [10], Proposition 14.11.) This gives the following lemma.

Lemma 6.2. Let ℓ be a prime with ℓ ∤ N such that a(ℓ) ≠ 0. Then for any prime λ not dividing ±a(ℓ) + ℓ^{k} + ℓ^{k−1} and for any square-free integer D not divided by λ, we have H^{0}(ℚ, A_{λ} ⊗ χ_{D}(k)) = 0.

Combining these fact, we have the following proposition.

Proposition 6.3. Let f = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}(Γ_{0}(N)) be a normalized newform. Assume the following conditions.

1. λ does not divide 2Nk!.

2. The residue representation of ρ_{f,λ} is irreducible.

3. λ satisfies SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f,λ} : Gal(\overline{ℚ}/ℚ(μ_{p^{∞}})) → GL_{2}(O_{λ})] .

4. There exists a prime number ℓ with ℓ ∤ N satisfying λ ∤ (ℓ^{k} + ℓ^{k−1} ± a(ℓ)).

5. |L(f^{∗}, k)/Ω_{f}|_{λ} = 1. Then we have #H_{f}^{1}(ℚ, A_{λ}(f, k)) = 1

Now we give the proof of Theorem 2.1 for non-CM case. Let

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+1/2}(N, χ)

be the eigenform given in Theorem 4.1 for f^{∗}.

By replacing f^{∗} by a suitable quadratic twist of f^{∗} if necessary, we assume that δ = 1, where δ is the sign of the functional equation of L(f^{∗}, s). By the result of Friedberg and Hoffstein [8], we can take a negative integer D such that (D, 2N) = 1, and b(D_{0}) ≠ 0. Especially, for almost all finite places λ of K we have

|b(|D_{0}|)|_{λ} = 1.

By the Ribet’s result [18], if f does not have complex multiplication, for almost all finite place λ of K mod λ Galois representation

ρ_{f,λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{λ})

is surjective. Therefore by Lemma 5.4, we have

#\{D ∈ S(X) | p ∤ D, |b(D)|_{λ} = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

Then it is well-known that Ω(f ⊗ χ_{D}, k) = Ω(f, k)/\sqrt{D}. Therefore, for almost all finite places λ of K we have

#\{D ∈ S(X) | p ∤ D, |L(f^{∗} ⊗ χ_{D}, k)/Ω_{f⊗χ_{D}}|_{λ} = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

Moreover one can show that almost all finite places λ of K satisfies the assumption in Proposition 6.3. We fix such λ. Then for any squaer-free integer D with λ ∤ D, the following conditions hold.

1. λ does not divide 2NDk!.

2. The residue representation of ρ_{f⊗χ_{D},λ} is irreducible.

3. λ satisfies SL_{2}(ℤ_{p}) ⊂ Im[ρ_{f⊗χ_{D},λ} : Gal(\overline{ℚ}/ℚ(μ_{p^{∞}})) → GL_{2}(O_{λ})] .

4. There exists a prime number ℓ with ℓ ∤ ND satisfying λ ∤ (ℓ^{k} +ℓ^{k−1} +χ_{D}(ℓ)a(ℓ)). Then for almost all finite places λ of K, |b(D)|_{λ} = 1 leads #H_{f}^{1}(ℚ, A_{λ}(f ⊗χ_{D}, k)) = 1. Therefore by Proposition 6.3 we have

#\{D ∈ S(X) | #H_{f}^{1}(ℚ, A_{λ}(f ⊗ χ_{D}, k)) = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

Therefore we have the following.

Proposition 6.4. We assume f does not have complex multiplication and a finite place λ of K satisfies the conditions (1)–(5) in Proposition 6.3. Moreover we suppose that residue representation \overline{ρ}_{f,λ} is surjective and λ does not divide ϵ_{D} for any D as in Theorem 4.1. Then we have

#\{D ∈ S(X) | #H_{f}^{1}(ℚ, A_{λ}(f ⊗ χ_{D}, k)) = 1\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

Our main theorem follows from this proposition.

Remark 6.5. If N is odd and square-free, then we can use the Kohnen-Zagier formula instead of Theorem 4.1. By a result of Kohnen[11], there exists an eigenform

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+1/2}(Γ_{0}(4N))

such that if a square-free integer D satisfies (−1)^{k}D > 0 and (\frac{D}{p}) = ε_{p} for any

p | N,

|b(|D|)|^{2} = 2^{ν(N)} · \frac{(k − 1)!|D|^{k−1/2}}{π^{k}} · \frac{⟨g, g⟩}{⟨f, f⟩}L(f ⊗ χ_{D}, k),

where ε_{p} is the eigenvalue of Atkin-Lehner involution w_{p} and ⟨·, ·⟩ is the Petersson inner product and ν(N) is the number of prime divisors of N. If we use this formula and a certain normalization, then we don’t need a assumption that λ does not divide ϵ_{D} for any D.

7. Indivisibility for non-central values and congruences of modular forms

In this section, we consider a special case for non-central values of modular Lfunctions. We fix prime p greater than 7 and let f = ∑_{n=1}^{∞} a(n)q^{n} be a normalized Hecke eigenform of weight p + 1, level 1. We assume that the integer ring of an algebraic number field K contains all Fourier coefficients of f and choose a period

Ω(f) satisfying that for any Dirichlet character χ, the quotient τ(χ^{−1})\frac{L(f ⊗ χ, 1)}{Ω(f)^{±}} is

an algebraic integer in K(χ) where τ is the Gauss sum and ± = χ(−1).

Theorem 7.1. Let λ be a prime in K above p. We assume the following conditions.

1. There exists an unique eigenform F of weight 2, level p with trivial character

such that

F ≡ f mod λ.

2. The mod λ Galois representation associated to f is surjective.

3. There exists an square-free negative integer d_{0} such that (d_{0}, 2p) = 1, χ_{d_{0}}(p) = −ϵ(F), where ϵ(F) is the sign of functional equation of L(F, s) and

\frac{L(f ⊗ χ_{d_{0}}, 1)}{Ω(f)^{±}\sqrt{d_{0}}} ≢ 0 mod λ.

Then we have

#\{D ∈ S(X) | \frac{L(f ⊗ χ_{D}, 1)}{Ω(f)^{±}\sqrt{D}} ≢ 0 mod λ\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

For the proof, we recall a result of Ash and Stevens.

Theorem 7.2 (Ash-Stevens [1]). Let k be a positive integer less than p + 2 and f = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{k}(Γ_{0}(1)) an eigenform. We fix a prime λ above p in a number field K which contains all Fourier coefficients of f. Assume that

1. There exists a prime l satisfying a(l) ≢ l^{k−1} + 1 mod λ.

2. There exists an unique eigenform F ∈ S_{2}(Γ_{1}(p)) such that f ≡ F mod λ. Then there exists a complex number Ω(f)^{±} such that for any Dirichlet character χ satisfying (cond χ, p) = 1, we have

\frac{τ(χ^{−1})L(f ⊗ χ, 1)}{Ω(f)^{±}} ≡ \frac{τ(χ^{−1})L(F ⊗ χ, 1)}{Ω(F)^{±}} mod λ.

Now we prove Theorem 7.1. By the Kohnen-Zagier formula [11], there exists an eigenform

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{3/2}(Γ_{0}(4p))

such that for any negative square-free integer D satisfying (\frac{D}{p}) = −ϵ(F),

|b(|D|)|^{2} = 2 · \frac{\sqrt{D}}{π} · \frac{⟨g, g⟩}{⟨F, F ⟩}L(F ⊗ χ_{D}, 1),

where ⟨·, ·⟩ is the Petersson inner product. The surjectivity of mod λ Galois representation associated to f gives the existence of prime l satisfying a(l) ≢ l^{k−1} + 1 mod λ, so the assumption of Theorem 7.1 holds the assumption of theorem 7.2. Since τ(χ_{D}^{−1}) = ±1/\sqrt{D}, one can see

\frac{L(f ⊗ χ, 1)}{Ω(f)^{±}\sqrt{D}} ≡ \frac{L(F ⊗ χ, 1)}{Ω(F)^{±}\sqrt{D}} = |b(|D|)|^{2} · c mod λ

for some algebraic number c. By the assumption, one can show

ord_{λ} (\frac{L(f ⊗ χ_{d_{0}}, 1)}{Ω(f)^{±}\sqrt{d_{0}}}) = 1,

therefore s := ord_{λ}(b(d_{0})) is equal to min{ord_{λ}(b(n)) | n : square-free, (2p, n) = 1}, χ_{D}(p) = −ϵ(f). Hence by Lemma 5.4,

#\{D ∈ S(X) | χ_{D}(p) = −1, ord_{λ}(b(D)) = s\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

So we have

#\{D ∈ S(X) | \frac{L(f ⊗ χ_{D}, 1)}{Ω(f)^{±}\sqrt{D}} ≢ 0 mod λ\} ≫_{f,λ} \frac{\sqrt{X}}{log X} .

Example 7.3. Let

f = Δ = q ∏_{n=1}^{∞}(1 − q^{n})^{24} ∈ S_{12}(Γ_{0}(1))

and

F = q ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{11n})^{2} ∈ S_{2}(Γ_{0}(11)).

Then it is well-known that f ≡ F mod 11, dimS_{2}(Γ_{0}(11)) = 1 and the mod 11 Galois representation associated to f is surjective. Moreover one can check

\frac{L(Δ ⊗ χ_{−3}, 1)}{Ω(Δ ⊗ χ_{−3})^{+}} = 36741600 ≢ 0 mod 11

by using MAGMA. So the assumption of Theorem 7.1 is satisfied for f = Δ. Hence we have

#\{D ∈ S(X) | \frac{L(Δ ⊗ χ_{D}, 1)}{Ω(Δ)^{±}\sqrt{D}} ≢ 0 mod 11\} ≫ \frac{\sqrt{X}}{log X} .

Acknowledgment . The author is supported in part by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists.

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  E-mail address: sa2m22@math.tohoku.ac.jp

Institute of Mathematics, Tohoku University, Aoba, Sendai, Japan, 980-8578