MODULAR FORM CONGRUENCES AND SELMER GROUPS

William J. McGraw and Ken Ono

April 8, 2002 (Not for distribution)

1. Introduction and Statement of Results.

Motivated by Cremona and Mazur’s notion of visibility of elements in Shafarevich-Tate groups [C-Ma, Ma2], there have been a number of recent works which test its compatibility with the Conjectures of Birch and Swinnerton-Dyer and Bloch-Kato, conjectures which provide formulas for the orders of Shafarevich-Tate groups in terms of values of L-functions. For example, one may see recent work of Agashe and Stein [A-S] and Dummigan [D]. In the examples they consider, they find that the presence of visible elements agrees with the expected divisibility properties of the relevant L-values.

In this paper we aim to provide further results in this direction. We begin by briefly recalling the relevant objects. Suppose that F(z) = ∑_{n=1}^{∞} A(n)q^{n} ∈ S_{2k}^{new}(Γ_{0}(N)) is an even weight newform whose coefficients are algebraic integers in a number field K. If D is the fundamental discriminant of a quadratic field, then let F_{D}(z) denote its D-quadratic twist. Similarly, if E/ℚ is an elliptic curve, then let E(D) denotes its D-quadratic twist.

By a classical theorem of Deligne [De1], for each finite prime λ of K there is a two dimensional vector space V_{λ} over K_{λ}, and a continuous Galois representation

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

with the property that for primes p ∤ Nℓ (where λ | ℓ) we have

Tr(ρ_{λ}(Frob_{p})) = A(p) and det(ρ_{λ}(Frob_{p})) = p^{2k−1}.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11F33, 11F37, 11F67, 11G40.

Key words and phrases. Congruences, Selmer groups, modular forms.

The first author thanks the Number Theory Foundation for their generous support. The second author is grateful for the support of the Alfred P. Sloan, David and Lucile Packard, and H. I. Romnes Fellowships, and the support of the National Science Foundation.

By work of Scholl [Sch], the spaces V_{λ} are λ-adic realizations of a Grothendieck motive, say M_{F}. If j is a positive integer, then let M_{F}^{(j)} denote the usual jth Tate twist of M_{F}. Furthermore, if D is a fundamental discriminant, then let M_{F}^{(j,D)} denote the motive

M_{F}^{(j,D)} := M_{F}^{(j)} ⊗_{ℚ} D,  (1.1)

where D denotes the motive associated to the usual Kronecker character χ_{D}. As usual, let S (M_{F}^{(j,D)}) (resp. � (M_{F}^{(j,D)})) denote the associated Selmer group (resp. Shafarevich-

Tate group). By the Bloch-Kato Conjecture, if L(F_{D}, k) ≠ 0, then #� (M_{F}^{(k,D)}) is predicted by the formula [B-K]

L(F_{D}, k) = \frac{(∏_{p} c_{p}(χ_{D}, k)) Ω(F_{D}, k) · #� (M_{F}^{(k,D)})}{#Γ_{ℚ}(F_{D}, k)^{2}}.  (1.2)

Here the c_{p}(χ_{D}, k)’s are the Tamagawa numbers for M_{F}^{(k,D)}, Ω(F_{D}, k) is its Deligne period [De2], and Γ_{ℚ}(F_{D}, k) denotes its global points. For convenience, we define the algebraic part of L(F_{D}, k) by

L^{alg}(F_{D}, k) = \frac{L(F_{D}, k)}{Ω(F_{D}, k)}.  (1.3)

Throughout we adopt the following notation. If p is an odd prime and D is a fundamental discriminant coprime to p, then let D(p) denote the fundamental discriminant

D(p) := (−1)^{(p−1)/2}Dp.  (1.4)

Furthermore, we shall refer to a prime p as a congruence prime for a newform f(z) ∈ S_{2k}^{new}(Γ_{0}(M)) if there is another newform f_{1}(z) ∈ S_{2k}^{new}(Γ_{0}(M)) for which

f(z) ≡ f_{1}(z) (mod 𝔭),

for some prime ideal 𝔭 above p in the ring of algebraic integers of a suitably large number field.

Using this notation, we are able to describe a natural strategy for visualizing elements in Selmer groups of motives of higher weight modular forms using elliptic curves. Suppose that E/ℚ is an elliptic curve, and that f(z) is the usual weight 2 newform associated to E. Suppose that p is a prime of bad reduction that is not a congruence prime for f(z). Then there may be a newform of weight p + 1, say F(z), and a prime ideal 𝔭 above p in a suitable number field for which

f(z) ≡ F(z) (mod 𝔭)

and

L^{alg}(f_{D}, 1) ≡ L^{alg}(F_{D(p)}, (p + 1)/2) (mod 𝔭),  (1.5)

for certain fundamental discriminants D. For such D, if E(D) has positive rank over ℚ, then the p-part of the Selmer group of E(D) is non-trivial, and one then hopes to visualize the existence of some corresponding elements of order p in S (M_{F}^{(k,D(p))}). Furthermore, we

expect to visualize 𝔭-divisibility of L^{alg}(F_{D(p)}, (p + 1)/2).

Examples of this strategy can be worked out explicitly. For example, Dummigan [D] analyzes the classical mod 11 congruence (q = e^{2πiz} throughout)

Δ(z) ≡ f(z) = q − 2q^{2} − q^{3} + … (mod 11),  (1.6)

where Δ(z) is the unique normalized weight 12 cusp form on SL_{2}(ℤ), and f(z) is the unique normalized form in S_{2}(Γ_{0}(11)) which is associated to the elliptic curve E = X_{0}(11)

E : y^{2} + y = x^{3} − x^{2} − 10x − 20.

He shows how (1.6) naturally provides relations between the 11-parts of the corresponding Selmer groups. He then confirms the expected congruences between the relevant L-values by employing work of Kohnen and Zagier [K2, K-Z] on Shimura’s correspondence [Sh]. These works show that the relevant L-values are essentially the squares of the coefficients of the Kohnen newforms

g(z) = q^{3} − q^{4} − q^{11} − q^{12} + q^{15} + ⋯ ∈ S_{3/2}^{new}(Γ_{0}(44)),
G(z) = q − 56q^{4} + 120q^{5} − 240q^{8} + ⋯ ∈ S_{13/2}^{new}(Γ_{0}(4)).

The conjectured L-value congruences follow from the fact that

g(z) ≡ −G(z) | U(11) (mod 11),  (1.7)

where U(d) denotes the usual Atkin U-operator

(∑_{n=0}^{∞} c(n)q^{n}) | U(d) = ∑_{n=0}^{∞} c(dn)q^{n}.  (1.8)

One naturally wonders how general these beautiful observations are; after all, the heuristics motivating these calculations suggest that such relations should be quite common. Indeed, works by Koblitz, Mazur and Vatsal [Kb1, Ma1, V] and others already consider questions related to congruences between critical values of congruent L-functions. In this direction, Dummigan generalizes his mod 11 example using works of Hida [H1] and Stevens [Ste] on Λ-adic modular forms of integral and half-integral weight. He shows that if E is an elliptic curve with prime conductor p ≡ 3 (mod 4) where p ∤ ord_{p}(j(E)), then there are Hecke eigenforms of some large weight which play the role of Δ(z) as above.

Here we concentrate on proving general theorems regarding congruences like (1.7). We obtain such general p-adic results on Kohnen’s theory of half-integral weight modular forms [K1, K2]. As a consequence, this allows us to obtain some further results on the visibility of elements of Selmer groups. As a special case, combining our observations with Dummigan’s computations [Th. 8.1, D] leads to the following theorem.

Theorem 1. Suppose that E/ℚ is an elliptic curve with prime conductor p, and let f(z) ∈ S_{2}^{new}(Γ_{0}(p)) be its associated newform. If p is not a congruence prime for any newform in S_{p+1}(Γ_{0}(1)), then there is a newform

F(z) ∈ S_{p+1}(Γ_{0}(1))

and a prime ideal 𝔭 in a suitably large number field which satisfies all of the following:

(1) We have that

f(z) ≡ F(z) (mod 𝔭).

(2) If w_{p} ∈ {±1} is the eigenvalue of f(z) with respect to the Atkin-Lehner involution W (p) and D is a negative fundamental discriminant D with (\frac{D}{p}) = w_{p}, then the sign of the functional equation of L(F_{D(p)}, s) is +1.

(3) There is a negative fundamental discriminant, say D_{0}, with the property that

𝕃^{alg}(F_{D(p)}, (p + 1)/2) = \frac{L^{alg}(F_{D(p)}, (p + 1)/2)}{L^{alg}(F_{D_{0}(p)}, (p + 1)/2)}

and

𝕃^{alg}(f_{D}, 1) = \frac{L^{alg}(f_{D}, 1)}{L^{alg}(f_{D_{0}}, 1)}

are both 𝔭-integral for every negative fundamental discriminant D with (\frac{D}{p}) = w_{p}. Moreover, for such D we have that

𝕃^{alg}(F_{D(p)}, (p + 1)/2) ≡ 0 (mod 𝔭) if and only if 𝕃^{alg}(f_{D}, 1) ≡ 0 (mod 𝔭).

(4) If D is a negative fundamental discriminant for which (\frac{D}{p}) = w_{p} and E(D) has rank ≥ 2 over ℚ, then

𝕃^{alg}(F_{D(p)}, (p + 1)/2) ≡ 0 (mod 𝔭) and S (M_{F}^{((p+1)/2,D(p))}) [p] ≠ {0}.

Moreover, the number of such −X < D < 0 is ≫_{E} \frac{X^{1/7}}{(log X)^{2}} .

Remarks. 1) Let K_{1} (resp. K_{2}) be the number field obtained by adjoining all the Fourier coefficients of the newforms (resp. Kohnen newforms) in S_{2}^{new}(Γ_{0}(p)) and S_{p+1}^{new}(Γ_{0}(1)) (resp. Snew_{\frac{3}{2}}(Γ_{0}(4p)) and Snew_{\frac{(p+2)}{2}} (Γ_{0}(4))). Any number field containing K = K_{1}K_{2} is sufficiently large for Theorem 1.

2) The conclusion of Theorem 1 is valid provided that there is at most one newform F(z) ∈ S_{p+1}(Γ_{0}(1)) for which F(z) ≡ f(z) (mod 𝔭). The existence of a single such F(z) is guaranteed by a well known lemma of Deligne and Serre [Lemma 6.11, De-S].

3) W. Stein has kindly informed us that the only prime p < 10^{4} that is a congruence prime for any newform in S_{p+1}(Γ_{0}(1)) is p = 389.

4) Assuming the Parity Conjecture for elliptic curves over ℚ, a result of Stewart and Top [§12, S-T], extending work of Gouvêa and Mazur [G-Ma], allows us to improve the lower bound in Theorem 1 (4) to X^{\frac{1}{2}} .

5) By the proof of [Th. 1, J-O], it follows that there are ≫_{E} \sqrt{X}/ log X many choices for the D_{0} in Theorem 1 (3) with −X < |D_{0}| < 0. Similarly, it also implies that

# \{−X < D < 0 : (\frac{D}{p}) = w_{p} and 𝕃^{alg}(F_{D(p)}, (p + 1)/2) ≢ 0 (mod 𝔭)\} ≫_{E} \frac{\sqrt{X}}{log X} .

It is natural to ask how often the elements in these Selmer groups are elements of Shafarevich-Tate groups. In this direction, we state a widely believed conjecture [CKRS].

Conjecture Z. If F(z) = ∑_{n=1}^{∞} A(n)q^{n} ∈ S_{2k}^{new}(Γ_{0}(N)) is a newform with 2k ≥ 6, then there are at most finitely many fundamental discriminants D for which

ord_{s=k}(L(F_{D}, s)) ≡ 0 (mod 2) and L(F_{D}, k) = 0.

The Bloch-Beilinson Conjecture, the conjectured injectivity of the p-adic Abel-Jacobi map, and Conjecture Z imply that the fundamental discriminants D in Theorem 1 (4), apart from finitely many exceptions, yield nontrivial elements in � (M_{F}^{((p+1)/2,D(p))}) [p]. In particular, we have

# \{−X < D < 0 : � (M_{F}^{((p+1)/2,D(p))}) [p] ≠ \{0\}\} ≫_{E} \frac{X^{1/7}}{(log X)^{2}} .  (1.9)

As in Remark 4, the truth of the Parity Conjecture for elliptic curves would allow us to replace this lower bound by one of the form X^{\frac{1}{2}} .

As mentioned above, these results follow from a general study of the p-adic properties of Kohnen’s theory of half-integral weight modular forms (see §3 for notation and essential facts). For odd primes p and non-negative integers j, let κ(p, j) denote the integer

κ(p, j) := \frac{p^{j}(p − 1)}{2}.  (1.10)

If p, k and j satisfy some mild inequalities, then we obtain the following commutative diagram on modular forms modulo p^{j+1} when N is an odd-square free integer coprime to p:

S_{2k}^{new}(Γ_{0}(Np)) ⟶^{≡} S_{2(k+κ(p,j))}^{+}(Γ_{0}(N))
↑  ↑
S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4Np)) ⟶^{≡′} S_{k+κ(p,j)+ \frac{1}{2}}^{+} (Γ_{0}(4N))

(1.11)

The vertical arrows denote the Shimura correspondence modulo p^{j+1}. The two horizontal arrows denote the reduction of p-integral modular forms modulo p^{j+1}. However, the ≡′ above the bottom arrow denotes a congruence with a form hit with the U(p) operator. Notice that (1.7) is the case of this picture where N = 1, p = 11, j = 0 and k = 1.

Since there may be congruences between newforms of weight 2(k + κ(p, j)), it is often difficult to tease arithmetic information out of (1.11). However, the next result provides a clear picture in some cases where this difficulty is not present.

Theorem 2. Suppose that p ≥ 5 is prime and that Np is an odd square-free integer. Furthermore, suppose that j ≥ 0 and k ≥ 1 are integers for which p^{j} ≥ k, and that

f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}^{new}(Γ_{0}(Np)

and

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4Np))

are newforms that are associated by Kohnen’s isomorphism. Let K be the number field obtained by adjoining all the coefficients of all the newforms in the spaces S_{2k}^{new}(Γ_{0}(N′)) and S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4N′)) where N′ | N. Let O_{K} denote its ring of algebraic integers, and suppose that 𝔭 ⊂ O_{K} is a prime ideal above p. If there is exactly one newform

F(z) = ∑_{n=1}^{∞} A(n)q^{n} ∈ S_{2(k+κ(p,j))}^{new}(Γ_{0}(N′))

with N′ | N with the property that

a(ℓ) ≡ A(ℓ) (mod 𝔭)

for every prime ℓ ∤ Np, then N′ = N implies that there is a unique Kohnen newform G(z) = ∑_{n=1}^{∞} B(n)q^{n} ∈ S_{k+κ(p,j)+ \frac{1}{2}}^{new} (Γ_{0}(4N)) for which

g(z) ≡ G(z) | U(p) (mod 𝔭).

Moreover, F(z) is the image of G(z) under the Shimura correspondence.

Theorem 2 is quite useful when combined with works by Kohnen and Zagier and Waldspurger (see [K1, K2, K-Z, W]) on Shimura’s correspondence [Sh]. They provide useful formulae for the central critical values of quadratic twists of the L-functions of even weight newforms. These numbers are essentially the squares of suitable Fourier coefficients of halfintegral weight eigenforms. As a consequence, we often obtain congruences between the algebraic parts of L-values associated to congruent modular forms.

We recall a well known formula due to Kohnen. Suppose that N is odd and square-free, and suppose further that

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

is a Kohnen newform. By Kohnen’s isomorphism, there is a unique newform, say f(z) ∈ S_{2k}^{new}(Γ_{0}(N)), associated to g(z) under Shimura’s correspondence. The coefficients of g(z) determine the central critical values of many of the quadratic twists L(f_{D}, s). To state these formulae, let ν(N) denote the number of distinct prime divisors of N, and let ⟨f, f⟩ (resp. ⟨g, g⟩) denote the usual Petersson inner product on S_{2k}(Γ_{0}(N)) (resp. S_{k+ \frac{1}{2}} (Γ_{0}(4N))). If ℓ | N is prime, then let w_{ℓ} ∈ {±1} be the eigenvalue of the usual Atkin-Lehner involution

f |_{2k}W (ℓ) = w_{ℓ}f.

If (−1)^{k}D > 0 and D has the additional property that (\frac{D}{ℓ}) = w_{ℓ} for each prime ℓ | N, then Kohnen proves [Cor. 1, K2] that

L(f_{D}, k) = \frac{⟨f, f⟩ · π^{k}}{2^{ν(N)}(k − 1)!|D|^{k− \frac{1}{2}} ⟨g, g⟩} · b(|D|)^{2}.  (1.12)

For all other fundamental discriminants D with (−1)^{k}D > 0, it turns out that b(|D|) = 0. For those D for which (1.12) holds, we define L_{K}^{alg}(f_{D}, k), the Kohnen algebraic part of L(f_{D}, k), by

L_{K}^{alg}(f_{D}, k) := b(|D|)^{2}.  (1.13)

This discussion together with Theorem 2 immediately implies the following corollary.

Corollary 3. Assume the hypotheses and notation from Theorem 2. If D is a fundamental discriminant with (−1)^{k}D > 0 with the additional property that (\frac{D}{ℓ}) = w_{ℓ} for every prime ℓ | Np, then

L_{K}^{alg}(f_{D}, k) ≡ L_{K}^{alg}(F_{D(p)}, k + κ(p, j)) (mod 𝔭).

In §2 we prove certain p-adic facts regarding integral weight newforms, and in §3 we prove their analogs in Kohnen’s theory of half-integral weight modular forms. In §4 we prove Theorems 1 and 2.

Acknowledgements.

The authors thank...

2. p-adic properties of integer weight newforms.

We begin this section by recalling some preliminary facts about modular forms. For more general information, consult [Kb2, M]. Let k be a positive integer. The group

GL_{2}^{+}(ℝ) = \{γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) | a, b, c, d ∈ ℝ and ad − bc > 0\}

acts on functions f : 𝔥 → ℂ by the operator

(f|_{k}γ) (z) = (det γ)^{k/2}(cz + d)^{−k}f(γz),  (2.1)

where γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Here, 𝔥 = \{x + iy ∈ ℂ | y > 0\}. For a prime divisor p of N for which

gcd(p, N/p) = 1, define the operator |_{k}W (p) on M_{k}(Γ_{0}(N)) by the matrix

W (p) = ( \begin{matrix}p & a \\ N & pb \end{matrix} ) ∈ M_{2}(ℤ)  (2.2)

with determinant p. Since W (p) is unique up to left multiplication by an element of Γ_{0}(N), this indeed defines an action on M_{k}(Γ_{0}(N)). It turns out that W (p) is an involution on M_{k}(Γ_{0}(N)), a so-called Atkin-Lehner involution. Notice that if N = p, then we may select the following representative for W (p)

W (p) = ( \begin{matrix}0 & −1 \\ p & 0 \end{matrix} ) .  (2.3)

Similarly, if d is a positive integer, then the V -operator V (d) is defined by

(∑_{n=0}^{∞} c(n)q^{n}) | V (d) = ∑_{n=0}^{∞} c(n)q^{dn}.  (2.4)

Our main result (i.e. Theorem 2.3) in this section shall depend on these operators, the Uoperators defined in (1.8), the Atkin-Lehner theory of newforms, and the p-adic properties of certain Eisenstein series.

We now briefly recall some essential features of the Atkin-Lehner theory of newforms [At-L]. For a positive integer ℓ, let δ_{ℓ} = ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ). Then we define the subspace

S_{k}^{old}(Γ_{0}(N)) := ⊕_{ℓM|N} S_{k}(Γ_{0}(M))|_{k}δ_{ℓ},

where the sum runs over ℓM | N with M ≠ N. Define S_{k}^{new}(Γ_{0}(N)) to be the orthogonal complement of S_{k}^{old}(Γ_{0}(N)) with respect to the Petersson inner product. The Hecke operators are defined as usual on f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{k}(Γ_{0}(N)); for primes ℓ ∤ N, T (ℓ) is given by

f(z)|_{k}T (ℓ) = ∑_{n=1}^{∞} c(n)q^{n},

where

c(n) = a(ℓn) + ℓ^{k−1}a (\frac{n}{ℓ}) .

A newform of level N is a normalized cusp form that is an eigenform of all the Hecke operators and all the Atkin-Lehner involutions. It is well-known that the coefficients of a newform are algebraic integers in a number field, and S_{k}^{new}(Γ_{0}(N)) has a basis consisting of newforms.

If k ≥ 2 is even, then let E_{k}(z) denote the usual Eisenstein series

E_{k}(z) = 1 − \frac{2k}{B_{k}} ∑_{n=1}^{∞} σ_{k−1}(n)q^{n},  (2.5)

As usual, B_{k} denotes the kth Bernoulli number and σ_{r}(n) := ∑_{d|n} d^{r}. For k ≥ 4, E_{k}(z) ∈ M_{k}(Γ_{0}(1)). If p ≥ 5 is prime, then define the modular form E_{p}(z) ∈ M_{p−1}(Γ_{0}(p)) by

E_{p}(z) := E_{p−1}(z) − p^{\frac{p−1}{2}} (E_{p−1}|_{p−1}W (p)) (z).  (2.6)

Although E_{2}(z) = 1 − 24 ∑_{n=1}^{∞} σ_{1}(n)q^{n} is not a modular form, it plays an important role here. In particular, if p = 3, then define E_{3}(z) ∈ M_{2}(Γ_{0}(3)) [p. 817, He] by

E_{3}(z) := E_{2}(z) − 3E_{2}(3z).  (2.7)

Lemma 2.1. If p is an odd prime, then we have

E_{p}(z) ≡ 1 (mod p),  (2.8)
(E_{p}|_{p−1}W (p))(z) ≡ 0 (mod p^{κ(p,0)+1}).  (2.9)

Proof. By (2.3), we first note that, for p ≥ 5,

E_{p−1}|_{p−1}W (p) = E_{p−1}|_{p−1} ( \begin{matrix}0 & −1 \\ p & 0 \end{matrix} )

= E_{p−1}|_{p−1} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )

= p^{\frac{p−1}{2}} E_{p−1}|V (p).  (2.10)

This is clear for p = 3 by definition. In particular E_{p}(z) ≡ E_{p−1}(z) (mod p), but E_{p−1}(z) ≡ 1 (mod p) since the von Staudt-Clausen congruences imply that B_{p−1}^{−1} ≡ 0 (mod p). This gives (2.8).

By (2.10), we have

E_{p}|_{p−1}W (p) = E_{p−1}|_{p−1}W (p) − p^{\frac{p−1}{2}} E_{p−1}
= p^{\frac{p−1}{2}} (E_{p−1}|V (p) − E_{p−1}).

This formula together with (2.5) and the von Staudt-Clausen congruences immediately implies (2.9).

□

If M and N are relatively prime positive integers, we define the trace operator

Tr_{N}^{MN} : M_{k}(Γ_{0}(MN)) → M_{k}(Γ_{0}(N))

by

Tr_{N}^{MN}(f) = ∑_{i=1}^{r} f|_{k}γ_{i},  (2.11)

where {γ_{1}, …, γ_{r}} is a complete set of coset representatives for Γ_{0}(NM)\Γ_{0}(N). Obviously, these trace operators have the property that they take cusp forms to cusp forms. The next lemma provides a closed formula for these operators.

Lemma 2.2. Suppose that p is an odd prime and that p ∤ N. If f ∈ M_{k}(Γ_{0}(Np)), then

Tr_{N}^{Np}(f) = f + p^{1−k/2}f|_{k}W (p)U(p).

Proof. A complete set of representatives for Γ_{0}(Np)\Γ_{0}(N) is given by the matrices

\{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )\} ∪ \{( \begin{matrix}1 & 0 \\ N & 1 \end{matrix} ) ( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} )\}_{j=0}^{p−1} .

However,

( \begin{matrix}1 & 0 \\ N & 1 \end{matrix} ) ( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}\frac{1}{p} & 0 \\ 0 & \frac{1}{p} \end{matrix} ) ( \begin{matrix}p & a \\ Np & pb \end{matrix} ) ( \begin{matrix}1 & j−a \\ 0 & p \end{matrix} ) ,

where ( \begin{matrix}p & a \\ Np & pb \end{matrix} ) is the matrix for W (p). Since scalar matrices act trivially on M_{k}(Γ_{0}(Np)),

Tr_{N}^{MN}(f) = f + ∑_{j=1}^{p−1} f|_{k}W (p) ( \begin{matrix}1 & j \\ 0 & p \end{matrix} ) .

A simple calculation shows that, for g ∈ M_{k}(Γ_{0}(Np)),

∑_{j=0}^{p−1} g (\frac{z + j}{p} ) = p(g|U(p))(z).

This proves the assertion.

□

Theorem 2.3. Suppose that p is an odd prime and that

f(z) = ∑_{n=1}^{∞} a(n)q^{n} ∈ S_{2k}^{new}(Γ_{0}(Np))

is an even integer weight newform. If N is coprime to p and j is a non-negative integer for which M := min(j + 1, p^{j} − k + 1) is positive, then there is a cusp form

F (z) = ∑_{n=1}^{∞} A(n)q^{n} ∈ S_{2(k+κ(p,j))}(Γ_{0}(N))

which satisfies the congruence

f(z) ≡ F (z) (mod p^{M}).

Theorem 2.3 is a straightforward generalization of work of Serre in the N = 1 case [§3, S].

Proof. For convenience, we define ν_{p}(P), for a power series P (z) = ∑ c(n)q^{n} by

ν_{p}(P) := inf{ord_{p}(c(n))},  (2.13)

where ord_{p} is the usual p-adic valuation.

Define the cusp form f_{j}(z) ∈ S_{2(k+κ(p,j))}(Γ_{0}(Np)) by

f_{j}(z) := f(z)E_{p}(z)^{p^{j}}.  (2.14)

By Lemma 2.2, we have

Tr_{N}^{Np}(f_{j}) = f_{j} + p^{1−k−κ(p,j)}f_{j}|_{2(k+κ(p,j))} W (p)U(p)
= f_{j} + p^{1−k−κ(p,j)} ((f|_{2k} W (p)) (E_{p}|_{2κ(p,0)}W (p))^{p^{j}}) | U(p).  (2.15)

Since f is a newform, [Th.3, At-L] implies that there an eigenvalue w_{p} ∈ {±1} of the Atkin-Lehner involution W (p) for which

f|_{2k}W (p) = w_{p}f.

In particular it follows that ν_{p} (f|_{2k}W (p)) = 0. Therefore by (2.9) it follows that

ν_{p} ((f|_{2k}W (p) · E_{p}|_{2κ(p,0)}W (p))^{p^{j}}) ≥ p^{j}(κ(p, 0) + 1) = κ(p, j) + p^{j}.

Therefore by (2.15), we have

ν_{p} (Tr_{N}^{Np}(f_{j}) − f_{j}) ≥ p^{j} − k + 1.

Since f_{j} ≡ f (mod p^{j+1}), this completes the proof.

□

3. p-adic properties of Kohnen newforms.

Here we obtain a result which is analogous to Theorem 2.3, but we need, first, to discuss the work of Kohnen on newforms of half-integral weight. For a general reference on half-integral weight modular forms, see [Kb2]. Shimura [Sh] defines a correspondence between certain integral weight and half-integral weight modular forms. This correspondence commutes with the respective actions of the Hecke algebras. However, it is not, in general, an isomorphism. Kohnen [K1] clarified this problem by defining suitable subspaces of cusp forms for which the Shimura correspondence is an isomorphism of Hecke modules.

Suppose that N is an positive odd square-free integer, and that k is a positive integer. Moreover, suppose that χ is a Dirichlet character modulo N and that ε = χ(−1). Kohnen [K1] defines the space S_{k}^{+}(Γ_{0}(4N), χ) to be the space of cusp forms g(z) = ∑_{n=1}^{∞} b(n)q^{n} of weight k + \frac{1}{2} and character (\frac{4ε}{·}) χ on Γ_{0}(4N) with the property that

b(n) = 0 whenever ε(−1)^{k}n ≡ 2, 3 (mod 4).  (3.1)

In Kohnen’s notation, this space is S_{k}^{+}(N, χ) (or simply S_{k}(N, χ)). If χ is trivial, we suppress χ from the notation and refer to the space as S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)).

As in the integral weight case, S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)) decomposes into subspaces (note. we suppress the + in the notation below)

S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)) = S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4N)) ⊕ S_{k+ \frac{1}{2}}^{old} (Γ_{0}(4N)).  (3.2)

Kohnen proves the beautiful fact that S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4N)) is isomorphic to S_{2k}^{new}(Γ_{0}(N)). This isomorphism is given by an explicit combination of Shimura lifts. We remark that we abuse terminology throughout and refer to this linear combination as the Shimura correspondence.

As before, we can define Hecke operators on S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)). However, in this case, we

have only the Hecke operators T (n^{2}) for positive integers n. If ℓ ∤ 4N is prime, T (ℓ^{2}) acts on g(z) = ∑ b(n)q^{n} ∈ S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)) by

g(z)|_{k}T (ℓ^{2}) = ∑_{n=1}^{∞} c(n)q^{n},

where

c(n) = b(ℓ^{2}n) + (\frac{ε(−1)^{k}n}{ℓ}) ℓ^{k−1}b(n) + ℓ^{2k−1}b (\frac{n}{ℓ^{2}} ) .

As in the integer weight theory due to Atkin and Lehner, the spaces S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4N)) have a basis of cusp forms which are eigenforms of the Atkin-Lehner involutions and the halfintegral weight Hecke operators. It is well known that such forms may be normalized so that they have coefficients which are in O_{K}, the ring of algebraic integers in some fixed number field K. Moreover, they can be normalized so that the ideal generated by its coefficients is the full ring O_{K}. We shall refer to such a cusp form as a Kohnen newform of level 4N. By Kohnen’s isomorphism, the image of a Kohnen newform is a newform in S_{2k}^{new}(Γ_{0}(N)).

Now we recall the explicit operators we require to prove the main result of this section (i.e. Theorem 3.3). If k is a positive integer, then, following Kohnen, define the group

G̃L_{2}^{+}(ℝ)_{k} = \{(γ, ϕ) | γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(ℝ) and |ϕ(z)| = (det γ)^{− \frac{k}{2} − \frac{1}{4}} |cz + d|^{k+ \frac{1}{2}} \} .

(3.3)

The functions ϕ above are holomorphic functions on 𝔥. This group acts on functions f : 𝔥 → ℂ by the operator

(f|_{k+ \frac{1}{2}} γ) (z) = ϕ(z)^{−1}f(γz).  (3.4)

For a prime divisor p of N such that gcd(p, 4N/p) = 1, there is an operator W (p)_{k} given by

W (p)_{k} = (( \begin{matrix}p & a \\ 4N & pb \end{matrix} ) , (\frac{−4}{p})^{−(k+ \frac{1}{2} )} p^{− \frac{1}{2} (k+ \frac{1}{2} )}(4Nz + pb)^{k+ \frac{1}{2}} ) ,  (3.5)

with a, b ∈ ℤ and determinant p. We note that W (p)_{k} is unique up to left multiplication by an element of Γ_{0}(4N) and therefore as an action on M_{k+ \frac{1}{2}} (Γ_{0}(4N)). The map

W (p)_{k} : S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)) → S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N), (\frac{·}{p}))  (3.6)

is an isomorphism and W (p)_{k}^{2} acts as the identity [p. 39, K1]. We will suppress k in this notation as it will be clear in context. If p | N, then the operator U(p) defined by (1.8) also maps [Prop. 1.5, Sh]

U(p) : S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N)) → S_{k+ \frac{1}{2}}^{+} (Γ_{0}(4N), (\frac{·}{p})) .  (3.7)

Therefore, for every prime p | N this defines a Hermitian involution 𝔴_{p} given by

𝔴_{p} = p^{− \frac{k}{2} + \frac{1}{4}} U(p)W (p).  (3.8)

By Shimura’s correspondence and Kohnen’s isomorphism, the ±1-eigenspaces of 𝔴_{p} correspond to the ±1-eigenspaces under the W (p) operator on S_{2k}^{new}(Γ_{0}(N)) [Prop. 4, K1].

Suppose that k is a positive integer. If p is an odd prime, and a character χ_{p} denotes the Legendre symbol modulo p, then there is an Eisenstein series E_{k,χ_{p}}(z) ∈ M_{k}(Γ_{0}(p), χ_{p}), whose Fourier expansion is given by

E_{k,χ_{p}}(z) = 1 − \frac{2k}{B_{k,χ_{p}}} ∑_{n=1}^{∞} σ_{k−1}(n, χ_{p})q^{n},  (3.9)

where B_{k,χ_{p}} is a generalized Bernoulli number and σ_{k}(n, χ_{p}) = ∑_{d|n} χ_{p}(d)d^{k}. It is well known that E_{k,χ_{p}}(z) is the weight k Eisenstein series which is 1 at infinity and vanishes at the cusp 0. The complementary Eisenstein series which vanishes at ∞ and is non-zero at 0 is (see [5.1, H2])

(E_{k,χ_{p}}|_{k}W (p)) (z) = −χ_{p}(−1)p^{k−2}𝔤(χ_{p})\frac{2k}{B_{k,χ_{p}}} ∑_{n=1}^{∞} σ′_{k−1}(n, χ_{p})q^{n},  (3.10)

where 𝔤(χ_{p}) = ∑_{a (mod p)} χ_{p}(a)e^{\frac{2πia}{p}} is the Gauss sum and σ′_{k}(n, χ_{p}) = ∑_{d|n} χ_{p}(d) (\frac{n}{d})^{k}. If p is an odd prime, then define Ẽ_{p}(z) ∈ M_{κ(p,0)}(Γ_{0}(p), χ_{p}) by

Ẽ_{p}(z) = E_{\frac{p−1}{2} ,χ_{p}}(z).  (3.11)

Lemma 3.1. If p is an odd prime, then we have

Ẽ_{p}(z) ≡ 1 (mod p),  (3.12)
(Ẽ_{p}|_{\frac{p−1}{2}} W (p)) (z) ≡ 0 (mod p^{κ(p,0)−1}).  (3.13)

Proof. Property (3.12) follows from the von Staudt-Clausen congruences; in this case, the generalized Bernoulli number

(B_{\frac{p−1}{2} ,(\frac{·}{p})})^{−1} ≡ 0 (mod p).  (3.14)

This congruence, combined with (3.10), gives (3.13).

□

In light of (3.13), we will write

(Ẽ_{p}|_{\frac{p−1}{2}} W (p)) (z) = p^{κ(p,0)−1}Ẽ′_{p}(z),  (3.15)

where Ẽ′_{p}(z) is clear from (3.10).

As in the last section, if M and N are relatively prime positive integers, we define the trace operator

Tr_{4N}^{4MN} : M_{k+ \frac{1}{2}} (Γ_{0}(4MN)) → M_{k+ \frac{1}{2}} (Γ_{0}(4N))

by

Tr_{4N}^{4MN}(f) = ∑_{i=1}^{r} f|_{k+ \frac{1}{2}} γ_{i},  (3.16)

where {γ_{1}, …, γ_{r}} is a complete set of coset representatives for Γ_{0}(4NM)\Γ_{0}(4N). Clearly, the trace operator sends cusp forms to cusp forms. The following lemma shows that it preserves the +-space, while providing a closed formula for these traces. Its proof is analogous to the proof of Lemma 2.2, and so we omit it for brevity (note. this calculation may also be found on [p. 67, K1]).

Lemma 3.2. If p is an odd prime and N is an odd square-free integer such that p ∤ N, then, for f ∈ M_{k+ \frac{1}{2}} (Γ_{0}(Np)),

Tr_{4N}^{4Np}(f) = f + (\frac{−4}{p})^{−(k+ \frac{1}{2} )} p^{− \frac{k}{2} + \frac{3}{4}} f|_{k+ \frac{1}{2}} W (p)U(p).

Theorem 3.3. Suppose that Np is an odd square-free integer, where p is an odd prime, and that

g(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4Np))

is a Kohnen newform. If j is a non-negative integer for which

M = min(j − k + 2, κ(p, j) − p^{j} − k + 3/2)

is positive, then there is a cusp form G(z) = ∑_{n=1}^{∞} B(n)q^{n} ∈ S_{k+κ(p,j)+ \frac{1}{2}}^{+} (Γ_{0}(4N)) with p-integral coefficients which satisfies the congruence

g(z) ≡ G(z) | U(p) (mod p^{M}).

Proof. Mimicking the proof of Theorem 2.3, we let

g_{j}(z) = (g(z)|U(p)) · Ẽ_{p}(4z)^{p^{j}} ∈ S_{k+κ(p,j)+ \frac{1}{2}}^{+} (Γ_{0}(4Np)).  (3.17)

In this case, we take the trace of g_{j}|U(p) ∈ S_{k+κ(p,j)+ \frac{1}{2}}^{+} (Γ_{0}(4Np)). We have, by Lemma 3.2,

Tr_{4N}^{4Np}(g_{j})
= g_{j} + (\frac{−4}{p})^{−k−κ(p,j)− \frac{1}{2}} p^{−(k+κ(p,j))/2+ \frac{3}{4}} g_{j}|_{k+κ(p,j)+ \frac{1}{2}} W (p)U(p).  (3.18)

By Lemma 3.1, we have

g_{j}(z)|_{k+κ(p,j)+ \frac{1}{2}} W (p)U(p) = ((g(z)|_{k+ \frac{1}{2}} U(p)W (p)) (Ẽ_{p}^{p^{j}}(4z)|_{κ(p,j)}W (p))) |U(p)
= (p^{\frac{k}{2} − \frac{1}{4}} (g(z)|_{k+ \frac{1}{2}} 𝔴_{p}) p^{κ(p,j)−p^{j}} (Ẽ′_{p}(4z)^{p^{j}})) |U(p)
= w_{p}p^{\frac{k}{2} − \frac{1}{4} +κ(p,j)−p^{j}} (g(z)Ẽ′_{p}(4z)) |U(p).

where w_{p} ∈ {±1} is the eigenvalue of the Atkin-Lehner 𝔴_{p} operator ([Th. 1, K1]). Moreover, by (3.10) Ẽ′_{p}(4z) has p-integral Fourier coefficients. Therefore, it turns out that

ν_{p} (Tr_{4N}^{4Np}(g_{j}) − g_{j}) ≥ \frac{k}{2} − \frac{1}{4} + κ(p, j) − p^{j} − \frac{(k + κ(p, j))}{2} + \frac{3}{4}

≥ p^{j} (\frac{p − 3}{2}) + \frac{1}{2}.

By (3.12), g_{j}(z) ≡ g(z)|U(p) (mod p^{j+1}); thus

Tr_{4N}^{4Np}(g_{j}) ≡ g|U(p) (mod p^{m}),

where m = min (j + 1, p^{j} (\frac{p−3}{2}) + \frac{1}{2}). Since g is a newform [p. 64, K1],

f|U(p)U(p) = g|U(p^{2}) = −w_{p}p^{k−1}g.  (3.19)

Therefore,

−w_{p}Tr_{4N}^{4Np}(g_{j})|U(p) ≡ g (mod p^{M}),

where

M = m − (k − 1) = min (j − k + 2, p^{j} (\frac{p − 3}{2}) − k + \frac{3}{2}) .

□

4. Proof of Theorems 1 and 2.

In this section we prove Theorems 1 and 2. We begin with the proof of Theorem 2 using Theorems 2.3 and 3.3.

Proof of Theorem 2. The hypotheses of Theorems 2.3 and 3.3 are satisfied when p^{j} ≥ k, and so there are forms F (z) ∈ S_{2(k+κ(p,j))}(Γ_{0}(N)) and G(z) ∈ S_{k+κ(p,j)+ \frac{1}{2}}^{+} (Γ_{0}(4N)) for which

f(z) ≡ F (z) (mod p),
g(z) ≡ G(z) | U(p) (mod p).

By a famous lemma of Deligne and Serre [Lemma 6.11, De-S], there is an eigenform in O_{K}[[q]], say F(z) (resp. G(z)), of the Hecke operators T (ℓ) (resp. T (ℓ^{2})) for primes ℓ ∤ Np with the same eigenvalues as f(z) (resp. G(z)) modulo 𝔭.

If F(z) ∈ S_{2(k+κ(p,j))}^{new}(Γ_{0}(4N)) is a newform, then Kohnen’s isomorphism illustrates that G(z) ∈ S_{k+ \frac{1}{2}}^{new} (Γ_{0}(4N)) must be the Kohnen newform corresponding to F(z). Otherwise, there would be at least two newforms of weight 2(k+κ(p, j)) with the same Hecke eigenvalues modulo 𝔭 for the Hecke operators T (ℓ) for primes ℓ ∤ Np.

□

We shall use Corollary 3 to prove Theorem 1. In addition, we shall require results giving a lower bound for the number of suitable quadratic twists of a fixed elliptic curve E with rank ≥ 2 over ℚ. We also require the following result of Dummigan [Th. 8.1, D] describing certain instances where elements of order p in the Selmer group of an elliptic curve can be visualized in the Selmer group of a congruent motive.

Theorem. (Dummigan) Let E/ℚ be an elliptic curve with prime conductor p and let D be a negative fundamental discriminant with the property that the rank of E(D) over ℚ is ≥ 2. If there is an eigenform F(z) ∈ S_{k}(Γ_{0}(1)) for which k ≡ 2 (mod p^{t−1}(p−1)) and k−2 is an odd multiple of p^{t}−p^{t−1}, then S (M_{F}^{(k/2,D(p))}) contains a subgroup isomorphic to ℤ/p^{t}ℤ.

Proof of Theorem 1. This is the N = 1, k = 1 and j = 0 case of Theorem 2. Begin by noticing that there are no elliptic curves of prime conductor p < 5 over ℚ. The hypothesis that p is not a congruence prime for the newforms in S_{p+1}(Γ_{0}(1)) together with the aforementioned lemma of Deligne and Serre [Lemma 6.11, De-S] implies that there is a unique newform F(z) ∈ S_{p+1}(Γ_{0}(1)) for which

f(z) ≡ F(z) (mod 𝔭).  (4.1)

This is claim (1).

If D′ is a fundamental discriminant, then it is well known that the sign of the functional equation of L(F_{D′}, s) is (−1)^{(p+1)/2}χ_{D′}(−1) = (−1)^{(p+1)/2} · sign(D′). If D is a negative fundamental discriminant coprime to p, then by (1.4) we have that D(p) is positive if and only if p ≡ 3 (mod 4). Therefore, it follows that the sign of the functional equation of L(F_{D(p)}, s) is +1 for every negative fundamental discriminant D. This is claim (2).

Suppose that g(z) ∈ Snew_{\frac{3}{2}}(Γ_{0}(4p)) and G(z) ∈ S_{(p+2)/2}^{new}(Γ_{0}(4p)) are the Kohnen newforms in Theorem 2. Using the notation defined by (1.13), Corollary 3 allows us to conclude that if D is a negative fundamental discriminant with (\frac{D}{p}) = w_{p}, then

L_{K}^{alg}(f_{D}, 1) ≡ L_{K}^{alg}(F_{D(p)}, (p + 1)/2) (mod 𝔭).  (4.2)

Since g(z) and G(z) are Kohnen newforms, there is a fundamental discriminant D_{0} for which

0 ≢ b(|D_{0}|) ≡ B(|D_{0}(p)|) (mod 𝔭).

Therefore by (1.12) and (1.13), if D is a negative fundamental discriminant with (\frac{D}{p}) = w_{p}, then

\frac{L(F_{D(p)}, (p + 1)/2) · |D(p)|^{p/2}}{L(F_{D_{0}(p)}, (p + 1)/2) · |D_{0}(p)|^{p/2}} = \frac{L_{K}^{alg}(F_{D(p)}, (p + 1)/2)}{L_{K}^{alg}(F_{D_{0}(p)}, (p + 1)/2)}

= \frac{B(|D(p)|)^{2}}{B(|D_{0}(p)|)^{2}} ≡ \frac{b(|D|)^{2}}{b(|D_{0}|)^{2}} (mod 𝔭)

= \frac{L(f_{D}, 1)|D|^{1/2}}{L(f_{D_{0}}, 1)|D_{0}|^{1/2}} .  (4.3)

A standard calculation [§6, De2] reveals that the quotients

\frac{Ω(F_{D(p)}, (p + 1)/2)}{Ω(F_{D_{0}(p)}, (p + 1)/2)} and \frac{Ω(f_{D}, 1)}{Ω(f_{D_{0}}, 1)}

are both coprime to 𝔭. Therefore the expression in (4.3) agrees with 𝕃^{alg}(F_{D(p)}, (p + 1)/2) (mod 𝔭) up to a 𝔭-unit. Therefore we have

𝕃^{alg}(F_{D(p)}, (p + 1)/2) ≡ 0 (mod 𝔭) ⟺ 𝕃^{alg}(f_{D}, 1) ≡ 0 (mod 𝔭).  (4.4)

This proves (3).

Now suppose that D is a negative fundamental discriminant for which both (\frac{D}{p}) = w_{p} and E(D) has rank ≥ 2 over ℚ. By Dummigan’s Theorem, we have that

S (M_{F}^{((p+1)/2,D(p))}) [p] ≠ {0}.

Since j(E) ≠ 0, 1728, a simple modification of the proof of [Th. 3, S-T] implies that the number of such D ≫_{E} X^{1/7}/(log X)^{2}. This follows from the observation that [Th. 2, S-T], the main ingredient in the proof of [Th. 3, S-T], is general enough to ensure that the D which are constructed may be chosen to have arbitrary values of (\frac{D}{p}). For these D, Kolyvagin’s famous theorem on the Birch and Swinnerton-Dyer Conjecture implies that

L(f_{D}, 1) = 0,

and so b(|D|) = 0 which by (4.4) implies that

𝕃^{alg}(F_{D(p)}, (p + 1)/2) ≡ 0 (mod 𝔭).

This proves (4).

□

References

[A-S] A. Agashe and W. Stein, Visible elements for the Birch and Swinnerton-Dyer Conjecture for

rank 0 modular abelian varieties, to appear, Math. Comp..

[At-L] A. O. L. Atkin and J. Lehner, Hecke operators on Γ_{0}(m), Math. Annalen 185 (1970), 134-160. [B-K] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck

Festchrift, vol. 1, Prog. in Math., Birkhäuser, Boston (1990), 333-400.

[CKRS] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular L-functions, preprint.

[C-Ma] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment.

Math. 9 (2000), 13-28.

[De1] P. Deligne, Formes modulaires et représentations ℓ-adiques, Sém. Bourbaki, éxp. 355, Springer Lect. Notes Math. 179 (1969), 139-172.

[De2] P. Deligne, Valeurs de fonctions L et périodes d’intégrales, AMS Proc. Sympos. Pure Math. 33 (1979), 313-346.

[De-S] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 7 (1974),

507-530.

[D] N. Dummigan, Congruences of modular forms and Selmer groups, Math. Res. Letters 8 (2001), 479-494.

[G-Ma] F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math.

Soc. 4 (1991), 1-23.

[He] E. Hecke, Mathematische Werke, Vandenhoeck und Ruprecht, Göttingen, 1959.

[H1] H. Hida, Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), 545-613.

[H2] H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge Univ. Press, Cambridge, 1993.

[J-O] K. James and K. Ono, Selmer groups of quadratic twists of elliptic curves, Math. Ann. 314

(1999), 1-17.

[Kb1] N. Koblitz, p-adic congruences and modular forms of half integer weight, Math. Ann. 274 (1986), 199-220.

[Kb2] N. Koblitz, Introduction to elliptic curves and modular forms, Springer Verlag, New York, 1984.

[K1] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32-72.

[K2] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268.

[K-Z] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip,

Invent. Math. 64 (1981), 173-198.

[Ma1] B. Mazur, On the arithmetic of special values of L-functions, Invent. Math. 55 (1979), 207-240.

[Ma2] B. Mazur, Visualizing elements of order three in the Shafarevich-Tate group, Asian J. Math. 3 (1999), 221-232.

[M] T. Miyake, Modular forms, Springer Verlag, New York, 1989.

[Sch] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430.

[S] J.-P. Serre, Formes modulaires et fonctions zêta p-adiques, Springer Lect. Notes in Math. 350 (1973), 191-268.

[Sh] G. Shimura, On modular forms of half-integral weight, Ann. Math. 97 (1973), 440-481.

[Ste] G. Stevens, Λ-adic modular forms of half-integral weight and a Λ-adic Shintani lifting, Arith. Geometry (Tempe, Az. 1993) Contemp. Math., Amer. Math. Soc. 174 (1994), 129-151.

[S-T] C. Stewart and J. Top, On ranks of twists of elliptic curves and power-free values of binary

forms, J. Amer. Math. Soc. 8 (1995), 943-973.

[V] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397-419.

[W] J.-L. Waldspurger, Sure les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures et Appl. 60 (1981), 375-484.

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

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

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

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