VANISHING OF L-FUNCTIONS OF ELLIPTIC CURVES OVER NUMBER FIELDS

CHANTAL DAVID, JACK FEARNLEY, AND HERSHY KISILEVSKY

Abstract. Let E be an elliptic curve over ℚ, with L-function L_{E}(s). For any primitive Dirichlet character χ, let L_{E}(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions L_{E}(s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that L_{E}(1, χ) = 0, but that for any fixed prime k ≥ 7, there are only finitely many character of order k such that L_{E}(1, χ) vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions K/ℚ of prime degree such that E acquires new rank over K.

1. Introduction

Let E be an elliptic curve defined over ℚ with conductor N_{E}. For any number field K/ℚ, let E(K) be the group of points of E defined over K. By the Mordell-Weil Theorem, E(K) is a finitely generated abelian group. Let L_{E}(s, K) be the L-function of E over the field K.

Conjecture 1.1 (Birch and Swinnerton-Dyer conjecture over number fields). L_{E}(s, K) has analytic continuation to the whole complex plane, and

ord_{s=1}L_{E}(s, K) = r_{K}(E)

where r_{K}(E) is the rank of E(K).

In this paper, we fix E an elliptic curve over ℚ, and we study how the rank varies over abelian fields K/ℚ of fixed prime degree. For example, are there infinitely many such number fields where E acquires new rank over K (i.e. r_{K}(E) > r_{ℚ}(E))? With the Birch and Swinnerton-Dyer conjecture, one can rephrase the question in terms of vanishing of the L-function L_{E}(s, K) at s = 1. Let K be an abelian extension of ℚ with Galois group G and conductor m. Let Ĝ be the group of characters of G which can be identified with a set of Dirichlet characters

The first and third authors are partially supported by grants from NSERC and FCAR.

χ : (ℤ/mℤ)^{∗} → ℂ^{∗}.

Let

L_{E}(s) = L_{E}(s, ℚ) = ∑_{n≥1} \frac{a_{n}}{n^{s}}

be the L-function of E over ℚ. For each primitive Dirichlet character χ, let

L_{E}(s, χ) = ∑_{n≥1} \frac{χ(n) a_{n}}{n^{s}}

be the L-function of E over ℚ twisted by the character χ. By the work of [16, 15, 1], L_{E}(s) and L_{E}(s, χ) have analytic continuation to the whole complex plane. It also follows from properties of number fields that

L_{E}(s, K) = ∏_{χ∈Ĝ} L_{E}(s, χ), (1)

and the vanishing of the twisted L-functions L_{E}(s, χ) at s = 1 is equivalent, via the Birch and Swinnerton-Dyer conjecture, to the existence of rational points of infinite order on E(K).

In this paper, we use random matrix theory to study the vanishing of the twisted L-functions L_{E}(s, χ) at s = 1. It has been known since the work of Montgomery [13] that certain statistics on probability spaces of random matrices (as pair correlation between the eigenangles of the matrices) are similar to the same statistics on the zeroes of the Riemann zeta function. This intuition is supported by the extensive computations of Odlyzko [14] on the critical zeroes of the Riemann zeta function.

This was explored further in the work of Katz and Sarnak [8, 9], who extend the analogy between other probability spaces of matrices, and families of L-functions. Katz and Sarnak also studied the case of function fields, where they can actually prove some of those mysterious connections between random matrices and families of L-functions.

In order to study different statistics of number theoretic objects, Keating and Snaith [10, 11] introduced a new random variable on spaces of random matrices, the characteristic polynomial of the matrix evaluated at a given point. They computed the probability distribution of this new variable, which led to striking conjectures for the asymptotic behavior of the moments of the Riemann zeta function on the critical line. The ideas of Keating and Snaith have been applied to study vanishing of L-functions in families [3, 4, 17]. Families of quadratic twists are studied in [3], where a conjectural asymptotic for the number of quadratic twists with even non-zero rank is presented. This refines a conjecture of Goldfeld [6] which predicts that quadratic twists with rank greater than one have density zero. In [4], the authors used the ideas of Keating and Snaith to obtain a conjectural asymptotic for the number of cubic Dirichlet characters χ such that L_{E}(1, χ) vanishes. We present in this paper the case of characters of order k, for k any odd prime. More precisely, the conjectures that we obtain from the random matrix model are (the case k = 3 of [4] is included for completeness)

Conjecture 1.2. Let k be an odd prime, let E be an elliptic curve defined over ℚ, and let

N_{E,k}(X) = # {χ of order k : cond(χ) ≤ X and L_{E}(1, χ) = 0} .

If k = 3, then

log N_{E,k}(X) ∼ \frac{1}{2} log X as X → ∞.

If k = 5, then N_{E,k}(X) is unbounded, but N_{E,k}(X) ≪ X^{ϵ} for any ϵ > 0 as X → ∞.

If k ≥ 7, then N_{E,k}(X) is bounded.

In the light of (1), and under the Birch and Swinnerton-Dyer conjecture, one can rewrite N_{E,k}(X) as

N_{E,k}(X) = (k − 1) # {K/ℚ cyclic of degree k :
cond(K) ≤ X and r_{K}(E) > r_{ℚ}(E)} .

The structure of the paper is as follows. In the second section, we use modular symbols to rewrite the special values L_{E}(1, χ) as a product of terms depending only on E, and some algebraic integer n_{E}(χ) depending on the character. In the third section, we use the embedding of number fields as lattices in ℂ to give a discretisation of the algebraic integers n_{E}(χ). In the fourth section, we use this discretisation and the work of Keating and Snaith to obtain conjectures on the asymptotic behavior of N_{E,k}(X). Finally, the last section presents some experimental results.

2. Special values and modular symbols

The notation of this section follows the introduction of [12]. Let E be an elliptic curve over ℚ, and let f(z) = ∑_{n≥1} a_{n} e^{2πinz} be the Fourier expansion of E. Then, the L-function L_{E}(s) is the Mellin transform

(2)  L_{E}(s) = \frac{(2π)^{s}}{Γ(s)} ∫_{0}^{∞} f(it)t^{s−1} dt.

By the work of [16, 15, 1], f is modular and L_{E}(s) has analytic continuation to the whole complex plane, and satisfies the functional equation

(3)  Λ_{E}(s) = (\frac{\sqrt{N_{E}}}{2π} )^{s} Γ(s)L_{E}(s) = w_{E}Λ_{E}(2 − s)

where w_{E} = ±1 is called the root number. From (2), we have that L_{E}(1) = 2π ∫_{0}^{∞} f(it) dt. For a, m ∈ ℚ, m > 0, one defines the modular symbols

(4)  λ(a, m; E) = λ(a, m; f) = 2π ∫_{0}^{∞} f (it − \frac{a}{m}) dt.

Let χ be a primitive character of modulus m with Gauss sum

τ(χ) = ∑_{a mod m} χ(a)e^{2πia/m}.

The twisted L-function L_{E}(s, χ) satisfies the functional equation

Λ_{E}(s, χ) = (\frac{m\sqrt{N_{E}}}{2π} )^{s} Γ(s)L_{E}(s, χ)
= \frac{w_{E}χ(N_{E})τ(χ)^{2}}{m} Λ_{E}(2 − s, \overline{χ}).

From the identity

χ(n) = \frac{1}{τ(\overline{χ})} ∑_{a mod m} \overline{χ}(a)e^{2πian/m},

we have

f_{χ}(z) = ∑_{n≥1} χ(n)a_{n}e^{2πinz}

= \frac{1}{τ(\overline{χ})} ∑_{a mod m} \overline{χ}(a)f (z + a/m)

by rearranging the sums (Birch’s lemma). It then follows that

(5)  L_{E}(1, χ) = \frac{1}{τ(\overline{χ})} ∑_{a mod m} \overline{χ}(a)λ(a, m; E).

We define

λ^{+}(a, m; E) = λ(a, m; E) + λ(−a, m; E)
= 2π ∫_{0}^{∞} ∑_{n≥1} a_{n} e^{−2πnt} (e^{2πian/m} + e^{−2πian/m}) dt

= 4π ∑_{n≥1} a_{n} (Re {e^{2πian/m}}) ∫_{0}^{∞} e^{−2πnt}dt
= 2 ∑_{n≥1} \frac{a_{n}}{n} Re \{e^{2πian/m}\}

which is a real number as E is defined over ℚ. Then, all λ^{+}(a, m; E) can be written as a rational multiple of the real period, which we denote by Ω_{E}. We define

Λ(a, m; E) = \frac{λ^{+}(a, m; E)}{Ω_{E}} ∈ ℤ.

In this paper, χ has prime order k ≥ 3. Then, χ(−1) = 1, and we can rewrite (5) as

(6)  L_{E}(1, χ) = \frac{Ω_{E}}{2τ(\overline{χ})} ∑_{a mod m} \overline{χ}(a)Λ(a, m; E)

where the integers Λ(a, m; E) do not depend on the character χ, but only on the conductor m. Then, we see from (6) that

\frac{2τ(\overline{χ})L_{E}(1, χ)}{Ω_{E}}

is an algebraic integer in the field obtained by adding a kth root of unity. In fact, one can prove the stronger result which is critical to the discretisation of the next section. The particular case k = 3 was proven in [4]. For any odd prime k, let ℚ(ξ_{k}) be the cyclotomic field obtained by adding a primitive kth root of unity ξ_{k}, and let ℚ(ξ_{k})^{+} be the maximal real extension ℚ ⊆ ℚ(ξ_{k})^{+} ⊆ ℚ(ξ_{k}) of degree (k − 1)/2 over ℚ. The ring of integers of ℚ(ξ_{k})^{+} will be denoted by ℤ[ξ_{k}]^{+}.

Theorem 2.1. Let k be an odd prime, and let χ be a primitive character of order k. Then,

\frac{2τ(\overline{χ})L_{E}(1, χ)}{Ω_{E}} = \{ \begin{matrix}χ(N_{E})^{(k+1)/2} n_{E}(χ) & when w_{E} = 1 \\ (ξ_{k}^{−1} − ξ_{k})^{−1} χ(N_{E})^{(k+1)/2} n_{E}(χ) & when w_{E} = −1 \end{matrix}

where n_{E}(χ) ∈ ℤ[ξ_{k}] ∩ ℝ = ℤ[ξ_{k}]^{+}.

Proof: From the functional equation, we have

L_{E}^{alg}(1, χ) = \frac{2τ(\overline{χ})L_{E}(1, χ)}{Ω_{E}} = \frac{2τ(\overline{χ})w_{E}χ(N_{E})τ(χ)^{2}}{mΩ_{E}} L_{E}(1, \overline{χ})
= w_{E}χ(N_{E})\overline{\frac{2τ(\overline{χ})L_{E}(1, χ)}{Ω_{E}}} = w_{E}χ(N_{E})\overline{L_{E}^{alg}(1, χ)}

Let z ∈ ℂ^{∗} satisfying z = w_{E}χ(N_{E})\overline{z}. Then, L_{E}^{alg}(1, χ) = α\overline{z}^{−1} with α real. If w_{E} = 1, we take z = χ(N_{E})^{(k+1)/2}, and L_{E}^{alg}(1, χ) = α\overline{z}^{−1} with α ∈ ℝ ∩ ℤ[ξ_{k}] = ℤ[ξ_{k}]^{+}, which gives the result. If w_{E} = −1, we take z = (ξ_{k} − ξ_{k}^{−1}) χ(N_{E})^{(k+1)/2}, and L_{E}^{alg}(1, χ) = α\overline{z}^{−1} with α ∈ ℝ ∩ ℤ[ξ_{k}] = ℤ[ξ_{k}]^{+}, which gives the result. □

3. Discretisation

By Theorem 2.1, n_{E}(χ) is an algebraic integer in ℤ[ξ_{k}]^{+}, and there is then a natural discretisation on the algebraic integer n_{E}(χ) given by the geometry of numbers. Let ϕ be the map

ϕ : ℤ[ξ_{k}]^{+} → ℝ^{(k−1)/2}
α ↦ (σ_{1}(α), σ_{2}(α), …, σ_{(k−1)/2}(α))

where Gal(ℚ(ξ_{k})^{+}/ℚ) = {σ_{1} = 1, σ_{2}, …, σ_{(k−1)/2}}. Let α_{1}, …, α_{(k−1)/2} be an integral basis for ℤ[ξ_{k}]^{+}. The image of ℤ[ξ_{k}]^{+} in ℝ^{(k−1)/2} is the lattice generated by the linearly independent vectors

ω_{1} = ϕ(α_{1}), …, ω_{(k−1)/2} = ϕ(α_{(k−1)/2}).

Let R ⊆ ℝ^{(k−1)/2} be the region

R = { a_{1}ω_{1} + a_{2}ω_{2} + ⋯ + a_{(k−1)/2}ω_{(k−1)/2} :
−1 < a_{i} < 1 for 1 ≤ i ≤ (k − 1)/2 } .

The discretisation given by the embedding of ℤ[ξ_{k}]^{+} in ℝ^{(k−1)/2} is then

n_{E}(χ) = 0 ⟺ ϕ(n_{E}(χ)) ∈ R. (7)

Let χ be any character of conductor m and order k. For any automorphism σ ∈ Gal(ℚ(ξ_{k})/ℚ), let χ^{σ} be the character

χ^{σ} : (ℤ/mℤ)^{∗} → ⟨ξ_{k}⟩ ⊆ ℂ^{∗}
a ↦ σ (χ(a))

Then, χ^{σ} is also a character of conductor m and order k.

Lemma 3.1. Let k be an odd prime, and χ a character of order k and conductor m. For any σ in Gal(ℚ(ξ_{k})/ℚ), we have

|L_{E}(1, χ^{σ})| = \frac{c_{E,k}}{m^{1/2}} |n_{E}(χ)^{σ}|

where c_{E,k} is an explicit constant depending only on E and k.

Proof: Using (6), we have

L_{E}^{alg}(1, χ)^{σ} = (\frac{2τ(\overline{χ})L_{E}(1, χ)}{Ω_{E}})^{σ}
= ∑_{a mod m} \overline{χ}^{σ}(a)Λ(a, m; E) = L_{E}^{alg}(1, χ^{σ}).

Suppose first that ω_{E} = 1. Then,

n_{E}(χ)^{σ} = L_{E}^{alg}(1, χ)^{σ} (χ(N_{E})^{(k+1)/2})^{σ}
= \frac{2τ(\overline{χ}^{σ})L_{E}(1, χ^{σ})}{Ω_{E}} (χ(N_{E})^{(k+1)/2})^{σ} ,

and taking absolute values we get the result with c_{E,k} = Ω_{E}/2. The proof for ω_{E} = −1 is similar, with a different explicit constant c_{E,k}. □

We first consider the case k = 5. We have that ℤ[ξ_{5}]^{+} = ℤ [α] with α = (1 + \sqrt{5})/2, and G_{5} = ⟨1, τ⟩, where the non-trivial automorphism τ sends \sqrt{5} to −\sqrt{5}. Then, the lattice of ℤ [α] in ℝ^{2} is generated by ω_{1} = (α, α^{τ}) and ω_{2} = (α^{τ}, α). Let R be the region

R = {aω_{1} + bω_{2} : −1 < a < 1, −1 < b < 1} .

By (7), n_{E}(χ) = 0 if and only if ϕ(n_{E}(χ)) = (n_{E}(χ), n_{E}(χ)^{τ}) ∈ R. As the region R is not symmetric with respect to the absolute value, and we have a probability model for |L_{E}(1, χ)|, we also consider the two regions of ℝ^{2}

R_{1} = {(x, y) : −1 < x, y < 1}
R_{2} = \{(x, y) : −\sqrt{5} < x, y < \sqrt{5}\}

with the property that R_{1} ⊆ R ⊆ R_{2}, and

(8) (n_{E}(χ), n_{E}(χ)^{τ}) ∈ R_{i} ⟺ (|n_{E}(χ)|, |n_{E}(χ)^{τ}|) ∈ |R_{i}|

where

|R_{i}| = {(x, y) ∈ R_{i} : x, y ≥ 0} .

The following lemma is now immediate from (8) and Lemma 3.1 Lemma 3.2. Let σ ∈ Gal(ℚ(ξ_{5})/ℚ) be an automorphism which restricts to the non-trivial automorphism of ℚ(\sqrt{5}). For i = 1, 2, we have

(n_{E}(χ), n_{E}(χ)^{σ}) ∈ R_{i} ⟺ |L_{E}(1, χ)|, |L_{E}(1, χ^{σ})| ≤ \frac{c_{i}}{\sqrt{m}}

where c_{1}, c_{2} are explicit constants depending only on E.

We now suppose that k ≥ 7. Let B be the non-zero constant

B = max_{1≤i≤(k−1)/2} ∑_{j=1}^{(k−1)/2} |σ_{i}(α_{j})|,

and let R ⊆ R′ ⊆ ℝ^{(k−1)/2} be the region

R′ = {(x_{1}, …, x_{(k−1)/2}) : −B ≤ x_{i} ≤ B for 1 ≤ i ≤ (k − 1)/2.}

Then, n_{E}(χ) = 0 ⇒ ϕ(n_{E}(χ)) ∈ R′ by (7). The following lemma is now immediate from Lemma 3.1

Lemma 3.3. Let σ_{1}, …, σ_{(k−1)/2} ∈ Gal(ℚ(ξ_{k})/ℚ) be a set of representatives for G_{k} = Gal(ℚ(ξ_{k})^{+}/ℚ). Then, ϕ(n_{E}(χ)) ∈ R′ if and only if

|L_{E}(1, χ^{σ_{i}})| ≤ \frac{c_{k}}{m^{1/2}} for 1 ≤ i ≤ (k − 1)/2

where c_{k} is an explicit constant depending only E and k.

4. Unitary Random Matrices

Let U(N) be the set of unitary N × N matrices with complex coefficients which forms a probability space with respect to the Haar measure. For each A ∈ U(N), let

P_{A}(λ) = det(A − λI)

be the characteristic polynomial of A. For any s ∈ ℂ, let

M_{U}(s, N) = ∫_{U(N)} |P_{A}(1)|^{s} dHaar

be the moments for the distribution of |P_{A}(1)| in U(N) with respect to the Haar measure. In [10], Keating and Snaith proved that

(9)  M_{U}(s, N) = ∏_{j=1}^{N} \frac{Γ(j)Γ(j + s)}{Γ^{2}(j + s/2)},

and then M_{U}(s, N) is analytic for Re(s) > −1, and has meromorphic continuation to the whole complex plane. By Fourier inversion, the probability density of |P_{A}(1)| is

p(x) = \frac{1}{2πi} ∫_{(c)} M_{U}(s, N)x^{−s−1}ds

for some c > −1. Then, for any I ⊆ ℝ,

Prob (|P_{A}(1)| ∈ I) = ∫_{I} p(x) dx.

In our application to the vanishing of twisted L-functions, we will be interested only in small values of x where the value of p(x) is determined by the first pole of M_{U}(s, N) at s = −1. More precisely, for

x ≤ N^{−1/2},

one can show that

p(x) ∼ G^{2}(1/2)N^{1/4} as N → ∞,

where G(z) is the Barnes G-function, with special value

G(1/2) = exp (\frac{3}{2}ζ′(−1) − \frac{1}{4} log π + \frac{1}{24} log 2)

(see [10, p. 81] or [7, p. 58] for more details).

We now consider the moments for the special values of L-functions in families of twists. Fix k ≥ 3, and let

S_{k}(X) = {χ of order k and conductor ≤ X}
N_{k}(X) = #S_{k}(X) ∼ b_{k}X

with an explicit constant b_{k} (see for example [2]). We then define for any s ∈ ℂ

(10)  M_{E}(s, X) = \frac{1}{N_{k}(X)} ∑_{χ∈S_{k}(X)} |L_{E}(1, χ)|^{s}

The family of twists of order k has unitary symmetry, as the values |ζ(1/2 + it)| on the critical line. Then

Conjecture 4.1 (Keating and Snaith Conjecture for twists of order k).

M_{E}(s, X) ∼ a_{E}(s/2)M_{U}(s, N) as N = 2 log X → ∞,

where a_{E}(s/2) is an arithmetic factor depending only on the curve E.

In the conjecture, the relation between N and X is obtained by equating the mean density of eigenangles of matrices in the unitary group, and the mean density of non-trivial zeroes of the twisted Lfunctions L_{E}(s, χ) at a fixed height (see [4]). The arithmetic factor a_{E}(s) can not be obtained from the random matrix theory, and has to be determined separately for each family from its arithmetic. This was done for the family of cubic twists in [4], and could be done for the family of twists of order k for each k. The arithmetic factor a_{E}(s) would then be a meromorphic function for all s ∈ ℂ. As it will be seen below, the only influence of the arithmetic factor a_{E}(s) in our application is that the special value a_{E}(−1/2) will be part of the constant of the conjectural asymptotic of N_{E,k}(X). This would not provide any further information to the cases k ≥ 5 considered in this paper considered in this paper in view of Conjecture 1.2.

From Conjecture 4.1, the probability density p_{E}(x) for the distribution of the special values |L_{E}(1, χ)| for characters of order k is

p_{E}(x) = \frac{1}{2πi} ∫_{(c)} M_{E}(s, X) x^{−s−1} ds
(11)  ∼ \frac{1}{2πi} ∫_{(c)} a_{E}(s/2) M_{U}(s, N) x^{−s−1} ds

as N = 2 log X → ∞. As above, when x ≤ N^{−1/2}, the value of p_{E}(x) is determined by the residue of M_{U}(s, N) at s = −1, and it follows from (11) that

p_{E}(x) ∼ C_{E} log^{1/4} X (12)

for x ≤ (2 log X)^{−1/2}, X → ∞, and C_{E} = 2^{1/4}a_{E}(−1/2)G^{2}(1/2).

Let χ be a character of order k ≥ 3 and conductor m. We apply the above model to find the probability that |L_{E}(1, χ)| < cm^{−1/2}, for some constant c > 0. For x < cm^{−1/2} < (2 log m)^{−1/2} (for m large enough), we have p_{E}(x) ∼ C_{E} log^{1/4} m, and then

Prob (|L_{E}(1, χ)| < cm^{−1/2}) ∼ ∫_{0}^{cm^{−1/2}} C_{E} log^{1/4} m dx
(13)  = c C_{E} \frac{log^{1/4} m}{m^{1/2}}.

We now use the probability density of the random matrix model with the discretisation of Section 3 to obtain conjectures for the vanishing of the L-values L_{E}(1, χ). We first suppose that k = 5, and as in the previous section, let σ ∈ Gal(ℚ(ξ_{5})/ℚ) which restricts to the nontrivial automorphism of ℚ(\sqrt{5}). We saw in the previous two sections that

L_{E}(1, χ) = 0 ⟺ ϕ(n_{E}(χ)) = (n_{E}(χ), n_{E}(χ)^{σ}) ∈ R.

As R_{1} ⊆ R ⊆ R_{2}, and using Lemma 3.2, the probability that L_{E}(1, χ) is zero is bounded below by

Prob (|L_{E}(1, χ)| < \frac{c_{1}}{\sqrt{m}}) Prob (|L_{E}(1, χ^{σ})| < \frac{c_{1}}{\sqrt{m}})

and bounded above by

Prob (|L_{E}(1, χ)| < \frac{c_{2}}{\sqrt{m}}) Prob (|L_{E}(1, χ^{σ})| < \frac{c_{2}}{\sqrt{m}}) .

Assuming that |L_{E}(1, χ)| and |L_{E}(1, χ^{σ})| are independent identically distributed random variables, and using (13), we get that the probability that L_{E}(1, χ) is zero is about

\frac{log^{1/2} m}{m},

neglecting all constants which are not significant here. The sum of the probabilities is

(14)  ∑_{χ∈S_{3}(X)} \frac{log^{1/2} m}{m} ∼ \frac{2b_{3}}{3} log^{3/2} X.

As discussed in [4], the exact power of log X that is obtained with the random matrix approach depends subtly on the discretisation, and is difficult to predict. For examples, rational torsion of order three on the elliptic curve seemed to cause extra vanishing of the twisted L-values L_{E}(1, χ) for cubic characters, and changed the power of logarithm in the conjectural asymptotic for N_{E,k}(X) of [4]. For k = 5, the sum of the probabilities is just on the border between convergence and divergence, and the random matrix model seems to indicate that the number of quintic twists such that L_{E}(1, χ) vanishes is infinite, but that N_{E,k}(X) ≪ X^{ϵ} for any ϵ > 0. This agrees with the empirical evidence of Section 5.

We now suppose that k ≥ 7. Let σ_{1} = 1, …, σ_{(k−1)/2} be elements of the Galois group of ℚ(ξ_{k})/ℚ which form a set of representatives for the Galois group of ℚ(ξ_{k})^{+}/ℚ. As we saw in the two previous sections,

L_{E}(1, χ) = 0 ⟺ ϕ(n_{E}(χ)) = (n_{E}(χ)^{σ_{1}}, …, n_{E}(χ)^{σ_{(k−1)/2}}) ∈ R.

As R ⊆ R′, and using Lemma 3.3, the probability that L_{E}(1, χ) is zero is bounded by

Prob (|L_{E}(1, χ^{σ_{1}})| < \frac{c_{k}}{\sqrt{m}}) , …, Prob (|L_{E}(1, χ^{σ_{(k−1)/2}})| < \frac{c_{k}}{\sqrt{m}}) .

Assuming that |L_{E}(1, χ^{σ_{i}})| are independent identically distributed random variables for 1 ≤ i ≤ (k − 1)/2, and using (13), we get that the probability that L_{E}(1, χ) is zero is

\frac{log^{(k−1)/8} m}{m^{(k−1)/4}}.

neglecting all constants which are not significant here. Summing the probabilities, this gives for k ≥ 7

(15)  ∑_{χ∈S_{3}(X)} \frac{log^{(k−1)/8} m}{m^{(k−1)/4}} = O(1).

From the random matrix model, we then conjecture that the number of twists of order k ≥ 7 such that L_{E}(1, χ) vanishes is bounded. This also agrees with the empirical evidence of Section 5.

5. Numerical Evidence

The following table shows the observed number of vanishing twists L_{E}(1, χ) for characters of orders three, five and seven, and for the first three elliptic curves in the Cremona catalogue [5]. For each elliptic curve E, the characters with conductor prime to N_{E} and less than two million were considered. Any two characters of conductor m and order k generating the same cyclic subgroup of the character group are conjugate, and hence the special values L_{E}(1, χ) vanish simultaneously by Lemma 3.1. The number in the table records one of each class of conjugate characters for which the special value vanishes, which is 1/(k −1) of the number of characters with vanishing special value. The twists of order eleven in the same range for the curve E14 were also computed, and no vanishing were found.

Curve Cubic Quintic Septic
vanishing vanishing vanishing
E11  1152  15  2
E14  4347  10  0
E15  2050  11  0

The results for cubic twists have been analyzed in [4] and support Conjecture 1.2. The results for quintic and septic twists are too sparse to either support or refute Conjecture 1.2, but they nevertheless illustrate the extreme scarcity of vanishing in higher order twists which is predicted by the conjecture.

Acknowledgments This paper was first presented at the “Ranks of Elliptic Curves and Random Matrix Theory” workshop held at the Isaac Newton Institute in February 2004. The first and second authors would like to thank the organizers of the workshop and the Isaac Newton Institute for their hospitality and financial support. The first author would also like to thank B. Birch, B. Conrey and C. Hughes for helpful discussions.

References

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Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montréal (Québec), Canada H3G 1M8

E-mail address, David: cdavid@mathstat.concordia.ca
E-mail address, Fearnley: jack@mathstat.concordia.ca
E-mail address, Kisilevsky: kisilev@mathstat.concordia.ca