manuscripta math. 122, 289−304 (2007)

© Springer-Verlag 2007

Kazuo Matsuno

Construction of elliptic curves with large Iwasawa λ-invariants and large Tate-Shafarevich groups Received: 17 September 2005 / Revised: 18 October 2006 Published online: 5 January 2007 Abstract. In this paper we construct elliptic curves defined over the rationals with arbitrarily large Iwasawa λ-invariants for primes p satisfying p ≤ 7 or p = 13. We use this to obtain that the p-rank of the Tate-Shafarevich group can be arbitrarily large for such primes p.

1. Introduction Let E be an elliptic curve defined over Q. Assume that E has good ordinary reduction at a prime p. Let λ E, p and μ E, p be the Iwasawa invariants associated with the p ∞ -Selmer groups of E over the cyclotomic Z p -extension Q∞ /Q. It is natural to ask which values do these invariants take when E varies. As for the μ-invariant, it is known that there exist elliptic curves E with μ E, p > 0 for small p. This is in contrast to the classical Iwasawa μ-invariants attached to ideal class groups. It is believed, however, that μ E, p takes only small values (say μ E, p ≤ 4), and in fact, it is conjectured that μ E, p = 0 for any E if p > 37. On the other hand, Greenberg proved that λ E, p + μ E, p can be arbitrarily large as E varies for any p ([9, Remark in p. 111], see also Proposition 3.2). This implies the unboundedness of λ E, p provided that μ E, p behaves as conjectured. The aim of this paper is to prove the unboundedness of λ E, p for small p. The main result is as follows. Theorem (Theorem 3.4). Suppose p is a prime number, p ≤ 7 or p = 13. Then λ E, p can be arbitrarily large as E varies over all elliptic curves with good ordinary reduction at p. Moreover, if p = 2 or 3 then there exists an elliptic curve E/Q with λ E, p = n for any non-negative integer n. This result was already proved for p = 2 in [15] by showing a Kida-type formula, which describes the variation of the λ-invariants in Galois p-extensions. Moreover, Greenberg and Vatsal mentioned in [10, p. 22] that the first assertion K. Matsuno: Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan. e-mail: [email protected] Mathematics Subject Classification (2000): Primary 11R23, 11G05

DOI: 10.1007/s00229-006-0068-9

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of this theorem can be verified for p = 3, 5 by applying their result [10, Proposition 2.8] to an infinite family of elliptic curves with isomorphic mod- p representations. Their argument, however, cannot be applied to the case p ≥ 7 since such an infinite family does not exist. We prove the first assertion of the above theorem (for odd p) by applying an argument that is used by Greenberg to prove that λ E, p + μ E, p can be arbitrarily large to a family of elliptic curves with an isogeny of degree p, which turn out to have μ E, p = 0. The second assertion of the theorem for p = 3 is proved by using the result of Greenberg–Vatsal mentioned previously and also using some results concerning the ideal class groups of imaginary quadratic fields. As a consequence of the above theorem and its proof, we obtain the following result. Corollary (Theorem 5.1). For p ≤ 7 or 13, the p-rank dimF p X(E/Q)[ p] of the Tate-Shafarevich group X(E/Q) of an elliptic curve E over Q can be arbitrarily large as E varies. The assertion of this result is expected to be true for any prime p. This problem has been treated by many authors and already been proved in the case p ≤ 5 by Cassels [3] ( p = 3), B¨olling [1] ( p = 2), and Fisher [7] ( p = 5). For p = 7 and 13, a partial result, either of the Mordell-Weil rank or the p-rank of the Tate-Shafarevich group being arbitrarily large, was obtained by Kloosterman and Schaefer [13] (see also a remark in the end of Sect. 5). Our proof is based on an investigation of the Iwasawa modules attached to two isogenous elliptic curves. This also gives another proof for the known cases p = 3, 5. We would like to mention that Steve Donnelly [6] proved (independently) the above result by a different method. See also [12] and [16] for related results. This paper is organized as follows. In Sect. 2, we fix notation and recall some basic facts about Iwasawa invariants associated with elliptic curves over Q. Our main result, Theorem 3.4, is stated in Sect. 3. We also give a brief sketch of the proof of a result of Greenberg, Lemma 3.5, which is used to prove the unboundedness of λ E, p + μ E, p . In Sect. 4, we prove the first half of the main theorem, Theorem 3.4A. For elliptic curves with isogenies of degree p over Q, Greenberg gave a sufficient condition for the μ-invariant being trivial (Proposition 4.1). If p ≤ 7 or p = 13, then there exists an infinite family of such elliptic curves since the modular curve X 0 ( p) has genus 0. We show that there indeed exist curves in this family satisfying the conditions for two results of Greenberg simultaneously, i.e., λ E, p + μ E, p being arbitrarily large and μ E, p = 0. In Sect. 5, we discuss the unboundedness of the p-rank of the Tate-Shafarevich groups. The second half of the main theorem, Theorem 3.4B, is proved in Sect. 6. We recall a formula given by Greenberg–Vatsal [10] and apply it to a family of elliptic curves with a Galoissubmodule isomorphic to Z/ pZ ⊕ μ p . In Sect. 7, we present some numerical examples. 2. Iwasawa invariants Let E be an elliptic curve defined over Q. In this paper, we treat only the case where E satisfies the following condition (Ord) at a prime p.

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(Ord) E has good ordinary reduction at p. Let Q∞ be the cyclotomic Z p -extension of Q. Let Sel p∞ (E/Q∞ ) be the p ∞ -Selmer group of E over Q∞ , i.e.,    1 ∞ 1 Sel p∞ (E/Q∞ ) := Ker H (Q∞ , E[ p ]) → H (Q∞,v , E) , v

E[ p ∞ ]

is the group of all p-power torsion points of E and v runs over all where primes of Q∞ . Let X p (E/Q∞ ) := HomZ p (Sel p∞ (E/Q∞ ), Q p /Z p ) be the Pontryagin dual of Sel p∞ (E/Q∞ ). The natural action of the Galois group  := Gal(Q∞ /Q) on Sel p∞ (E/Q∞ ) enables us to consider X p (E/Q∞ ) as a module over the completed group ring  := Z p [[]]. Under the assumption (Ord), X p (E/Q∞ ) is known to be a finitely generated torsion -module by results of Kato, Rohrlich and Rubin (cf. [9, Theorem 1.5]). Definition 2.1. We denote by g(X ) (respectively, λ(X ), μ(X )) the minimum number of generators (respectively, λ-invariant, μ-invariant) of a finitely generated torsion -module X . For an elliptic curve E over Q satisfying (Ord) at p, put g E, p = g(X p (E/Q∞ )),

λ E, p = λ(X p (E/Q∞ )),

μ E, p = μ(X p (E/Q∞ )).

Lemma 2.2. Let X be a finitely generated torsion -module. (i) (ii) (iii) (iv)

λ(X ) = dimQ p (X ⊗Z p Q p ). μ(X ) = 0 if and only if X is finitely generated as a Z p -module. g(X ) = dimF p (X/MX ), where M is the maximal ideal of . If X has no nonzero finite -submodule, then λ(X ) + μ(X ) ≥ g(X ).

Proof. (i) and (ii) are immediate from the definition of λ and μ. (iii) is a part of the assertion of Nakayama’s lemma (cf. [20, Lemma 13.16]). See [9, p. 111] for a proof of (iv).  It is known that X p (E/Q∞ ) satisfies the assumption of Lemma 2.2(iv). Proposition 2.3 (cf. [9, Proposition 4.15], [11]). The -module X p (E/Q∞ ) has no nonzero finite submodule. In particular, we have λ E, p + μ E, p ≥ g E, p . The following theorem, a consequence of Mazur’s control theorem, says that the invariants λ E, p and μ E, p describe the behavior of the size of the Selmer group of E in Q∞ /Q. Define K n as the unique subfield of Q∞ such that [K n : Q] = p n . Let Sel p∞ (E/K n ) denote the p ∞ -Selmer group of E over K n . Theorem 2.4 (Mazur, cf. [9, Theorem 1.10]). Let rn (respectively tn ) be the Z p rank (respectively the order of the Z p -torsion subgroup) of the Pontryagin dual of Sel p∞ (E/K n ). Then rn is bounded as n varies, and there exist integers ν and n 0 such that tn = p (λ E, p −r )n+μ E, p p for all n ≥ n 0 . Here, we put r = sup{rn | n ≥ 0}.

n +ν

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Remark. Consider the exact sequence 0 −→ E(K n ) ⊗ Q p /Z p −→ Sel p∞ (E/K n ) −→ X(E/K n )[ p ∞ ] −→ 0. It is conjectured that the Tate-Shafarevich group X(E/K n ) of E over K n is finite for any n. Under this conjecture, the above sequence implies that rn = rank Z E(K n ) and tn = #X(E/K n )[ p ∞ ]. In any case, we have λ E, p ≥ rn ≥ rank Z E(K n ) for any n. 3. Results For a fixed prime p, let L p be the set of the Iwasawa λ-invariants of all elliptic curves E defined over Q satisfying (Ord): L p := {λ E, p | E satisfying (Ord) at p}. We have the following conjecture on the size of the set L p . Conjecture 3.1. The set L p is unbounded, i.e., λ E, p can be arbitrarily large as E varies over elliptic curves over Q satisfying (Ord) at p. Related to this conjecture, Greenberg proved the following. Proposition 3.2 (Greenberg [9, Remark after Corollary 5.6]). For any n ∈ Z, there exists an elliptic curve E satisfying (Ord) such that E[ p] is irreducible as a Gal(Q/Q)-module and λ E, p + μ E, p ≥ n. It is expected that the μ-invariant of an elliptic curve does not become so large. In fact, it is conjectured that μ E, p = 0 if E[ p] is irreducible (cf. [9, Conjecture 1.11]). In particular, Greenberg’s result reduces Conjecture 3.1 to this conjecture on the μ-invariants. More optimistically, we have the following conjecture. Conjecture 3.3. The set L p coincides with Z≥0 , the set of all non-negative integers. That is, λ E, p takes any non-negative integer value as E varies. The aim of this paper is to prove the above conjectures for small p. The main theorem is stated as follows. Theorem 3.4. (A) Assume that p ≤ 7 or p = 13. Then Conjecture 3.1 is valid, i.e., sup L p = +∞. (B) Conjecture 3.3 is valid for p ≤ 3, i.e., L2 = L3 = Z≥0 . As mentioned in Sect. 1, this theorem has already been proved in [15] for p = 2. We will give the proof for odd primes in the following sections. At the end of this section, we recall a key fact used in the proof of Greenberg’s result Proposition 3.2. This is also needed to prove Theorem 3.4A. Lemma 3.5 (Greenberg). Assume that E satisfies (Ord) at p. If E satisfies the conditions (H1)–(H4), then we have λ E, p + μ E, p ≥ g E, p ≥ n.

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(H1) E(Q)[ p] = 0. (H2) The Hasse-Weil L-function L(E, s) of E is non-zero at s = 1. (H3) E has split multiplicative reduction at primes 1 , . . . , n . (H4) ordi ( j E ) = − p for each i = 1, . . . , n, where j E is the j-invariant of E. Remark. The assertion of this lemma can be proved without assuming (H2). See the proof of [9, Corollary 5.6]. Sketch of the proof of Lemma 3.5. Let S be a finite set of primes of Q containing p, the archimedean prime, and the primes at which E has bad reduction. Then the p ∞ -Selmer group Sel p∞ (E/Q) of E over Q is the kernel of the localization map ϕ : H 1 (Q S /Q, E[ p ∞ ]) −→



H 1 (Qv , E)[ p ∞ ],

v∈S

where Q S is the maximal extension of Q unramified outside S. By the CasselsTate-Poitou global duality, one can prove under the assumption (H1) that dimF p Coker(ϕ)[ p] ≤ s, where s is the Z p -rank of the Pontryagin dual of Sel p∞ (E/Q) (cf. [9, Proposition 4.13]). Under (H2), Sel p∞ (E/Q) is finite by a result of Kolyvagin [14]. In particular, ϕ is surjective. By this fact and the inflation–restriction sequence, we obtain an exact sequence 0 −→ Sel p∞ (E/Q) −→ Sel p∞ (E/Q∞ ) −→



W E, −→ 0,

(1)

∈S

where W E, is the kernel of the map H 1 (Q , E)[ p ∞ ] → H 1 (Q∞,v , E)[ p ∞ ] for a prime v of Q∞ above  ∈ S. The group W E, is known to be finite for any  ∈ S (cf. [9, Lemma 3.5]). If  = p, ∞, then the order of W E, is equal to the highest power of p dividing the Tamagawa factor c E, of E at  (cf. [9, p. 74]). By the theory of Tate curves (cf. [19, Remark IV.9.6]), we have #W E,i = c E,i = p for i = 1, . . . , n by (H3) and (H4). The exact sequence (1) implies that S := Sel p∞ (E/Q∞ ) is finite and dimF p S[ p] = dimF p S/ pS ≥



dimF p W E, / pW E, ≥ n.

∈S

By definition, S[ p] is isomorphic to the Pontryagin dual of X p (E/Q∞ )/ MX p (E/Q∞ ), where M is the maximal ideal of . Hence we have g E, p = dimF p S[ p] ≥ n by Lemma 2.2. By Proposition 2.3, this implies the assertion of the lemma.  In order to complete the proof of Proposition 3.2, it suffices to construct for any n an elliptic curve E such that E[ p] is an irreducible Galois module and E satisfies the conditions (H1)–(H4) for a set of n primes {1 , . . . , n }. See [9, Proposition 5.4] for the details of construction. We will apply Lemma 3.5 to curves E with reducible E[ p] in the next section to prove Theorem 3.4A.

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4. Proof of Theorem 3.4A We prove Theorem 3.4A in this section. Assume p ≥ 3 in what follows. We use the following result on the μ-invariant due to Greenberg. Proposition 4.1 ([9, Proposition 5.10]). Assume that E[ p] has a Gal(Q/Q)-invariant subgroup  of order p satisfying either of the following conditions (OU) or (ER). (OU)  is odd and unramified at p. (ER)  is even and ramified at p. Then we have μ E, p = 0. Remark. Let  be a cyclic group of order p on which Gal(Q/Q) acts by a primitive Dirichlet character ϕ. When p is odd, we say  is odd (respectively even) if ϕ(−1) = −1 (respectively ϕ(−1) = 1), and we say  is ramified (respectively unramified) at p if ϕ( p) = 0 (resp. ϕ( p) = 0). Although the statement of the above proposition also holds for p = 2, we have to define the terms “ramified” and “odd” more carefully (cf. [9, p. 131]). non-trivial Dirichlet charFor a non-zero integer D, we denote by χ D a unique √ acter corresponding to the quadratic extension Q( D)/Q, and by E D the χ D -twist of an elliptic curve E. If E[ p] has a Galois-invariant cyclic subgroup corresponding to a Dirichlet character ϕ, then E D [ p] has a subgroup corresponding to ϕχ D . Since χ D (−1) is the sign of D and we have χ D ( p) = 0 if D is prime to p, the following fact is an immediate consequence of Proposition 4.1. Corollary 4.2. Assume that E[ p] has a Gal(Q/Q)-invariant subgroup of order p. Then at least one of the following assertions holds. (i) (ii)

μ E D , p = 0 for all positive integers D prime to p. μ E D , p = 0 for all negative integers D prime to p.

In order to prove Theorem 3.4A, it suffices to find for any n an elliptic curve E with μ E, p = 0 satisfying all assumptions of Lemma 3.5. We first show that, if E[ p] is reducible as a Galois module, then the assumption (H4) is the only essential one. The other assumptions can be satisfied by replacing E with its quadratic twists. Proposition 4.3. Let E and E be elliptic curves over Q satisfying (Ord) at p and having an isogeny E → E of degree p defined over Q. Suppose there exist primes 1 , . . . , n satisfying (H4) in Lemma 3.5 for E. Then there exists an integer D prime to p such that the quadratic twists E D and E D satisfy all the following conditions. (T1) E D (Q)[ p] = E D (Q)[ p] = 0. (T2) L(E D , 1) = 0. (T3) E D has split multiplicative reduction at 1 , . . . , n−1 .

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(T4) μ E D , p = 0. (T5) The Tamagawa factor of E D at any prime  = 1 , . . . , n is not divisible by p. (T6) E D does not have anomalous reduction at p, i.e., the reduction of E D modulo p has no F p -rational point of order p. Remark. (i) If A is an elliptic curve over Q p having good reduction, then the kernel of the reduction map A(Q p ) → A(F p ) has no element of order p. Therefore (T6) implies (T1) automatically. (ii) The conditions (T5) and (T6) are not needed for the proof of Theorem 3.4A. We will use them in the next section. We use the following result of Waldspurger to prove this proposition. Theorem 4.4 (Waldspurger, cf. [2, Theorem in Sect. 0]). Let A be an elliptic curve defined over Q and S a finite set of primes containing all primes at which A has bad reduction. Let denote the sign in the functional equation for L(A, s). Then there exist infinitely many square-free integers d such that d > 0, L(Ad , 1) = 0 and χd () = 1 for all  ∈ S. Proof of Proposition 4.3. The proof is separated into several steps. In the following, we denote by Dk the square-free part of the product d1 · · · dk . Step 1: From Corollary 4.2, it follows that there is a d1 ∈ {−1, 1} such that μ E dd1 , p = 0 for any positive integer d prime to p. Step 2: For each i , there exists an extension K i /Qi with [K i : Qi ] ≤ 2 such reduction over K i by (H4). Let d2 be a positive that E D1 has split multiplicative √ integer such that p  d2 and Qi ( d2 ) = K i for all i. Then E D2 = E d1 d2 has split multiplicative reduction at each i . In particular, E D2 satisfies (T3). Step 3: Let d3 be the product of all primes  = 1 , . . . , n such that the Tamagawa factors of E D2 at  are divisible by p. Let d3 be a positive integer prime to p such that d3 |d3 and χd3 (i ) = 1 for any i. Then E D3 satisfies (T5). Step 4: Set e = −1 if E D3 has anomalous reduction at p, otherwise set e = 1. Take an odd prime d4 such that χd4 (i ) = 1 for any i, χd4 ( p) = e, and E D3 has good reduction at d4 . Then E D4 satisfies (T5) and (T6). Step 5: Set d5 = 1 if the sign in the functional equation for L(E D4 , s) is plus. Otherwise take d5 to be a prime number satisfying d5 ≡ 1 (mod 4), χd5 ( p) = 1 and  −1 if  = n , χd5 () = 1 if  = n

for any prime  at which E D4 has bad reduction. Then the sign in the functional equation for L(E D5 , s) is plus (cf. [5, Sects. 2.8 and 2.11]). Step 6: Take a positive integer d6 prime to p such that χd6 ( p) = 1, χd6 () = 1 for all  at which E D5 has bad reduction, and L(E D6 , 1) = 0. The existence of such d6 is ensured by Theorem 4.4. (Take A as E D5 and S as a finite set of primes containing p and all bad primes of E D5 .) Then E D6 satisfies (T2).

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Since E D2 satisfies (T3) and we have χ D6 /D2 (1 ) = · · · = χ D6 /D2 (n−1 ) = 1, E D6 also satisfies (T3). Similarly E D6 satisfies (T5) and (T6). Hence (T1) is also satisfied. Since D6 /D1 is positive, μ E D6 , p = 0 as remarked in Step 1, that is, E D6 satisfies (T4). Thus E D6 satisfies all (T1)–(T6). The proof has been completed.  Corollary 4.5. If E satisfies assumptions in Proposition 4.3, then we have λ E D , p ≥ n − 1 and μ E D , p = 0 for some D ∈ Z prime to p. Proof. Take an integer D such that the conditions (T1)–(T4) hold. Then E D satisfies the conditions (H1)–(H4) in Lemma 3.5 for 1 , . . . , n−1 since the j-invariant of E D is equal to that of E. Hence we have μ E d , p = 0 and λ E D , p = λ E D , p +  μ E D , p ≥ n − 1 by Lemma 3.5 and (T4). Thus, in order to complete the proof of Theorem 3.4A, it suffices to construct an elliptic curve E satisfying the assumptions of Proposition 4.3 for an arbitrarily large n. We prove the existence of such curves by providing an explicit model of a family of curves satisfying the conditions. We can easily give numerical examples using these explicit models (see Sect. 7). Suppose p = 3, 5, 7 or 13. Let a(t) and b(t) be the polynomials defined as follows. p

a(t)

b(t)

3 5 7 13

+ 33

t + 35 + 2 · 53 t + 55 2 t + 5 · 72 t + 74 4 3 t + 19 · 13t + 20 · 132 t 2 + 7 · 133 t + 134

t

1 t 2 + 13t + 72 t 2 + 5t + 13

t2

Let c(t), d(t) ∈ Z[t] be the irreducible monic polynomials satisfying a(t)b(t)3 − c(t)d(t)2 = 1728t p . For example, c(t) = t 2 + 6t + 13 and d(t) = t 6 − 38 · 13t 5 − 122 · 132 t 4 − 108 · 133 t 3 − 46 · 134 t 2 − 10 · 135 t − 136 when p = 13. Consider the following family1 of elliptic curves. A(t) : y 2 = x 3 − 27a(t)b(t)c(t)x − 54a(t)c(t)2 d(t).

(2)

The discriminant  A(t) of the Weierstrass equation (2) is 612 t p a(t)2 c(t)3 . Therefore A(t) is an elliptic curve for any t ∈ Q with ta(t)c(t) = 0. For such t, it is clas

sically known that there exist an elliptic curve A(t) and an isogeny A(t) → A(t)

(t) (t) p of degree p both defined over Q. The j-invariants of A and A are f (t)/t and t p f ( p/t), respectively, where f (t) = a(t)b(t)3 . For an arbitrary n, take primes 1 , . . . , n not equal to p. We can take an integer t satisfying ord p (ta(t)c(t)) = 0 and ordi (t) = 1 for any i = 1, . . . , n. Then A(t) has good reduction at p. In fact, the discriminant  A(t) is not divisible by p if p ≥ 5. In the case p = 3, the facts a(t)b(t)c(t) ≡ t 2 and a(t)c(t)2 d(t) ≡ t 3 1 The author learned the Weierstrass model of this type from Masato Kuwata.

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(mod 27) imply the Weierstrass equation (2) is not minimal over Q3 . Therefore the minimal discriminant of A(t) is a divisor of t /312 = 212 t 3 a(t)2 c(t)3 , which is prime to 3. Thus A(t) has good reduction at p even in the case p = 3. Moreover, A(t) has ordinary reduction at p since A(t) has an isogeny of degree p over Q. The assumption ordi (t) = 1 implies ordi ( f (t)) = 0 and ordi ( j A(t) ) = − p for each i. Therefore A(t) satisfies (Ord) at p and satisfies (H4) for 1 , . . . , n . By Corollary 4.5, there exists an integer D such that λ A(t) , p ≥ n − 1. The proof of D Theorem 3.4A has been completed. When p ≤ 7, the modular curve X 1 ( p) has genus 0. Hence there exists an infinite family of elliptic curves E (t) having a Q-rational point of order p: ⎧ 2 3 ⎪ ( p = 3), ⎨y + x y + t y = x (t) E : y 2 + (t + 1)x y + t y = x 3 + t x 2 ( p = 5), ⎪ ⎩ 2 2 2 3 2 2 ( p = 7). y − (t − t − 1)x y − t (t − 1)y = x − t (t − 1)x If we take t ∈ Z such that t ≡ −1 (mod p) and ordi (t) = 1 for primes 1 , . . . , n , then the above E (t) satisfies (Ord) at p and satisfies (H4) for 1 , . . . , n . Since E (t) has an isogeny of degree p whose kernel is E (t) (Q)[ p] ∼ = Z/ pZ, we can apply Corollary 4.5 to E (t) . Hence there exists an integer D such that λ E (t) , p ≥ n − 1. This also implies Theorem 3.4A for p = 3, 5, 7. The coefficients D

of the equation for E (t) have smaller degrees than those for A(t) . So these families will be more useful for giving examples. 5. Large Tate-Shafarevich group In this section, we give a proof of the following result.

Theorem 5.1. Let p be an odd prime such that p ≤ 7 or p = 13. Then, for any n ∈ Z, there exists an elliptic curve E defined over Q such that dimF p X(E/Q)[ p] ≥ n. Here X(E/Q) is the Tate-Shafarevich group of E over Q. Assume that p = 3, 5, 7, 13. Take an integer n ≥ 1 and let L = {1 , . . . , n } be a set of prime numbers not equal to p. Let E and E be elliptic curves define over Q satisfying (Ord) at p with an isogeny E → E of degree p also defined over Q. We assume that ord ( j E ) = − p and ord ( j E ) = −1 for any  ∈ L. (See the previous section for the existence of such curves. Note ord ( j A(t) ) = −1 if ord (t) = 1.) Since E satisfies the assumptions of Proposition 4.3, we can take an integer d prime to p such that E d and E d satisfy the conditions (T1)–(T6) in Proposition 4.3. Lemma 5.2. We have g E d , p = dimF p X(E d /Q)[ p]. Here g E d , p is the number of generators of X p (E d /Q∞ ) as defined in Sect. 2. Proof. By (T2) and Kolyvagin’s result [14], E d (Q) is finite and Sel p∞ (E d /Q) coincides with the p-primary part of X(E d /Q). In particular, we have dimF p X(E d /Q)[ p] = dimF p Sel p∞ (E d /Q)[ p].

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Moreover, under (T1) and (T2), the exact sequence (1) in the proof of Lemma 3.5 holds for E d . By (T3), (T5) and the assumption on the j-invariant of E , we have W E d , = 0 for any  = p since the order of W E d , is the highest power of p dividing the Tamagawa factor of E d at . We also have W E d , p = 0 by (T6) (cf. [9, Lemma 3.4]). Hence Sel p∞ (E d /Q) is isomorphic to Sel p∞ (E d /Q∞ ) by (1). We have g E d , p = dimF p Sel p∞ (E d /Q∞ ) [ p] by Lemma 2.2, as mentioned in the proof of Lemma 3.5. Thus the assertion is proved.  By this lemma, we have only to show that g E d , p can be arbitrarily large. We prove this by using Lemma 3.5 and the following fact on the difference between two pseudo-isomorphic -modules with certain conditions. Lemma 5.3. Let X be a finitely generated torsion -module with no nonzero finite submodule. Let Y be a -submodule of X such that Z := X/Y is of exponent p. ) Then we have g(X ) ≥ g(Y 2 . Proof. Take a non-zero element f ∈  such that X/ f X is finite and that f and p generate the maximal ideal of . Then X [ f ], the submodule of X consisting of the elements annihilated by f , is trivial since X has no nonzero finite submodule. Hence we have an exact sequence 0 → Z [ f ] → Y/ f Y → X/ f X → Z / f Z → 0.

(3)

Put W := Ker(X/ f X → Z / f Z ). Since X/ f X and W are finite, we have g(X ) = dimF p X/( f, p)X = dimF p (X/ f X )[ p] ≥ dimF p W [ p] = g(W ) by Lemma 2.2. Moreover, we have g(X ) ≥ g(Z ) and g(W ) ≥ g(Y ) − dimF p Z [ f ] ≥ g(Y ) − dimF p Z / f Z = g(Y ) − g(Z ) by (3) and the fact p Z = 0. Thus we have g(X ) ≥ max(g(Z ), g(Y ) − g(Z )), which implies the assertion of the lemma.  Corollary 5.4. We have g E d , p ≥

n−1 2 .

Proof. We have a -homomorphism ψ : X p (E d /Q∞ ) → X p (E d /Q∞ ) induced by an isogeny E d → E d of degree p. One sees that Ker(ψ) is Z p -torsion and Coker(ψ) is of exponent p. By Proposition 2.3, X p (E d /Q∞ ) has no nonzero finite submodule. This fact and (T4) imply that X p (E d /Q∞ ) is torsion-free as a Z p module (see Lemma 2.2). In particular, ψ is an injection. Hence we have g E d , p ≥ g Ed , p 2

by Lemma 5.3. Since we have g E d , p ≥ n − 1 by Lemma 3.5, the assertion is proved.  Thus we have dimF p X(E d /Q)[ p] ≥ Theorem 5.1 has been completed.

n−1 2

for an arbitrary n. The proof of

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Remark. We can also prove Theorem 5.1 by applying the argument given in the preceding section, especially applying results of Kolyvagin and Waldspurger, to a result of Cassels in [4]. As explained in [13, Sect. 2], Cassels gave a formula on the difference of Selmer groups associated with isogenies (cf. [13, Theorem 1]). This formula implies that the p-rank of the p-Selmer group Sel p (E d /Q) of an elliptic curve E d is not less than n − 2 if E d satisfies (T1), (T3) and (H4). If E d further satisfies (T2), then we have Sel p (E d /Q) = X(E d /Q)[ p]. 6. Greenberg–Vatsal’s method In this section, we give a proof of Theorem 3.4B for p = 3 by using a result of Greenberg and Vatsal [10]. We also give another proof of Theorem 3.4A for p = 5. We first recall Greenberg–Vatsal’s result. Definition 6.1. Let I (respectively Frob ) denote the inertia subgroup (respectively the Frobenius automorphism) at a prime  = p. For an elliptic curve A defined over Q, let PA, (T ) ∈ Z p [T ] be the characteristic polynomial of the action of Frob on the I -coinvariant quotient of V p A := (limn A[ p n ]) ⊗Z p Q p , i.e., ← −  PA, (T ) := det (1 − Frob T )|(V p A) I ∈ Z p [T ]. We denote by δ A, the multiplicity of T = −1 as a root of PA, (T ) mod p. Theorem 6.2 (Greenberg–Vatsal [10, Sect. 2]). Let E and E be elliptic curves satisfying (Ord) at an odd prime p. Assume that E[ p] ∼ = E [ p] as Gal(Q/Q)modules and that μ E, p = 0. Then we have μ E , p = 0 and  (δ E, − δ E , )s , λ E , p = λ E, p + = p

where  runs over all primes not equal to p and s denotes the number of the primes of Q∞ lying above . Remark. In the above setting, we have δ E, = δ E , for any  at which E and E have same reduction type. When p ≤ 5, it is known that, for any given elliptic curve E defined over Q, there exists an infinite family of elliptic curves E (t) such that E (t) [ p] ∼ = E[ p] as Gal(Q/Q)-modules. An explicit construction of such families, twists of the universal family over the modular curve X ( p), is given in [18]. One can find an elliptic curve with arbitrarily large λ-invariant by applying Theorem 6.2 to those families, as mentioned in [10, p. 22]. In order to prove Theorem 3.4B, we apply Theorem 6.2 to (a quadratic twist of) a family B (s) of elliptic curves such that B (s) [ p] ∼ = Z/ pZ ⊕ μ p as Gal(Q/Q)-modules. p=3 B (s) : y 2 + x y + s 3 y = x 3 − 5s 3 x − s 3 (7s 3 + 1),  B (s) = −s 3 (3s − 1)3 (9s 2 + 3s + 1)3 , j B (s) = (6s + 1)3 (36s 2 − 6s + 1)3 / B (s) .

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This B (s) is the universal family over X (3). For any s ∈ Q with s = 0, 13 , the curve B (s) is an elliptic curve over Q and B (s) [3] is isomorphic to Z/3Z ⊕ μ3 as a Gal(Q/Q)-module. Set B := B (−1) . Then B is an elliptic curve of conductor 14, the curve 14A1 in the table of [5]. Lemma 6.3. Let d be a square-free negative integer satisfying the following conditions. (B1) d ≡ 3 (mod 4) and χd (3) = χd (7) = −1. √ (B2) The class number of Q( d) is not divisible by 3. Then we have λ Bd ,3 = μ Bd ,3 = 0. Proof. By (B1), the Tamagawa factor of Bd at any prime is not divisible by 3 and Bd does not have anomalous reduction at 3. (Note B has anomalous reduction at 3.) Then we have g Bd ,3 = dimF3 Sel3∞ (Bd /Q)[3] by Lemma 2.2 and the exact sequence (1) (see the proof of Lemma 5.2). On the other hand, the order of the 3-Selmer group Sel3 (Bd /Q) of Bd divides the square of the class num√ ber of Q( d) under (B1) by a result of Frey [8, Corollary in p. 653]. Therefore Sel3 (Bd /Q) is trivial by (B1) and (B2). Since there is a surjection Sel3 (Bd /Q) → Sel3∞ (Bd /Q)[3], we have g Bd ,3 = 0, i.e., X3 (Bd /Q∞ ) = 0. (Recall g Bd ,3 is the number of generators of X3 (E d /Q∞ ).) In particular, we have λ Bd ,3 = μ Bd ,3 = 0 as desired.  Now fix an arbitrary positive integer n and let L := {1 , . . . , n } be a set of distinct prime numbers congruent to 4 modulo 9. Put B := B (s) , where s = 1 . . . n . One sees that B has split multiplicative reduction at each i . Take a square-free negative integer d satisfying (B1), (B2) and the following (B3). (B3) χd (i ) = 1 for each i ∈ L and χd () = 0 for all primes  ∈ L at which B has bad reduction. The existence of such an integer d is ensured by the following result of Nakagawa-Horie. Proposition 6.4 (Nakagawa–Horie [17, Theorem 1]). Let Q be a finite set of odd primes and fix q ∈ {−1, 0, 1} for each q ∈ Q. Then there exist infinitely many negative square-free integers d such √ that d ≡ 3 (mod 4), χd (q) = q for any q ∈ Q, and the class number of Q( d) is not divisible by 3. We have isomorphisms Bd [3] ∼ = Bd [3] ∼ = (Z/3Z ⊕ μ3 ) ⊗ χd of Gal(Q/Q)modules. Hence we have λ Bd ,3 = μ Bd ,3 = μ Bd ,3 = 0 and λ Bd ,3 =

 =3

 δ Bd , − δ Bd , s

(4)

by Theorem 6.2 and Lemma 6.3. By (B1) and (B3), Bd and Bd have same reduction type at any  not dividing 71 · · · n . Hence we have δ Bd , = δ Bd , for such

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primes. Moreover, we have δ Bd ,7 = δ Bd ,7 = 0. In fact, Bd has non-split multiplicative reduction at 7, which implies δ Bd ,7 = 0. As for Bd , one sees that Bd has good reduction or non-split multiplicative reduction at 7 and that Bd (Q7 )[3] ∼ = Bd (Q7 )[3] = 0. This implies δ Bd ,7 = 0. For an i ∈ L, we have δ Bd ,i = 2 and δ Bd ,i = 1 since Bd (respectively, Bd ) has good (respectively, split multiplicative) reduction at i and Bd (Qi )[3] ∼ = (Z/3Z)⊕2 by (B3). The assumption i ≡ 4 (mod 9) implies si = 1. Therefore we have λ Bd ,3 = n by (4), where n is an arbitrary non-negative integer. Thus Theorem 3.4B for p = 3 is proved. p=5 B (s) : y 2 + (s 5 + 1)x y + s 5 y = x 3 + s 5 x 2 + 5(s 15 − 2s 10 − s 5 )x +(s 25 − 10s 20 − 5s 15 − 15s 10 − s 5 ). For any non-zero s ∈ Q, B (s) is an elliptic curve defined over Q and B (s) [5] ∼ = Z/5Z ⊕ μ5 as Gal(Q/Q)-modules. Put B := B (1) , an elliptic curve of conductor 11 (11A1 in [5]). For fixed primes 1 , . . . , n congruent to 6 modulo 25, B := B (1 ···n ) has good (respectively, split multiplicative) reduction at 5 (respectively, at each i ). Similarly to the case p = 3, we have λ Bd ,5 = λ Bd ,5 + n if d is a negative integer satisfying χd (5) = χd (11) = −1, χd (i ) = 1 for each i , and χd () = 0 for any prime  ∈ {11, 1 , . . . , n } at which B has bad reduction. This proves Theorem 3.4A for p = 5. √ Remark. If we can take the above d so that the class number of Q( d) is not divisible by 5, then we have λ Bd ,5 = 0 and λ Bd ,5 = n by using the result of Frey [8] mentioned in the proof of Lemma 6.3. Therefore the assertion B of Theorem 3.4 holds for p = 5 if we have a result analogous to Proposition 6.4. 7. Examples In this section, we present some examples related to the results given in this paper. Computations below were done by using MAGMA2 and Risa/Asir.3 Example 1. Let A(t) be the family of elliptic curves defined in Sect. 4 for p = 13 and put A = A(−15) . Then A is an elliptic curve of conductor 26 · 3 · 5 · 372 · 1632 .

We have an isogeny φ : A → A = A(−15) of degree 13. By computing the 13-division polynomial of A, one sees that Ker(φ) ⊂ A[13] satisfies the condition (ER) in Proposition 4.1. Hence we have μ Ad ,13 = 0 for any positive integer d prime to 13. We have ord3 ( j A ) = ord5 ( j A ) = −13. Consider the quadratic twists of A and A by χ37 . These curves are given by the following equations: A37 : y 2 = x 3 + x 2 + 154549529495x + 21182222208207725, A 37 : y 2 = x 3 + x 2 − 21477749985x − 1211529110734587. 2 http://magma.maths.usyd.edu.au/ 3 http://www.math.kobe-u.ac.jp/Asir/ (Kobe distribution)

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These curves have split multiplicative reduction at 3 and 5. Moreover, we have L(A37 , 1) = 0 and A37 (Q)[13] = 0, i.e., A37 satisfies all assumptions of Lemma 3.5 for {1 , 2 } = {3, 5}. Hence we have λ A37 ,13 ≥ g A37 ,13 ≥ 2. Since A 37 has good or additive reduction outside {3, 5} and A 37 is non-anomalous at 13, we have dimF13 X(A 37 /Q)[13] ≥

g A37 ,13 ≥1 2

by an argument given in Sect. 5. Moreover, since X(A 37 /Q)[13] has even dimension, we have dimF13 X(A 37 /Q)[13] ≥ 2. We remark that the Birch and Swinnerton-Dyer conjecture predicts that X(A 37 /Q) ∼ = (Z/2Z)⊕2 ⊕ (Z/13Z)⊕2 . Example 2. Let E (t) be the family of elliptic curves with nontrivial Q-rational points of order 7 defined in Sect. 4. Put E = E (66) and let E denote an elliptic curve over Q with an isogeny ϕ : E → E of degree 7. Then E is an elliptic curve of conductor 2 · 3 · 5 · 11 · 13 · 252979. We have ord ( j E ) = −7 for any  ∈ L = {2, 3, 5, 11, 13} and ord252979 ( j E ) = −1. Then E −1751 satisfies the assumptions of Lemma 3.5 for primes in L. Since E −1751 [7] has a subgroup of order 7 satisfying (OU), we have μ E −1751 ,7 = 0 and λ E −1751 ,7 ≥ 5 by Lemma 3.5 and Proposition 4.1. It is worth mentioning that we can improve the bound given in Corollary 4.5 as λ E D , p ≥ n if there is an auxiliary prime q (say 252979 for this example) such that ordq ( j E ) is negative and prime to p. (Modify Steps 2, 4 and 5 in the proof of Proposition 4.3. For instance, replace  = n with  = q in Step 5.) Similarly, the result in Sect. 5 is improved as dimF p X(E d /Q)[ p] ≥ g E d , p ≥ n2 .

As for this example, we have dimF7 X(E −1751 /Q)[7] ≥ 4. (The smallest even 5 integer not less than 2 is 4.) The Birch and Swinnerton-Dyer conjecture predicts

that #X(E −1751 /Q) = 74 . Example 3. Let B (s) be the family of elliptic curves defined in Sect. 6 for p = 3. Then B := B (−1) is the curve 14A1 and B := B (1) is the curve  −1 26B1 in Cre−1 = = −1, mona’s table [5]. For d = −1, we have −1 3 7 13 = 1 and the √ class number of Q( −1) is 1. Then, by an argument given in Sect. 6, we have λ B−1 ,3 = μ B−1 ,3 = 0 and

,3 = λ B ,3 + s13 = 1. λ B−1 −1

,7 = 0, δ B ,13 = 1 and δ B ,13 = 2.) Similarly, for s = 2 (We have δ B−1 ,7 = δ B−1 −1 −1 and d = −37, we have λ B (2) ,3 = 2. −37

More generally, s prime to 3 and a negative integer √ d satisfying  for an integer d ≡ 3 (mod 4), d3 = d7 = −1 and the class number of Q( d) is not divisible by 3, one can show that λ B (s) ,3 is equal to the sum of s for all primes  at (s)

d

which Bd has split multiplicative reduction. We list elliptic curves whose 3-adic λ-invariant is each integer less than 10 obtained using this formula.

Elliptic curves with large Iwasawa λ-invariants and large X λ E,3

s

d

0 1 2 3 4 5 6 7 8 9

−1 1 2 −4 10 10 −19 −28 −28 17

−1 −1 −37 −37 −37 −421 −37 −37 −193 −85

303

(s)

E = Bd 2 3 y = x − x 2 + 72x + 368 y 2 = x 3 − x 2 − 72x + 496 y 2 = x 3 − x 2 − 789000x + 1416922352 y 2 = x 3 − x 2 + 6307896x + 89497287152 y 2 = x 3 − x 2 − 98568456x + 21884154972400 y 2 = x 3 − x 2 − 12761411080x + 32238164138716400 y 2 = x 3 − x 2 + 676077456x + 1029448361926592 2 y = x 3 − x 2 + 2163764280x + 10544724694952432 2 y = x 3 − x 2 + 58873671040x + 1496587192711577600 y 2 = x 3 − x 2 − 2555745008x + 6403866870552512

Acknowledgment. The author would like to thank Noboru Aoki, Masato Kuwata and the anonymous referee for many helpful suggestions and comments. He would also like to thank Yoshitaka Hachimori and Takae Tsuji for stimulating discussions. The author was partially supported by Grant-in-Aid for Young Scientists.

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15. Matsuno, K.: On the 2-adic Iwasawa invariants of ordinary elliptic curves. (2005, preprint) 16. Matsuno, K.: Elliptic curves with large Tate-Shafarevich groups over a number field. (2006, preprint) 17. Nakagawa, J., Horie, K.: Elliptic curves with no rational points. Proc. Am. Math. Soc. 104, 20–24 (1988) 18. Rubin, K., Silverberg, A.: Families of elliptic curves with constant mod p representations. In: Elliptic curves, modular forms, and Fermat’s Last Theorem, pp. 148–161. International Press, Cambridge (1995) 19. Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 151. Springer, Heidelberg (1994) 20. Washington, L.C.: Introduction to cyclotomic fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, Heidelberg (1997)