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. ARTICLES . March 2014 Vol. 57 No. 3: 649–658 doi: 10.1007/s11425-013-4705-y

On non-congruent numbers with 1 modulo 4 prime factors

OUYANG Yi & ZHANG ShenXing∗

Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China Email: [email protected], [email protected]

Received September 11, 2012; accepted December 28 2012; published online September 4, 2013

Abstract In this paper, we use the 2-descent method to find a series of odd non-congruent numbers ≡ 1 (mod 8) whose prime factors are ≡ 1 (mod 4) such that the congruent elliptic curves have second lowest Selmer groups, which include Li and Tian’s result as special cases.

Keywords non-congruent number, 2-descent, second 2-descent

MSC(2010) 11G05, 11D25

Citation: Ouyang Y, Zhang S X. On non-congruent numbers with 1 modulo 4 prime factors. Sci China Math, 2014, 57: 649–658, doi: 10.1007/s11425-013-4705-y

1 Introduction

The congruent number problem is about when a positive integer can be the area of a rational right triangle. A positive integer n is a non-congruent number if and only if the congruent elliptic curve

E := E(n) : y2 = x3 n2x (1.1) − has Mordell-Weil rank zero. In [3,4], Feng obtained several series of non-congruent numbers for E(n) with the lowest Selmer groups. In [5], Li and Tian obtained a series of non-congruent numbers whose prime factors are 1 (mod 8) such that E(n) has second lowest Selmer groups. The essential tool of the above ≡ results is the 2-descent method of elliptic curves. In this paper, we will use this method to get a series of odd non-congruent numbers whose prime factors are 1 (mod 4) such that E(n) has second lowest ≡ Selmer groups, which include Li and Tian’s result as special cases. Suppose n is a square-free integer such that n = p p 1 (mod 8) and primes p 1 (mod 4), 1 ··· k ≡ i ≡ then by quadratic reciprocity law ( pi ) = ( pj ). pj pi Definition 1.1. Suppose n = p p 1 (mod 8) and p 1 (mod 4). The graph G(n) := (V, A) 1 ··· k ≡ i ≡ associated with n is a simple undirected graph with vertex set V := prime p n and edge set A := { | } pq : ( p )= 1 . { q − } Recall for a simple undirected graph G = (V, A), a partition V = V V is called even if for any v V 0 ∪ 1 ∈ i (i =0, 1), # v V1 i is even. G is called an odd graph if the only even partition is the trivial partition { → − } V = V . Then our main result is: ∅ ∪ ∗Corresponding author

c Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com link.springer.com 650 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3

Theorem 1.2. Suppose n = p p 1 (mod 8) and p 1 (mod 4). If the graph G(n) is odd and 1 ··· k ≡ i ≡ δ(n) (as will be given by (4.5)) is 1, then for the congruent elliptic curve E = E(n),

X 2 rankZ(E(Q))=0 and (E/Q)[2∞] ∼= (Z/2Z) . As a consequence, n is a non-congruent number. The following corollary is Li and Tian’s result [5]:

Corollary 1.3. Suppose n = p1 pk and pi 1 (mod 8). If the graph G(n) is odd and the Jacobi 1+√ 1 ··· (n) ≡ symbol ( − )= 1, then for E = E , n − X 2 rankZ(E(Q))=0 and (E/Q)[2∞] ∼= (Z/2Z) . As a consequence, n is a non-congruent number.

2 Review of 2-descent method

In this section, we recall the 2-descent method of computing the Selmer groups of elliptic curves. This section follows [5, pp. 232–233], also cf. [1, Section 5] and [7, Chapter X. 4]. For an isogeny ϕ : E E′ of elliptic curves defined over a number field K, one has the following → fundamental exact sequence:

(ϕ) 0 E′(K)/ϕE(K) S (E/K) X(E/K)[ϕ] 0. (2.1) → → → →

Moreover, if ψ : E′ E is another isogeny, for the composition ψ ϕ : E E, then the following → ◦ → diagram of exact sequences commutes (cf. [8, p.5]):

0 0 0

ι1 ι2    / / (ϕ) / / 0 E′(K)/ϕE(K) S (E/K) X(E/K)[ϕ] 0

ψ    / / / / 0 / E(K)/ψϕE(K) / S(ψϕ)(E/K) / X(E/K)[ψϕ] / 0

   / / (ψ) / X / 0 E(K)/ψE′(K) S (E′/K) (E′/K)[ψ] 0.

 0

Now suppose n is a fixed odd positive square-free integer, K = Q, and E/Q, E′/Q, ϕ, ψ = ϕ∨ are given by

(n) 2 3 2 (n) 2 3 2 E = E : y = x n x, E′ = E : y = x +4n x, − y2 y(x2 + n2) ϕ : E E′, (x, y) , d , → 7→ x2 x2   y2 y(x2 4n2) ψ : E′ E, (x, y) , − . → 7→ 4x2 8x2   (ψ) (ψϕ) Then ϕψ = [2], ψϕ = [2]. In this case ι1 and ι2 are exact. Let S˜ (E′/Q) denote the image of S (E/Q) (ψ) in S (E′/Q). Then

(ϕ) (ψ) #S (E/Q) #S (E′/Q) #X(E/Q)[ϕ]= , #X(E′/Q)[ψ]= , #E′(Q)/ϕE(Q) #E(Q)/ψE′(Q) Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 651 and (ϕ) (ψ) #S (E/Q) #S˜ (E′/Q) #X(E/Q)[2] = · . (2.2) #E (Q)/ϕE(Q) #E(Q)/ψE (Q) ′ · ′ Similarly, (ψ) (ϕ) #S (E′/Q) #S˜ (E/Q) #X(E′/Q)[2] = · . (2.3) #E(Q)/ψE (Q) #E (Q)/ϕE(Q) ′ · ′ (ϕ) (ψ) The 2-descent method to compute the Selmer groups S (E/Q) and S (E′/Q) is as follows (cf. [7] for general elliptic curves): Let

2 S = prime factors of 2n , Q(S, 2) = b Q×/Q× :2 ord (b), p S . { }∪{∞} { ∈ | p ∀ 6∈ } Note that Q(S, 2) is represented by factors of 2n and we identify these two sets. By the exact sequence

i j 0 E′(Q)/ϕE(Q) Q(S, 2) W C(E/Q)[ϕ], → → → where

i : (x, y) x, O 1, (0, 0) 4n2, j : d C /Q 7→ 7→ 7→ 7→ { d } and Cd/Q is the homogeneous space for E/Q defined by the equation

2 2 2 4 Cd : dw = d +4n z , (2.4) the ϕ-Selmer group S(ϕ)(E/Q) is then

S(ϕ)(E/Q) = d Q(S, 2) : C (Q ) = , p S . (2.5) ∼ { ∈ d p 6 ∅ ∀ ∈ } Similarly, suppose 2 2 2 4 C′ : dw = d n z . (2.6) d − (ψ) The ψ-Selmer group S (E′/Q) is then

(ψ) S (E′/Q) = d Q(S, 2) : C′ (Q ) = , p S . (2.7) ∼ { ∈ d p 6 ∅ ∀ ∈ } The method to compute S˜(ϕ)(E/Q) follows from [1, Section 5, Lemma 10]: Lemma 2.1. Let d S(ϕ)(E/Q). Suppose (σ,τ,µ) is a nonzero integer solution of dσ2 = d2τ 2 +4n2µ2. ∈ 2 Let be the curve corresponding to b Q×/Q× given by Mb ∈ : dw2 = d2t4 +4n2z4, dσw d2τt2 4n2µz2 = bu2. (2.8) Mb − − Then d S˜(ϕ)(E/Q) if and only if there exists b Q(S, 2) such that is locally solvable everywhere. ∈ ∈ Mb Note that the existence of σ,τ,µ follows from Hasse-Minkowski theorem (cf. [6]).

3 Local computation

We need a modification of the Legendre symbol. For x Q or Q such that ord (x) is even, we set ∈ p ∈ p x xp ordp(x) := − . (3.1) p p     2 Thus ( · ) defines a homomorphism from x Q×/Q× : ord (x) is even to 1 . p { ∈ p } {± } 652 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3

3.1 Computation of Selmer groups

In this subsection, we will find the conditions when Cd or Cd′ is locally solvable. We will not give the details since we only need to consider the valuations and quadratic residue. Lemma 3.1. d S(ϕ)(E/Q) if and only if d satisfies ∈ (1) d> 0 has no prime factor p 3 (mod 4); ≡ (2)( n/d )=1 for all odd p d; p | (3) ( d )=1 for all odd p (2n/d); p | (4) if 2 d, n 1 (mod 8). | ≡± Proof. In this case C : dw2 = d2t4 +4n2z4. It is obvious that C (R) = d> 0. Assume d> 0. d d 6 ∅ ⇔ (i) If 2 ∤ d n, then C : w2 = d(t4 + 4(n/d)2z4). | d p = 2. C (Q ) = d 1 (mod 4). • d 2 6 ∅ ⇐⇒ ≡ p d. C (Q ) = ( n/d ) = 1 and p 1 (mod 4). • | d p 6 ∅ ⇐⇒ p ≡ p ∤ d. C (Q ) = ( d ) = 1. • d p 6 ∅ ⇐⇒ p (ii) If 2 d 2n, then C : w2 = d(t4 + (2n/d)2z4). | | d p = 2. C (Q ) = d 2 (mod 8), n 1 (mod 8). • d 2 6 ∅ ⇐⇒ ≡ ≡± 2 = p d. C (Q ) = ( n/d ) = 1 and p 1 (mod 4). • 6 | d p 6 ∅ ⇐⇒ p ≡ p ∤ d. C (Q ) = ( d ) = 1. • d p 6 ∅ ⇐⇒ p Combining (i) and (ii) completes the proof of the lemma. (ψ) Lemma 3.2. d S (E′/Q) if and only if d satisfies ∈ (1) d 1 (mod 8) or n/d 1 (mod 8); ≡± ≡± (2) ( n/d )=1 for all p d, p 1 (mod 4); p | ≡ (3) ( d )=1 for all p (n/d),p 1 (mod 4). p | ≡ 2 2 4 2 4 Proof. In this case C′ : dw = d t n z . d − (i) If 2 d, by considering the 2-valuation of each side, we see Cd′ (Q2)= . | 2 4 2 4 ∅ (ii) If 2 ∤ d n, then C′ : w = d(t (n/d) z ). | d − p = 2. Cd′ (Q2) = d 1 (mod 8) or n/d 1 (mod 8). • 6 ∅ ⇐⇒ n/d≡± n/d ≡± p d. Cd′ (Qp) = ( p )=1or( − p ) = 1. • | 6 ∅ ⇐⇒ d d p ∤ d. C′ (Q ) = ( )=1or( − ) = 1. • d p 6 ∅ ⇐⇒ p p Combining (i) and (ii) completes the proof of the lemma.

3.2 Computation of the images of Selmer groups

Suppose 0 < 2d S(ϕ)(E/Q), d is odd with no prime factor 3 (mod 4). We want to find a necessary ∈ ≡ condition for 2d S˜(ϕ)(E/Q). Write 2d = τ 2 + µ2 and select the triple (σ,τ,µ) in Lemma 2.1 to be ∈ (2n,nτ/d,µ). Then the defining equations of in (2.8) can be written as M4ndb w2 =2d(t4 + (n/d)2z4), w τt2 (n/d)µz2 = bu2. (3.2) − − By abuse of notations, we denote the above curve by . We use the notation O(pm) to denote a number Mb with p-adic valuation > m. The case p | d. For i τ/µ (mod pZ ), i Z and i2 = 1, then p ≡ p p ∈ p p − p (τ i µ), p ∤ (τ + i µ). | − p p It is easy to see v(t)= v(z), we may assume that z =1, t2 ipn (mod p), then is given by ≡± d Mb : w2 =2d(t4 + (n/d)2), w τt2 (n/d)µ = bu2. Mb − − 2 2 > 2 2 nµ m 2 4 n (i) If v(bu )= m 3, then by w = (τt + d + O(p )) =2d(t + d2 ), nτ 2 µt2 = O(pm). − d   Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 653

2 nτ > m Let t = dµ + β, where v(β)= α 2 , then

n 2 nτ 2 nτ 4n2 τµ dµ2 w2 =2d + +2 β + β2 = 1+ β + β2 . d dµ dµ µ2 n 2n2        Take the square root on both sides, then

2 2 2n 1 τµ dµ 2 1 τµ 3α 3 w = 1+ β + β β + O(p − ) ± µ 2 n 2n2 − 8 n       2 2n µβ 3α 2 = + τβ + nµ + O(p − ) , ± µ 2n     but on the other hand, nµ 2n w = τt2 + + bu2 = + τβ + bu2. d µ The sign must be positive and 2 2 µβ 3α 2 bu = nµ + O(p − ), 2n   thus p b, ( b/p ) = ( nµ/p ), ( n/b ) = ( µ ) = ( 2τ ). | p p p p p (ii) If v(bu2)= m 6 2 and t2 ipn (mod p), let t2 = ipn + pαi , then ≡ d d p 2i n nα pdα w2 =2d pαi p + pαi = 4p2 1+ , · p · d p − · p 2n     and w nα pdα w = = 2i 1+ + O(p2) , 1 p ± p p 4n   rnµ bu2 = w τt2 − − d nα pdα i τn nµ = 2pi 1+ p ταi p + O(p3) ± p p 4n − d − d − p r   p2i τ nα n 2 ni nα dα = p p (τ i µ)2 2p2i + O(p3). − n p ∓ pτ − 2dτ − p ± p p 4n r  r If v(bu2) = 2, then nα n (mod p), and p ≡± pτ q ni nα dα bu2 = p (τ i µ)2 2p2i + O(p3) − 2dτ − p ± p p 4n r ni (τ i µ)3(3τ + i µ) = − p − p p + O(p3) 8dτ 3 ni (τ i µ)3 = − p − p + O(p3)= O(p3), 2dτ 2 which is impossible. Thus v(bu2) = 1 and p b, | b/p pi τ/n 2pτ/n n/b 2τ = − p = , or = . p p p p p           (iii) If v(bu2)= m 6 2 and t2 i (n/d) (mod p), then ≡− p bu2 = w τt2 (n/d)µ = (τi µ)n/d + O(p) − − p − n =2i τn/d + O(p)=(1+ i )2 τ + O(p), p p · d · b τ n/d thus p ∤ b and ( p ) = ( p )( p ). Note that 2τ τ + µi (mod p) and ( 2n/d ) = 1, hence we have ≡ p p 654 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3

Lemma 3.3. The curve defined by (3.2) is locally solvable at p d if and only if Mb | n/b τ + µi b τ + µi either p b, = p ; or p ∤ b, = p . | p p p p         n The case p | d . In this case t is a p-adic unit if and only if w is so. (i) If v(w) = v(t) = 0, then w √2dt2 (mod p) and ( √2d τ)t2 bu2 (mod p). Since (√2d 2 2 ≡ ± ± − ≡ √2d τ − τ)(√2d + τ)=2d τ = µ and √2d τ are co-prime, ord (√2d τ) is even and ( − ) is well defined. − ± p − p b √2d τ We may assume p ∤ √2d + τ. If w √2d (mod p) or v(µ) = 0, then p ∤ b, ( ) = ( − − ). Otherwise ≡− p p v(µ) > 1, w √2d (mod p), v(bu2) > 1, ≡ w2 = (τt2 + µn/d + bu2)2 =2d(t4 + (n/d)2), (µt2 nτ/d)2 = bu2(2w bu2)= bu2(2τt2 + O(p)), − − thus p ∤ b, ( b ) = ( 2τ ) = ( √2d+τ ). Then is locally solvable if and only if p p p Mb 2d b (√2d τ) p ∤ b, = 1 and = ± − . p p p       (ii) If v(z) = 0 and w = pw ,t = pt , then w2 = 2d(p2t4 + ( n )2z4), w √2d n z2 (mod p) and 1 1 1 1 pd 1 ≡ ± pd bu2/p ( √2d µ) n z2 (mod p). Thus is locally solvable if and only if ≡ ± − pd Mb 2d n/(db) √2d µ p b, = 1 and = − . | p p p       Note that √2d µ 2(√2d τ) 2(√2d τ)(√2d µ) = (τ + µ √2d)2 − = − . − − − ⇒ p p     From now on, suppose n = p p 1 (mod 8) and p 1 (mod 4). Pick i Z such that i2 = 1, 1 ··· k ≡ i ≡ p ∈ p p − then 1 √2d 2 √2d τ = (τ + µi ) 1 . − − p · 2 − τ + µi  p  2d Note that ( p ) = 1, we have Lemma 3.4. defined by (3.2) is locally solvable at p n if and only if Mb | d 2d n/b τ + µi 2 p b, =1 and = p , | p p p p         2d b τ + µi 2 or p ∤ b, =1 and = p . p p p p         By Lemmas 2.1, 3.1, 3.3 and 3.4, we have Proposition 3.5. Suppose n = p p 1 (mod 8) and p 1 (mod 4), then 2d S(ϕ)(E/Q) if 1 ··· k ≡ i ≡ ∈ and only if d> 0 and ( 2n/d )=1 for p d, ( 2d )=1 for p n . In this case 2d S˜(ϕ)(E/Q) only if there p | p | d ∈ exists b Q(S, 2) satisfying: ∈ (1) if p d, i τ/µ (mod pZ ), i2 = 1, | p ≡ p p − n/b τ + µi b τ + µi p b, = p , or p ∤ b, = p ; | p p p p         (2) if p n , i2 = 1, | d p − n/b 2(τ + µi ) b 2(τ + µi ) p b, = p , or p ∤ b, = p . | p p p p         Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 655

4 Proof of the main result

4.1 Some facts about graph theory

We now recall some notations and results in graph theory, cf. [3, 4]. Definition 4.1. Let G = (V, A) be a simple undirected graph. Suppose #V = k. The adjacency matrix M(G) = (a ) of G is the k k matrix defined as ij ×

0, if vivj A; aij := 6∈ (4.1) (1, if vivj A. ∈ The Laplace matrix L(G) of G is defined as

L(G) = diag d ,...,d M(G), (4.2) { 1 k}− where di is the degree of vi. Theorem 4.2. Let G be a simple undirected graph and L(G) its Laplace matrix. k 1 r (1) The number of even partitions of V is 2 − − , where r = rankF2 L(G). (2) The graph G is odd if and only if r = k 1. − (3) If G is odd, then the equations c1 t1 . . L(G) . = . ck ! tk ! has solutions if and only if t + + t =0. 1 ··· k Proof. The proof of the first two parts follows from [3]. We have a bijection

Fk/ (0,..., 0), (1,..., 1) ∼ partitions of V 2 { } −→ { } (c ,...,c ) (V , V ), 1 k 7−→ 0 1 where V = v : c = i (1 6 j 6 k) , i 0, 1 . i { j j } ∈{ } Regard L(G) = diag d ,...,d (a ) as a matrix over F . If { 1 k}− ij 2

c1 b1 . L(G) . = . Fk, . . ∈ 2 ck ! bk ! then if v V ,t 0, 1 , i ∈ t ∈{ }

k k k bi = dici + aij cj = aij (ci + cj )= aij (t + cj )= aij = # vi V1 t F2. { → − } ∈ j=1 j=1 j=1 cj =1 t X X X X− (1) The number of even partitions is

c1 0 1 n . . k 1 r # (c ,...,c ) F : L(G) . = . =2 − − . 2 1 k ∈ 2 . . ( ck ! 0 !)

(2) follows from (1) easily. (3) Since L is of rank k 1, the image space of L is of dimension k 1, but it lies in the hyperplane − − x + + x = 0, thus they coincide and the result follows. 1 ··· k 656 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3

4.2 Graph G(n) and Selmer groups of E and E′

From now on, we suppose n = p p 1 (mod 8) and p 1 (mod 4). 1 ··· k ≡ i ≡ a a Recall for an integer a prime to n, the Jacobi symbol ( n )= p n( p ), which is extended to a multiplicative 2 | homomorphism from a Q×/Q× : ord (a) even for p n to 1 . Set { ∈ p | }Q {± } a 1 a := 1 . (4.3) n 2 − n      2 The symbol [ n· ] is an additive homomorphism from a Q×/Q× : ordp(a) even for p n to F2. { ∈ pi | } n By definition, the adjacency matrix M(G(n)) has entries aij = [ ]. For 0 < d n, we denote by d, pj | { d } the partition p : p d p : p n of G(n). { | }∪{ | d } The following proposition is a translation of results in Lemmas 3.1 and 3.2: Proposition 4.3. Given a factor d of n. (1) For the Selmer group S(ϕ)(E/Q), (1a) d S(ϕ)(E/Q) if and only if d> 0 and d, n/d is an even partition of G(n); ∈ { } (1b) Suppose

1, if pi d, 2 ci = | n ti = . 0, if pi ; pi  | d   Then 2d S(ϕ)(E/Q) if and only if d> 0 and ∈  c1 t1 . . L(G) . = . . ck ! tk ! (ψ) (2) For the Selmer group S (E′/Q), (ψ) (2a) d S (E′/Q) if and only if d 1 (mod 8) and d, n/d is an even partition of G(n); ∈ (ψ) ≡± { } (2b) 2d / S (E′/Q). ∈ Proof. One only shows (1b), the rest is easy. For any i, let [i] be the set of j such that pi and pj are both prime divisors of d or n/d. Then d n/d dici + aij cj = aij (ci + cj )= aij = or . pi pi j=i j=i j /[i]     X6 X6 X∈ Then (1b) follows from Lemma 3.1. Applying Theorem 4.2(3) to Proposition 4.3, we have Corollary 4.4. If G(n) is odd, there exists a unique factor 0

S(ϕ)(E/Q)= 1, 2d, 2n/d,n = Z/2Z Z/2Z, { } ∼ × and (ψ) S (E′/Q)= 1, n = Z/2Z Z/2Z. {± ± } ∼ × For the d given in Corollary 4.4, write 2d = τ 2 + µ2. If2d S˜(ϕ)(E/Q), we suppose b satisfies ∈ T the condition that b defined by (3.2) is locally solvable everywhere. Suppose c′ = (c1′ ,...,ck′ ) and T M t′ = (t1′ ,...tk′ ) are given by

τ + µipj , if pj d, 1, if pj b, pj | c′ = | t′ =  j j h2(τ + µipi ) n (0, if pj ∤ b;  j  , if pj . pj | d h i  By Proposition 3.5, Lc′ = t′, i.e., Lv = t′ has a solution v = c′, which means that the summation of tj′ must be zero in F2 by Theorem 4.2(3). Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 657

Definition 4.5. Suppose n is given such that G(n) is an odd graph. For the unique factor d given in 2 2 2n 2 2 Corollary 4.4, write 2d = τ + µ and = τ ′ + µ′ . Let i Z/nZ be defined by d ∈ τ τ n i (mod d), i ′ mod . (4.4) ≡ µ ≡ µ d ′   We define τ + µi 2 δ(n) := + F . (4.5) n d ∈ 2     Then the following is a consequence of Proposition 3.5. Corollary 4.6. If G(n) is odd and δ(n)=1, then

S˜(ϕ)(E/Q)= 1 . { } (ϕ) (ϕ) Proof. Let λ∗ be the F2-rank of S˜ (E/Q), λ be the F2-rank of S (E/Q), then λ = 2. The existence of the Cassels’ skew-symmetric bilinear form on X implies that the difference λ λ∗ is even. (ϕ) − By the above analysis, δ(n)= t′ = 0, thus 2d / S˜ (E/Q), we have λ∗ < λ, λ∗ = 0. j j 6 ∈ n 2 2 Remark 4.7. If we replace d Pby d in the definition, δ(n) is invariant. Indeed, [ d ] = [ n/d ]. For the other term, τ + µi τ + µi τ + µi = + ′ , n d n/d       where i τ/µ (mod d), i′ τ ′/µ′ (mod n/d). Let u = (ττ ′ µµ′)/2, v = (τµ′ µτ ′)/2, then ≡ ≡ − − τ u + vi = (τ + µi)(τ ′ + µ′i)/2 τ τ ′ + µ′ ≡ · µ   2 2 2 τµ(τ ′µ + τµ′)/µ (τ + µ) /µ v/2 (mod d). ≡ ≡ · 2 2 Similarly, u + vi′ (τ ′ + µ′) /µ′ v/2 (mod (n/d)). If we interchange d and n/d, δ(n) will differ ≡ · τ + µi τ + µi τ + µ i τ + µ i + ′ + ′ ′ ′ + ′ ′ d n/d n/d d         2(u + vi) 2(u + vi ) v v = + ′ = + d n/d d n/d         v n u2 + v2 = = = =0 F . n v v ∈ 2 h i h i   Thus δ(n) does not change, which implies that δ(n) does not depend on the choices of d,τ,µ and only depends on n.

4.3 Proof of the main result

Proof of Theorem 1.2. We shall use the fundamental exact sequence (2.1) and the commutative dia- gram in Section 2 frequently. Since E(Q)tor ψE′(Q) = O and #E(Q)tor = 4, #E(Q)/ψE′(Q) > 4. Since G(n) is odd, (ψ) ∩ { } (ψ) #S (E′/Q) = 4 and #E(Q)/ψE′(Q) = 4, by (2.1), X(E′/Q)[ψ] = 0. Apparently, S˜ (E′/Q) (ψ) ⊇ E(Q)/ψE′(Q) and thus #S˜ (E′/Q) = 4. (ϕ) By Corollary 4.6, S˜ (E/Q) = 1 , then #E′(Q)/ϕE(Q) = 1. The facts that #E(Q)/ψE′(Q)=4 2 { } and E(Q)tor ∼= (Z/2Z) imply that #E(Q)/2E(Q) = 4 and

rankZ E(Q) = rankZ E′(Q)=0.

From X(E′/Q)[ψ]= E′(Q)/ϕE(Q) = 0, the diagram tells us that

X(E/Q)[2] = X(E/Q)[ϕ] = S(ϕ)(E/Q) = Z/2Z Z/2Z, ∼ ∼ ∼ × 658 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 and (2.3) tells us that X X (E′/Q)[2] ∼= (E′/Q)[ψ] ∼= 0. k Hence X(E′/Q)[2∞] = 0 and X(E′/Q)[2 ψ] = 0. By the exact sequence

k k 1 0 X(E/Q)[ϕ] X(E/Q)[2 ] X(E′/Q)[2 − ψ], → → → we have for every k N+, ∈ X k X 2 (E/Q)[2 ] ∼= (E/Q)[ϕ] ∼= (Z/2Z) , X 2 and thus (E/Q)[2∞] ∼= (Z/2Z) . 1+√ 1 Proof of Corollary 1.3. In this case, d = 1 and τ = µ = 1, δ(n) = [ n− ], thus the result follows.

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 11171317) and National Key Basic Research Program of China (Grant No. 2013CB834202). This paper was prepared when the authors were visiting the Academy of Mathematics and Systems Science and the Morningside Center of Mathematics of Chinese Academy of Sciences, and was grew out of a project proposed by Professor Ye Tian to the second author. We would like to thank Professor Ye Tian for his vision, insistence and generous hospitality. We also would like to thank Jie Shu and Jinbang Yang for many helpful discussions.

References

1 Birch B, Swinnerton-Dyer H P F. Notes on ellptic curves (II). J Reine Angrew Math, 1965, 218: 79–108 2 Cassels J W S. Arithmetic on curves of genus 1, (IV) proof of the Hauptvermutung. J Reine Angrew Math, 1962, 211: 95–112 3 Feng K. Non-congruent Numbers and Elliptic Curves with Rank Zero. Hefei: Press of University of Science and Technology of China, 2008, 25–29 4 Feng K. Non-congruent number, odd graphs and the BSD conjecture. Acta Arith, 1996, 80 2 3 2 5 Li D, Tian Y. On the Birch-Swinnerton-Dyer conjecture of elliptic curves ED : y = x − D x. Acta Math Sinica, 2000, 16: 229–236 6 Serre J P. A Course in Arithmetic. Berlin: Springer-Verlag, 1973 7 Silverman J H. The Arithmetic of Elliptic Curves. GTM 106. New York: Springer-Verlag, 1986 8 Xiong M, Zaharescu A. Selmer groups and Tate-Shararevich groups for the congruent number problem. Comment Math Helv, 2009, 84: 21–56

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Mathematics

CONTENTS Vol. 57 No. 3 March 2014 Progress of Projects Supported by NSFC

From microscopic theory to macroscopic theory — symmetries and order parameters of rigid molecules ...... 443 XU Jie & ZHANG PingWen Articles

Left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra ...... 469 CHEN HongJia & LI JunBo Categories of exact sequences with projective middle terms ...... 477 SONG KeYan & ZHANG YueHui A note on the basic Morita equivalences ...... 483 HU XueQin On conjugacy class sizes of primary and biprimary elements of a finite group ...... 491 SHAO ChangGuo & JIANG QinHui ...... Thompson's conjecture for Lie type groups E 7 (q) 499 XU MingChun & SHI WuJie The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials ...... 515 XIAO QingHua, XIONG LinJie & ZHAO HuiJiang The critical case for a Berestycki-Lions theorem ...... 541 ZHANG Jian & ZOU WenMing Fujita phenomena in nonlinear pseudo-parabolic system ...... 555 YANG JinGe, CAO Yang & ZHENG SiNing Best constants for Hausdorff operators on n-dimensional product spaces ...... 569 WU XiaoMei & CHEN JieCheng Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra ...... 579 JI GuoXing On existence, uniqueness and convergence of multi-valued stochastic differential equations driven by continuous semimartingales ...... 589 REN JiaGang, WU Jing & ZHANG Hua Local linear estimator for stochastic differential equations driven by a-stable Lévy motions ...... 609 LIN ZhengYan, SONG YuPing & YI JiangSheng Robustness of orthogonal matching pursuit under restricted isometry property ...... 627 DAN Wei & WANG RenHong An improved nonlinear conjugate gradient method with an optimal property ...... 635 KOU CaiXia On non-congruent numbers with 1 modulo 4 prime factors ...... 649 OUYANG Yi & ZHANG ShenXing An interesting identity and asymptotic formula related to the Dedekind sums ...... 659 XU ZheFeng & ZHANG WenPeng

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