NONVANISHING OF QUADRATIC TWISTS OF MODULAR

L-FUNCTIONS AND APPLICATIONS TO ELLIPTIC CURVES

Ken Ono

Very preliminary

1. Introduction and Statement of Results

If F (z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}(Γ_{0}(M), χ_{0}) (note: q := e^{2πiz} throughout) is a newform of even integer weight 2k with trivial character χ_{0}, then let L(F, s) be its L-function

L(F, s) := ∑_{n=1}^{∞} \frac{a(n)}{n^{s}}.

If d is square-free or is a fundamental discriminant, then let χ_{d} denote the Kronecker character for ℚ(\sqrt{d}). Throughout D shall denote the fundamental discriminant of the quadratic field ℚ(\sqrt{D}). The D-quadratic twist of F, denoted F ⊗ χ_{D} is the newform corresponding to the twist of F by the character χ_{D}. In particular, if gcd(M, D) = 1, then (F ⊗ χ_{D})(z) = ∑_{n=1}^{∞} χ_{D}(n)a(n)q^{n} and

L(F ⊗ χ_{D}, s) = ∑_{n=1}^{∞} \frac{χ_{D}(n)a(n)}{n^{s}}.

We show, for those newforms satisfying a mild hypothesis, a curious multiplicative property for many D for which L(F ⊗ χ_{D}, k) ≠ 0.

Before we state our result we recall that a set of primes S is said to have Frobenius density if there is a Galois extension K/ℚ with the property that those primes p ∈ S are distinguished as those primes p for which F rob(p) constitutes a fixed conjugacy class in Gal(K/ℚ).

Typeset by AMS-TEX

The author thanks the National Science Foundation, the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation for their generous support.

Theorem 1. Let F (z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}(Γ_{0}(M), χ_{0}) be an even weight newform and let K be a number field containing the coefficients a(n). If v is a place of K over 2 and there is a prime p ∤ 2M for which

(1.1)  ord_{v}(a(p)) = 0,

then there is a discriminant D_{F} and a set of primes S_{F} with positive Frobenius density such that for every positive integer j we have

L(F ⊗ χ_{D_{F} p_{1}p_{2}···p_{2j}}, k) ≠ 0

whenever p_{1}, p_{2}, …, p_{2j} ∈ S_{F} are distinct primes not dividing D_{F}.

It is not difficult to see that the choice of D_{F} is not unique.

In an important paper [G], Goldfeld conjectured that

∑_{\begin{matrix}|D|≤X, \\ gcd(D,M)=1 \end{matrix}} ord_{s=k}(L(F ⊗ χ_{D}, k)) ∼ \frac{1}{2} ∑_{\begin{matrix}|D|≤X, \\ gcd(D,M)=1 \end{matrix}} 1.

(note: The original form of this conjecture was for weight 2 newforms F (z) associated to modular elliptic curves). Obviously, this conjecture implies the weaker conjecture that

(1.2)  #{|D| ≤ X : L(F ⊗ χ_{D}, k) ≠ 0 and gcd(D, M) = 1} ≫ X.

Recent important work by Katz and Sarnak [Ka-Sa] yields, among other results, conditional proofs of (1.2). Although there has been some recent success in proving (1.2) unconditionally for exceptional F (z) (those with exceptional mod 3 Galois representations) by the works of James, Kohnen, and Vatsal [J, Ko V], the best unconditional lower bound is due to the author and Skinner. They established [Cor. 3, O-Sk] that

(1.3)  #\{|D| ≤ X : L(F ⊗ χ_{D}, k) ≠ 0 and gcd(D, M) = 1\} ≫_{F} \frac{X}{log X} .

For most F (z) we obtain the following modest improvement to (1.3).

Corollary 2. If F (z) ∈ S_{2k}(Γ_{0}(M), χ_{0}) is a newform satisfying (1.1), then

#\{|D| ≤ X : L(F ⊗ χ_{D}, k) ≠ 0\} ≫_{F} \frac{X}{log^{1−α} X}

where 0 < α < 1 is the density of S_{F}.

Hypothesis (1.1) in Theorem 1.1 is a mild condition which is a satisfied by most newforms (there are exceptions like Δ(z)). For example, almost every weight 2 newform associated to an elliptic curve E/ℚ satisfies (1.1) (for example, see [Du]).

Suppose that E/ℚ is an elliptic curve

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

and let L(E, s) = ∑_{n=1}^{∞} a_{E}(n)n^{−s} be its Hasse-Weil L-function. In addition, for discriminants D let E(D) denote the D-quadratic twist of E

E : Dy^{2} = x^{3} + ax + b.

Moreover, if E is an elliptic curve defined over a number field K, then let rk(E, K) denote the rank of the Mordell-Weil group E(K). Using a celebrated theorem of Kolyvagin, we obtain the following corollary.

Corollary 3. If E/ℚ is an elliptic curve without a torsion point of order 2, then there is a number 0 < α(E) < 1 for which

#\{|D| ≤ X : rk(E(D), ℚ) = 0\} ≫_{E} \frac{X}{log^{1−α(E)} X} .

The most interesting consequence of Theorem 1 maybe the following result concerning the triviality of the rank of the Mordell-Weil group of most elliptic curves E over prescribed simple 2-extensions of ℚ of arbitrarily large degree.

Theorem 4. Let E/ℚ be an elliptic curve without a torsion point of order 2. Then there is a fundamental discriminant D_{E} and a set of primes S_{E} with Frobenius density 0 < α(E) < 1 with the property that for every positive integer j we have

rk(E(D_{E}), ℚ(\sqrt{m_{1}}, \sqrt{m_{2}}, …, \sqrt{m_{j}})) = rk(E(D_{E}), ℚ) = 0

whenever the numbers m_{1}, m_{2}, …, m_{j} satisfy the following conditions.

(1) If 1 ≤ i ≤ j and p | m_{i} is prime, then p ∈ S_{E}.

(2) If 1 ≤ s ≠ t ≤ j, then

gcd(m_{s}, m_{t}) = 1.

(3) If μ denotes the usual Möbius function and 1 ≤ i ≤ j, then

μ(m_{i}) = 1.

As in Theorem 1, the choice of D_{E} is not unique.

The last consequence of Theorem 1 involves Tate-Shafarevich groups of quadratic twists. By the works of Bölling, Cassels, Kramer, and Rohrlich [Bo, Ca, Kr, R], quite a bit is known about the non-triviality of the 2 and 3-parts of the Tate-Shafarevich groups of elliptic curves. However, much less is known about the non-triviality of p-parts of � (E) for primes p ≥ 5. In this direction, Wong [Wo] has proved that infinitely many quadratic twists of X_{0}(11) have elements of order 5 in their Tate-Shafarevich groups. We know of no other similar results when p = 5 for any other elliptic curve or any results for p ≥ 7.

Wong obtained his result by combining an old observation of the author and Frey with his own results regarding the prime factorization of the discriminants of certain quadratic fields whose class groups have elements of fixed order ℓ (i.e. in this case ℓ = 5). Using Theorem 1, a theorem of Frey, and a simple refinement of Wong’s strategy for X_{0}(11) we obtain a general result which probably holds for almost all (if not all) curves E whose Mordell-Weil group over ℚ has torsion subgroup ℤ/3ℤ, ℤ/5ℤ or ℤ/7ℤ. The hypotheses which need to be verified for any given E are cumbersome and unmotivating. For aesthetics, here we simply refer to an E which satisfies these hypotheses for a given prime ℓ ∈ {3, 5, 7} as an excellent elliptic curve at ℓ. We shall defer the explicit definition to §5.

Theorem 5. Suppose that E/ℚ is an elliptic curve whose torsion subgroup is ℤ/ℓℤ with ℓ ∈ {3, 5, 7}. If E is excellent at ℓ, then there are infinitely many negative fundamental discriminants D for which

rk(E(D), ℚ) = 0 and #� (E(D), ℚ) ≡ 0 (mod ℓ).

In §5 we illustrate this result for the elliptic curve E with conductor 26 given by

E : y^{2} + xy + y = x^{3} − x^{2} − 3x + 3

Its torsion subgroup is ℤ/7ℤ, and is excellent at 7. Consequently there are infinitely many negative fundamental discriminants D for which

rk(E(D), ℚ) = 0 and #� (E(D), ℚ) = 0 (mod 7).

2. Preliminaries

The arguments in this paper depend heavily on the combinatorics of the half-integer weight and integer weight Hecke operators combined with the intrinsic properties of Galois representations associated to newforms (see [K, Mi] for essential facts concerning modular forms). Here we briefly recall the definition of Hecke operators we need. If k is a positive integer and χ is a Dirichlet character modulo M, then for every prime p ∤ M the Hecke operator T_{p}^{k,χ} on S_{k}(Γ_{0}(M), χ) acts by sending the cusp form f(z) = ∑_{n=1}^{∞} a_{f}(n)q^{n} to

(2.1)  f(z) | T_{p}^{k,χ} := ∑_{n=1}^{∞}(a_{f}(np) + χ(p)p^{k−1}a_{f}(n/p))q^{n}.

Similarly, recall that if p ∤ 4N is prime and χ is a Dirichlet character modulo 4N, then the half-integer weight Hecke operator T_{k}^{χ}(p^{2}) on S_{k+ \frac{1}{2}} (Γ_{0}(4N), χ) is defined by sending a cusp form h(z) = ∑_{n=1}^{∞} a_{h}(n)q^{n} to

(2.2) h(z) | T_{k}^{χ}(p^{2}) := ∑_{n=1}^{∞} (a_{h}(np^{2}) + χ^{⋆}(p)p^{k−1}(\frac{n}{p})a_{h}(n) + χ^{⋆}(p^{2})p^{2k−1}a_{h}(n/p^{2})) q^{n},

where χ^{⋆}(p) := χ(p)(\frac{−1}{p})^{k}.

We also require Galois representations associated to modular forms due to the works of Deligne and Serre [D, D-S].

Theorem 2.1. Suppose that f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{k}(Γ_{0}(M), χ) is an integer weight newform and suppose that K is a number field whose ring of integers O_{K} contains the Fourier coefficients a(n) and the values of χ. If O_{v} is the completion of O_{K} at any finite place v of K, say with residue characteristic ℓ, then there is a continuous representation

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

with the property that if p ∤ ℓM is prime, then

Tr(ρ_{f,v}(F rob(p)) = a(p).

Using this theorem we prove the following crucial theorem.

Theorem 2.2. Let f_{1}(z), f_{2}(z), …, f_{s}(z) be integer weight cusp forms where

f_{i}(z) = ∑_{n=1}^{∞} a_{i}(n)q^{n} ∈ S_{k_{i}}(Γ_{0}(M_{i}), χ_{i}).

Suppose that the coefficients of all the f_{i}(z) and the values of all the χ_{i} are in O_{K}, the ring of algebraic integers of some number field K. Let v be a finite place of K with residue characteristic ℓ. If p_{0} ∤ lcm(ℓ, M_{1}, M_{2}, … M_{j}) is prime and j is a positive integer, then there is a set of primes p with positive Frobenius density such that for every 1 ≤ i ≤ s we have

ord_{v} (f_{i}(z) | T_{p_{0}}^{k_{i},χ_{i}} − f_{i}(z) | T_{p}^{k_{i},χ_{i}}) > j.

Proof. By the theory of newforms (a.k.a. primitive forms), each f_{i}(z) may be expressed as

(2.3)  f_{i}(z) = ∑_{δ|N_{i}} ∑_{s} β(i, s)h_{i,s}(δz)

where each β(i, s) is algebraic and the inner sum above is over the newforms h_{i,s}(z) with level dividing N_{i}/δ. For convenience, we denote the q-expansions of the h_{i,s}(z) by

(2.4)  h_{i,s}(z) = ∑_{n=1}^{∞} a_{i,s}(n)q^{n}.

Let K_{1} be a finite extension of K which contains all the β(i, s) and all the Fourier coefficients a_{i,s}(n) of all the newforms h_{i,s}(z). Let w be a place of K_{1} over v and let e be its ramification index. Moreover, let O_{w} be the completion of O_{K_{1}} at w and let λ be its uniformizer.

For each h_{i,s}(z) let ρ_{i,s,w} be the representation described in Theorem 2.1. If E is defined by

(2.5)  E := max_{i,s} |ord_{w}(β(i, s))|,

then consider the representation

ρ := ⊕_{i,s} ρ_{i,s,w} (mod λ^{E+je+1}).

Since the image of ρ is finite, the Chebotarev Density Theorem implies that there is a set of primes p with positive Frobenius density which have the property that

Tr(ρ_{i,s,w}(F rob(p)) ≡ a_{i,s}(p_{0}) (mod λ^{E+je+1})

for all i and s. Since a_{i,s}(p) is the eigenvalue of h_{i,s}(z) for the Hecke operator T_{p}^{k_{i},χ_{i}} for a prime p ∤ lcm(ℓ, M_{1}, M_{2}, …, M_{j}), it follows by (2.3-5) that

f_{i}(z) | T_{p}^{k_{i},χ_{i}} ≡ f_{i}(z) | T_{p_{0}}^{k_{i},χ_{i}} (mod λ^{je+1})

for all i for such primes p.

Q.E.D.

3. Fourier coefficients of half-integer weight cusp forms

In this section we prove a general nonvanishing theorem for the Fourier coefficients of most half-integer weight cusp forms. We begin with the following simple proposition.

Proposition 3.1. Suppose that k is a positive integer and that g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+ \frac{1}{2}} (Γ_{0}(4N), χ) is an eigenform of the Hecke operators T_{k}^{χ}(p^{2}). If p ∤ 4N is a prime and λ(p) is the eigenvalue of g(z) with respect to T_{k}^{χ}(p^{2}), then

b(np^{2}) = (λ(p) − χ^{⋆}(p)p^{k−1}(\frac{n}{p})) b(n) − χ^{⋆}(p^{2})p^{2k−1}b(n/p^{2}).

Lemma 3.2. Suppose that g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+ \frac{1}{2}} (Γ_{0}(4N), χ) is an eigenform of the Hecke operators T_{k}^{χ}(p^{2}) with eigenvalues λ(p) where k is a positive integer. If k = 1, then make the further assumption that the image of g(z) under the Shimura correspondence is a cusp form. Let G(z) ∈ S_{k+1}(Γ_{0}(4N), χχ_{−1}^{k+1}) be the cusp form defined by

G(z) = ∑_{n=1}^{∞} b_{g}(n)q^{n} := g(z) · (1 + 2 ∑_{n=1}^{∞} q^{n^{2}}) .

Let K be a number field with the property that all the coefficients b(n) and the values of χ are in O_{K}, the ring of integers of K. Let v be a place in O_{K} above 2. If there is a prime p_{0} ∤ 4N for which

ord_{v}(λ(p_{0})) = 0,

then for every positive integer j there is a set of primes S_{p_{0},j} with positive Frobenius density which satisfies the following conditions.

(1) If p ∈ S_{p_{0},j}, then

ord_{v}(λ(p)) = ord_{v}(λ(p_{0})) = 0.

(2) If p ∈ S_{p_{0},j} then

ord_{v} (G(z) | T_{p}^{k+1,χχ_{−1}^{k+1}} − G(z) | T_{p_{0}}^{k+1,χχ_{−1}^{k+1}}) > j.

Proof. If k > 1, then the image of g(z) under the Shimura correspondence is a newform f_{g}(z) of integer weight 2k and level dividing 4N with Nebenyptus character χ^{2}. Moreover, the eigenvalues of this newform with respect to T_{p}^{2k,χ^{2}} are the eigenvalues λ(p) for every prime p | 4N. The same is true for those g(z) with k = 1 whose image under the correspondence is a cusp form. Therefore we may apply Theorem 2.1 to the eigenvalues λ(p) for primes p ∤ 4N viewed as the Fourier coefficients a_{f_{g}}(p) of the newform f_{g}(z). The conclusion now follows immediately from Theorem 2.2 for the forms G(z) and f_{g}(z).

Q.E.D.

We shall require the following proposition.

Lemma 3.3. Assume the notation and hypotheses from Lemma 3.2, and define s_{0} by

s_{0} := min{ord_{v}(b(n))}.

Then the following are true.

(1) An integer n has the property that ord_{v}(b_{g}(n)) = s_{0} if and only if ord_{v}(b(n)) = s_{0}.

(2) Let n_{0} be an integer for which ord_{v}(b(n_{0})) = s_{0}. If j is a positive integer and p_{1}, p_{2}, …, p_{2j−1}, p_{2j} ∈ S_{p_{0},s_{0}} are distinct primes not dividing n_{0}, then

ord_{v}(b(n_{0}p_{1}p_{2} ⋯ p_{2j−1}p_{2j})) = s_{0}.

Proof. By the definition of G(z), it is easy to see that if n is a positive integer, then

b_{g}(n) = b(n) + 2 ∑_{n=1}^{∞} b(n − k^{2}).

Conclusion (1) follows easily from the definition of s_{0} and v.

Suppose that m is a positive integer for which

ord_{v}(b(m)) = s_{0}.

Let q_{1}, q_{2} ∈ S_{p_{0},s_{0}} be distinct primes which are coprime to m. The coefficient of q^{mq_{1}}, by

(2.1), in G(z) | T_{q_{1}}^{k+1,χχ_{−1}^{k+1}} is

= b_{g}(mq_{1}^{2}) + χ(q_{1})χ_{−1}^{k+1}(q_{1})q_{1}^{k}b_{g}(m)

= (λ(q_{1}) − χ^{⋆}(q_{1})q_{1}^{k−1}(\frac{m}{q_{1}})) b_{g}(m) + χ(q_{1})χ_{−1}^{k+1}(q_{1})q_{1}^{k}b_{g}(m)

= λ(q_{1})b_{g}(m) + b_{g}(m)χ^{⋆}(q_{1})q_{1}^{k−1} (χ_{−1}(q_{1})q_{1} − (\frac{m}{q_{1}}))

Since χ_{−1}(q_{1})q_{1} − (\frac{m}{q_{1}}) ≡ 0 (mod 2), by Lemma 3.2 we find that the coefficient of q^{mq_{1}} in

G(z) | T_{q_{1}}^{k+1,χχ_{−1}^{k+1}} has ord_{v} = s_{0}.

By Lemma 3.2 (2), the coefficient of q^{mq_{1}} in G(z) | T_{q_{2}}^{k,χχ_{−1}^{k+1}} also has ord_{v} = s_{0} if q_{2} ∈ S_{p_{0},s_{0}}. This then implies that

b_{g}(mq_{1}q_{2}) + χ(q_{2})χ_{−1}^{k+1}(q_{2})q_{2}^{k}b_{g}(mq_{1}/q_{2})
= b_{g}(mq_{1}q_{2})

has ord_{v} = s_{0}.

This shows, by (1), that if ord_{v}(b(m)) = s_{0} and q_{1}, q_{2} ∈ S_{p_{0},s_{0}} are distinct odd primes which do not divide m, then

(3.1)  ord_{v}(b(mq_{1}q_{2})) = s_{0}.

Iterating (3.1) yields (2).

Q.E.D.

Now we briefly recall Waldspurger’s theorem which gives a formula for many of the central values L(F ⊗ χ_{D}, k) in terms of the squares of the Fourier coefficients of cusp forms with half integer weight k + \frac{1}{2} which are related to F (z), or a twist of F (z), by Shimura’s correspondence [Sh]. For every D define D_{0} by

(3.2)  D_{0} := \{ \begin{matrix}|D| & if D is odd, \\ |D|/4 & if D if even. \end{matrix}

The following is a convenient reformulation of Waldspurger’s theorem.

Theorem 3.4. ([Th. 1, W1, §2 O-Sk]) If F (z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}(Γ_{0}(M), χ_{0}) is an even weight newform, then there is a δ ∈ {±1}, a positive integer N with M | N, a Dirichlet character modulo 4N, a non-zero complex number Ω_{F} and a non-zero eigenform

g_{F} (z) = ∑_{n=1}^{∞} b_{F} (n)q^{n} ∈ S_{k+ \frac{1}{2}} (Γ_{0}(4N), χ)

with the property that if δD > 0, then

b_{F} (D_{0})^{2} = \{ \begin{matrix}ϵ_{D} · ^{\frac{L(F ⊗χ_{D},k)D_{0}^{k− \frac{1}{2}}}{Ω_{F}}} & if gcd(D_{0}, 4N) = 1, \\ 0 & otherwise, \end{matrix}

where ϵ_{D} is algebraic. Moreover, the coefficients a(n), b_{F} (n) and the values of χ are in O_{K}, the ring of algebraic integers in some fixed number field K. In addition, if p ∤ 4N is prime, then

λ(p) = χ(p)a(p)

for every prime p ∤ N where λ(p) is the eigenvalue of g_{F} (z) for the half integer weight Hecke operator T_{k}^{χ}(p^{2}) on S_{k+ \frac{1}{2}} (Γ_{0}(4N), χ).

We now prove the following technical theorem which immediately implies Theorem 1.

Theorem 3.5. Assume the notation from Theorem 3.4. Let v be a place of K over 2 and let s_{0} be the non-negative integer

s_{0} := min{ord_{v}(b(n))}.

Suppose there is a prime p ∤ M for which ord_{v}(a(p)) = 0. Then there is a set of primes S_{F} with positive Frobenius density with the property that for every positive integer j we have

L(F ⊗ χ_{δn_{0}p_{1}p_{2}···p_{2j}}, k) ≠ 0

whenever n_{0} is a square-free integer with ord_{v}(b_{F} (n_{0})) = s_{0} and p_{1}, p_{2}, …, p_{2j} ∈ S_{F} are distinct primes not dividing n_{0}.

Proof of Theorem 3.5. We apply Lemma 3.3 to the eigenform g_{F} (z) given in Theorem 3.4. Since g_{F} (z) is an eigenform, by (2.2) there is a square-free integer n_{0} coprime to 4N for which ord_{v}(b(n_{0})) = s_{0}. Theorem 1 now follows from Lemma 3.3 and Theorem 3.4.

Q.E.D.

Lemma 3.6. Let T be a set of primes with Frobenius density 0 < α < 1. If ℕ_{T} denotes the set

ℕ_{T} := {n ∈ ℕ : n = ∏_{i} p_{i}, p_{i} ∈ T, and μ(n) = 1},

then

#\{n ≤ X : n ∈ ℕ_{T} \} ≫_{T} \frac{X}{log^{1−α} X} .

Proof. Generalizing an argument of Landau, Serre proved [Th. 2.8, S] that if X_{T} is defined by

N_{T} (X) := {n ≤ X : n = ∏_{i} p_{i} with p_{i} ∈ T },

then

(3.3)  #N_{T} (X) = c_{T} · \frac{X}{log^{1−α} X} + O (\frac{X}{log^{2−α} X} )

for some positive constant c_{T}. Therefore, we need to consider those n counted in (3.3) which are square-free and have an even number of prime factors. Now let M_{T} (X) denote the set

M_{T} (X) := {n ≤ X : n ∈ N_{T} (X) square-free}.

It is easy to see that

#M_{T} (X) ≥ #N_{T} (X) − ∑_{\begin{matrix}p_{i}≤\sqrt{X} \\ p_{i}∈T \end{matrix}} #N_{T} (X/p_{i}^{2})

≥ #N_{T} (X) − ∑_{\begin{matrix}p_{i}≤\sqrt{X} \\ p_{i}∈T \end{matrix}} \frac{1}{p_{i}^{2}} #N_{T} (X)

≫ #N_{T} (X). (3.4)

Define ℕ_{T}^{0} (X) by

ℕ_{T}^{o} (X) := #{n ≤ X : n ∈ M_{T} (X) with μ(n) = −1}.

Obviously, we have that

(3.5)  #M_{T} (X) = #ℕ_{T} (X) + #ℕ_{T}^{o} (X).

If p_{0} ∈ T is prime, then by multiplying elements in ℕ_{T}^{o} (X/p_{0}) by p_{0} we find that

#ℕ_{T} (X) ≥ #ℕ_{T}^{o} (X/p_{0}) − #{n ≤ X : n ∈ ℕ_{T}^{o} (X/p_{0}) and gcd(n, p_{0}) = p_{0}}

≫ #ℕ_{T}^{o} (X/p_{0}). (3.6)

So, if ℕ_{T} (X) = o (\frac{X}{log^{1−α} X} ), then by (3.5) and (3.6) we have that

M_{T} (X/p_{0}) = o (\frac{X}{log^{1−α} X} ) .

However, this contradicts (3.4) which asserts that

M_{T} (X/p_{0}) ∼ \frac{1}{p_{0}} \frac{X}{(log X − log p_{0})^{1−α}} .

Q.E.D.

Proof of Corollary 2. This result follows immediately by letting the set T in Lemma 3.6 be the set S_{p_{0}} in Theorem 1.

Q.E.D.

4. Applications to ranks of elliptic curves

We recall a celebrated result due to Kolyvagin [Kol] which depends on a nonvanishing theorem for modular L-functions which has been proved by Bump, Friedberg and Hoffstein, Iwaniec, and Murty and Murty [B-F-H, I, M-M].

Theorem 4.1. If E/ℚ is a modular elliptic curve and

L(E, 1) ≠ 0,

then E has rank zero.

Proof of Corollary 3. By the works of Wiles and Taylor, Diamond, Conrad, Diamond and Taylor, and Breuil, Conrad, Diamond and Taylor [W, T-W, Di, C-D-T, B-C-D-T], it is now known that every elliptic curve E/ℚ is modular. If E is an elliptic curve with conductor N(E) and D is coprime to N(E), then we have that L(E(D), s) is simply the D-quadratic twist of L(E, s). These are simply quadratic twists of modular L-functions. The conclusion of Theorem 3 now follows immediately by Theorem 4.1, Theorem 1, and Corollary 2. We use the fact that a_{E}(p) is even for all but finitely many primes p if and only if E has a rational point of order 2.

Q.E.D.

We now examine the ranks of E over simple 2-extensions. We begin with the following well known proposition.

Proposition 4.2. Suppose that K is a number field and that E/K is an elliptic curve. If d ∈ K^{⋆} has the property that L := K(\sqrt{d}) is a quadratic extension of K, then

rk(E, L) = rk(E, K) + rk(E(d), K).

As a corollary we obtain the following result.

Corollary 4.3. Let E/ℚ be an elliptic curve. Suppose that S := {m_{1}, m_{2}, …, m_{t}} is a set of square-free mutually coprime integers > 1 such that for each 1 ≤ s ≤ t and any d_{1}, d_{2}, …, d_{s} ∈ S distinct we have

rk(E(d_{1}d_{2} ⋯ d_{s}), ℚ) = 0.

Then for every integer 1 ≤ s ≤ t we have

rk(E, ℚ(\sqrt{m_{1}}, \sqrt{m_{2}}, …, \sqrt{m_{s}})) = rk(E, ℚ).

Proof. The proof will follow by induction. The case where u = 1 follows by Proposition 4.2 since

rk(E, ℚ(\sqrt{m_{1}})) = rk(E, ℚ) + rk(E(m_{1}), ℚ,) = rk(E, ℚ).

Assuming the u = n case we establish the u = n + 1 case. By Proposition 4.2 again, we have that

rk(E, ℚ(\sqrt{m_{1}}, …,\sqrt{m_{n+1}}))
= rk(E, ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n}})) + rk(E(m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n}}))
= rk(E, ℚ) + rk(E(m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n}})).

Therefore, it suffices to show that

(4.1)  rk(E(m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n}})) = 0.

By Proposition 4.2 again, we have that

rk(E(m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n}}))

= rk(E(m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n−1}})) + rk(E(m_{n}m_{n+1}), ℚ(\sqrt{m_{1}}, …, \sqrt{m_{n−1}}))

= rk(E(m_{n+1}), ℚ) + ∑_{i=1}^{n} rk(E(m_{i}m_{n+1}), ℚ) + ∑_{1≤i<j≤n} rk(E(m_{i}m_{j}m_{n+1}), ℚ) + ⋯

+ rk(E(m_{1}m_{2} ⋯ m_{n}m_{n+1}), ℚ).

By hypothesis, this is zero which proves (4.1).

Q.E.D.

Proof of Theorem 4. It is obvious that there are infinitely many ways of choosing the integers m_{i} in Corollary 4.3 using Theorem 1.

Q.E.D.

5. Applications to Tate-Shafarevich groups of elliptic curves

In this section we prove Theorem 5 and give one numerical example. If E is an elliptic curve over ℚ, then let N(E) denote its conductor, j(E) its j-invariant, and Δ(E) its discriminant. If p is prime, then let S(E, ℚ)_{p} be its pth-Selmer group and let � (E, ℚ) denote the Tate-Shafarevich group of E. We begin by recalling an important Theorem due to Frey [Fr] regarding the Selmer groups of quadratic twists of elliptic curves with a rational point of odd prime order p (note: By a theorem of Mazur this forces p = 3, 5, or 7).

Theorem 5.1. Let p be an odd prime such that E/ℚ is an elliptic curve with a point of odd prime order p, say P. Furthermore, assume that P does not reduce to infinity modulo p. Define the set of primes S(E, p) by

S(E, p) := {q | N(E) : q ≠ 2, q ≡ −1 (mod p) and ord_{q}(Δ(E)) ≢ 0 (mod p)}.

If d is an negative square-free integer satisfying the following conditions:

(1) We have gcd(d, pN(E)) = 1 and d ≡ 3 (mod 4).

(2) If ord_{p}(j(E)) < 0, then (\frac{d}{p}) = −1.

(3) If q | N(E) is prime but q ∉ {2, p} ∪ S(E, p) then

(\frac{d}{q}) = \{ \begin{matrix}−1 & if ord_{q}(j(E)) ≥ 0, \\ −1 & if ord_{q}(j(E)) < 0 and E/ℚ_{q} is a Tate curve, \\ 1 & otherwise. \end{matrix}

If Cl(d)_{p} denotes the p-part of the ideal class group of ℚ(\sqrt{d}), then

#Cl(d)_{p} | #S(E(d), ℚ)_{p}.

Now we are in a position to give the precise definition of an excellent elliptic curve.

Definition 5.2. Let E/ℚ be an elliptic curve with a rational point of odd prime order ℓ which does not reduce to infinity modulo ℓ. Moreover, suppose that

g_{E}(z) = ∑_{n=1}^{∞} b_{E}(n)q^{n} ∈ S_{3/2}(Γ_{0}(4N, χ)

is an eigenform, for some N, satisfying the conclusion of Theorem 3.4 for the weight 2 newform

F_{E}(z) = ∑_{n=1}^{∞} a_{E}(n)q^{n} ∈ S_{2}(Γ_{0}(N(E)), χ_{0})

associated to E with δ = −1. Moreover, suppose that v is a place of K over 2 and that

s_{0} := min{ord_{v}(b_{E}(n))).

We say that E is excellent at ℓ if there is a negative square-free integer d_{E} which satisfies all of the following conditions:

(1) This integer d_{E} is coprime to ℓN(E) and satisfies the congruence

d_{E} ≡ 3 (mod 4).

(2) If ord_{ℓ}(j(E)) < 0, then (\frac{d_{E}}{ℓ}) = −1.

(3) If q | N(E) is prime but q ∉ {2, ℓ} ∪ S(E, ℓ), then

(\frac{d_{E}}{q}) = \{ \begin{matrix}−1 & if ord_{q}(j(E)) ≥ 0, \\ −1 & if ord_{q}(j(E)) < 0 and E/ℚ_{q} is a Tate curve, \\ 1 & otherwise. \end{matrix}

(4) We have that

ord_{v}(b_{E}(|d_{E}|)) = s_{0}.

Remark. Condition (4) above is not as restrictive as it may appear since it is typical that a suitable modification of g_{E}(z) satisfies it. Sprecifically, suppose that d_{E} has the property that b_{E}(|d_{E}|) ≠ 0. By using combinations of twists and twists of twists of g_{E}(z), one may replace g_{E}(z) by the cusp form

g̃_{E}(z) = ∑_{n≡|d_{E}| mod (ℚ_{q}^{⋆})^{2} for each prime q|N(E)} b_{E}(n)q^{n}.

It is now easy to see that g̃_{E}(z) is non-zero. and has a d̃_{E} which satisfies (5).

Wong [Wo] has proved a nice theorem regarding the existence of quadratic fields whose ideal class groups contain elements of order ℓ with the additional property that all of the prime factors p of the discriminant of these fields have F rob(p) lying in some prescribed conjugacy class in the Galois group of any fixed Galois extension K/ℚ. A straightforward modification of Wong’s arguments yields the following theorem.

Theorem 5.3. Let K/ℚ be a finite Galois extension, and let c be a conjugacy class in Gal(K/ℚ). Moreover, let S(K, c) denote the set of negative square-free integers which are divisible by only primes p which are unramified in K/ℚ and whose F rob(p) lies in c. Let M_{0} and M_{1} be positive odd square-free coprime integers. For every odd prime ℓ there are infinitely many numbers −d ∈ S(K, c) for which

(i) − dM_{1} ≡ 3 (mod 4),
(ii) gcd(d, M_{0}) = 1,
(iii) #Cl(−dM_{1})_{ℓ} ≡ 0 (mod ℓ),
(iv) μ(d) = 1.

Before we sketch the proof of Theorem 5.3, we use it to prove Theorem 6.

Proof of Theorem 5. Since E is excellent at ℓ, we may apply Theorem 5.1 by hypothesis to E(d_{E}). However, much more is true. Let S_{E} be the set of primes with positive density given in Theorem 3.5. By considering a sufficiently large Galois extension, one which contains the ℓN(E)th roots of unity, we may assume that every prime p ∈ S_{E} enjoys the following properties:

(5.1) (\frac{p}{q}) = (\frac{p_{0}}{q}) for every prime q | N(E),

(\frac{p}{ℓ}) = (\frac{p_{0}}{ℓ}). (5.2)

By (5.1-2) and Theorem 5.1 we find that if j is a positive integer and p_{1}, p_{2}, …, p_{2j} ∈ S_{E} are distinct primes not dividing d_{E}, then

L(E(d_{E}p_{1}p_{2} ⋯ p_{2j}), 1) ≠ 0, (5.3) #Cl(d_{E}p_{1}p_{2} ⋯ p_{2j})_{ℓ} | #S(E(d_{E}p_{1}p_{2} ⋯ p_{2j}). (5.4)

This follows from the fact that the conditions in Frey’s theorem are quadratic residue conditions which d_{E} already satisfies and that there are an even number of p_{i}.

However, by Kolyvagin’s theorem (5.3) implies that E(d_{E}p_{1}p_{2} ⋯ p_{2j}) has rank zero. Therefore, we find that if p_{1}, p_{2}, …, p_{2j} ∈ S_{E} are distinct primes which do not divide d_{E}, then

(5.5) #Cl(d_{E}p_{1}p_{2} ⋯ p_{2j})_{ℓ} ≡ 0 (mod ℓ) ⇒ #� (E(d_{E}p_{1}p_{2} ⋯ p_{2j}) ≡ 0 (mod ℓ).

The theorem now follows from Theorem 5.3.

Q.E.D.

Now we briefly show how to modify the Wong’s argument to deduce Theorem 5.3.

Sketch of Proof of Theorem 5.3. Suppose that p_{0} is a prime whose F rob(p_{0}) lies in c. Now pick a sufficiently large odd prime g for which (\frac{p_{0}}{g}) = −1 and (\frac{M_{1}}{g}) = (\frac{−1}{g}). Consider the Galois closure K′ of the extension of K which includes the gth roots of unity. Moreover, let c′ be the conjugacy class of p_{0} in Gal(K′/ℚ). It is easy to see that every prime p whose F rob(p) is in c′ has the property that its F rob(p) in Gal(K/ℚ) is in c and has the additional property that

(5.6)  (\frac{p}{g}) = −1.

If Δ_{K′} is the discriminant of K′, then define the positive integer C by

(5.7)  C := ∏_{q|M_{0}·Δ_{K′} odd prime, q∤M_{1}} q.

For positive integers G, define the set of integers S(G) by (5.8)

S(G) := \{0 < x ≤ \frac{G}{g} : x ∈ 2ℤ, gcd(x, g) = 1, x ≢ ±1 (mod q) for every prime q | C\} .

Let δ be any fixed positive multiple of ℓ with the property that with the property that q − 1 divides δ for every prime q | C. If λ is sufficiently large and x ∈ S(g^{δλ}) and satisfies the property that

(5.9)  d := g^{2δλ} − x^{2}

is square-free, then we easily see that d ≡ 1 (mod 4) and is a positive integer with no prime factors in common with C. By [Lemma 1, Wo], it follows that #Cl(−d)_{ℓ} ≡ 0 (mod ℓ).

Suppose that d is such a number which is a multiple of M_{1} with the property that all the prime factors p of d/M_{1} have F rob(p) in c′. Since (\frac{d/M_{1}}{g}) = 1, by (5.9) and the choice of g, and since every prime p whose F rob(p) in c′ has the property that (\frac{p}{g}) = −1 by (5.6), there must be an even number of p. This yields (iv). Hence, any such d satisfies the conclusion of the theorem.

Therefore it suffices to show that one can construct infinitely many such d. If λ is sufficiently large, then there always is such a d, and the proof of this assertion follows virtually mutatis mutandis the proof of [Lemma 2, Wo]. Since selecting larger and larger λ′s uncovers quadratic fields whose class groups contains elements with orders that are larger and larger multiples of ℓ, the infinitude of the number of d obtained in this way is obvious.

Q.E.D.

Example. Let E/ℚ be the elliptic curve with conductor 26 given by

E : y^{2} + xy + y = x^{3} − x^{2} − 3x + 3.

Its torsion subgroup is ℤ/7ℤ. In his thesis [Ba], Basmaji finds the cusp form g_{E}(z)

g_{E}(z) = ∑_{n=1}^{∞} b_{E}(n)q^{n} := \frac{1}{2} ∑_{x,y,z∈ℤ} (q^{x^{2}+13y^{2}+13z^{2}} − q^{3x^{2}+9y^{2}+9z^{2}−8yz+2xz+2xy})

= q − q^{3} + q^{4} − 2q^{9} − 3q^{10} − q^{12} + 2q^{13} + q^{14} + q^{16} + q^{17} + ⋯ ∈ S_{3/2}(Γ_{0}(52), χ_{0})

which has the property that if n_{1} and n_{2} are square-free positive integers for which gcd(n_{1}n_{2}, 52) = 1 and \frac{n_{1}}{n_{2}} ∈ (ℚ_{p}^{⋆})^{2} for p = 2 and 13, then

b_{E}(n_{2})^{2} · L(E(−n_{1}), 1) · \sqrt{n_{1}} = b_{E}(n_{1})^{2} · L(E(−n_{2}), 1) · \sqrt{n_{2}}.

Notice that this form is slightly different from the form prescribed by Theorem 3.4. For example, there are nonzero coefficients b_{E}(n) for which gcd(n, 52) ≠ 1. Their presence has no impact on the theoretical arguments (e.g. one can simply annihilate them by twisting by the trivial character modulo 52). However, here we maintain Basmaji’s form because of its convenient presentation.

To justify that E is excellent at 7, notice that s_{0} = 1 and that b_{E}(17) = 1. Using standard facts about elliptic curves it is a straightforward exercise to show that d_{E} = −17 satisfies conditions (1-4) in Definition 5.2.

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Department of Mathematics, Penn State University, University Park, Pa. 16802

   E-mail address: ono@math.psu.edu

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

   E-mail address: ono@math.wisc.edu