A graphical method to calculate Selmer groups of several families of non-CM elliptic curves
aa r X i v : . [ m a t h . N T ] D ec A graphical method to calculate Selmer groups ofseveral families of non-CM elliptic curves
Fei Li and Derong Qiu ∗ (School of Mathematical Sciences,Institute of Mathematics and Interdisciplinary Science,Capital Normal University, Beijing 100048, P.R.China ) Abstract
In this paper, we extend the ideas of Feng [F1], Feng-Xiong [FX]and Faulkner-James [FJ] to calculate the Selmer groups of elliptic curves y = x ( x + εpD )( x + εqD ) . Key words: elliptic curve, Selmer group, directed graph2000 Mathematics Subject Classification:
In this paper, we consider the following elliptic curves E = E ε : y = x ( x + εpD )( x + εqD ) , (1 . E ′ = E ′ ε : y = x − ε ( p + q ) Dx + 4 m D x, (1 . ε = ± , p and q are odd prime numbers with q − p = 2 m , m ≥ D = D · · · D n is a square-free integer with distinct primes D , · · · , D n . Moreover,2 ∤ D, p ∤ D and q ∤ D. For each D i , denote c D i = D/D i ( c D = 1 if D = D ) . Wewrite E = E + , E ′ = E ′ + if ε = 1 , and E = E − , E ′ = E ′− if ε = − . ∗ E-mail: [email protected] ϕ of degree 2 between E and E ′ as follows: ϕ : E −→ E ′ , ( x, y ) ( y /x , y ( pqD − x ) /x ) . The kernel is E [ ϕ ] = { O, (0 , } , and the dual isogeny of ϕ is b ϕ : E ′ −→ E, ( x, y ) ( y / x , y (4 m D − x ) / x )with kernel E ′ [ b ϕ ] = { O, (0 , } (see [S, p.74]).In this paper, we extend the ideas of Feng [F1], Feng-Xiong [FX] and Faulkner-James [FJ] to calculate the ϕ ( b ϕ ) − Selmer groups S ( ϕ ) ( E/ Q ) and S ( b ϕ ) ( E ′ / Q ) . Themain results are as follows:
Theorem 1.1
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. If m, p and D satisfyone of the following conditions:(1) m = 1 , pD ≡ D ≡ m = 1 , pD ≡ D ≡ m = 2;(4) pD ≡ m = 3 , pD ≡ m = 4 , pD ≡ , then ♯S ( ϕ ) ( E/ Q ) = ♯ { ( V , V ) e G (+ D ) : − , p, q, D k / ∈ V ; s < k ≤ n } . Inthe other cases, ♯S ( ϕ ) ( E/ Q ) = ♯ { ( V , V ) e G (+ D ) : − , p, q, D k ∈ V ; s < k ≤ n } + ♯ { ( V , V ) qe G (+ D ) : − , p, q, D k / ∈ V ; s < k ≤ n } . Here G (+ D ) is thedirected graph (see the following Definition 2.5). Theorem 1.2.
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n, then ♯S ( b ϕ ) ( E ′ / Q ) =2 ♯ { ( V , V ) e g (+ D ) : ± V } . Here g (+ D ) is the directed graph (see thefollowing Definition 2.8). 2 heorem 1.3. Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. If m, p and D satisfyone of the following conditions:(1) m = 1 , pD ≡ , D ≡ m = 1 , pD ≡ ± D ≡ m = 2; (4) m = 3 , pD m = 4 , pD ≡ , (6) m ≥ , pD ≡ , then ♯S ( ϕ ) ( E/ Q ) = ♯ { ( V , V ) e G ( − D ) : p, q, D k ∈ V ; s < k ≤ n } ; Inother cases, ♯S ( ϕ ) ( E/ Q ) = ♯ { ( V , V ) e G ( − D ) : p, q, D k ∈ V ; s < k ≤ n } + ♯ { ( V , V ) qe G ( − D ) : p, q, D k ∈ V ; s < k ≤ n } . Here G ( − D ) is the directedgraph (see the following Definition 2.10). Theorem 1.4.
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n, then ♯S ( b ϕ ) ( E ′ / Q ) =2 ♯ { ( V , V ) e g ( − D ) : − , ± / ∈ V } . Here g ( − D ) is the directed graph (see thefollowing Definition 2.13).Moreover, another result about the Selmer group of elliptic curves in (1.1) forall integers m ≥ Let M Q be the set of all places of Q , including the infinite ∞ . For each place p, denote by Q p the completion of Q at p , and if p is finite, denote by v p the correspond-ing normalized additive valuation, so v p ( p ) = 1 . Let S = {∞ , , p, q, D , · · · , D n } , and define a subgroup of Q ⋆ / Q ⋆ as follows: Q ( S,
2) = < − > × < > × < p > < q > × < D > × · · · × < D n > ∼ = ( Z / Z ) n +4 . For any subset A ⊂ Q ⋆ , wewrite < A > for the subgroup of Q ⋆ / Q ⋆ generated by all the elements in A. Foreach d ∈ Q ( S, , define the curves C d : dw = d − ε ( p + q ) Ddz + 4 m D z , and C ′ d : dw = d + ε ( p + q ) Ddz + pqD z . We have the following propositions 2.1 ∼ C d and C ′ d . The proofs are similar to those in [LQ], so we omit thedetails.
Proposition 2.1
We assume ε = 1 and the elliptic curve E = E + be as in(1.1).(A) For d ∈ Q ( S, , if one of the following conditions holds:(1) d <
0; (2) p | d ; (3) q | d. Then d / ∈ S ( ϕ ) ( E/ Q ) . Moreover, if d > , then C d ( R ) = ∅ . (B) For each d > , | d | D, d ∈ Q ( S, , we have(1) if m = 1 , then C d ( Q ) = ∅ ⇐⇒ d − D ( p + 1) + D d ≡ m = 2 , then C d ( Q ) = ∅ ;if m = 3 , then C d ( Q ) = ∅ ⇐⇒ d − D ( p + 4) + D d ≡ m = 4 , then C d ( Q ) = ∅ ⇐⇒ d − Dp ≡ m ≥ , then C d ( Q ) = ∅ ⇐⇒ Dp ≡ d − Dp ≡ . (2) For each odd prime number l | pqDd , C d ( Q l ) = ∅ ⇐⇒ (cid:0) dl (cid:1) = 1 . (3) For each odd prime number l | d, C d ( Q l ) = ∅ ⇐⇒ (cid:16) pDdl − l (cid:17) = (cid:16) qDdl − l (cid:17) =1 . d > , d | D, d ∈ Q ( S, , we have(1) if m = 1 , then C d ( Q ) = ∅ ⇐⇒ d ≡ m = 2 , then C d ( Q ) = ∅ ⇐⇒ d ≡ d − D ( p + 2) ≡ m ≥ , then C d ( Q ) = ∅ ⇐⇒ d ≡ d − Dp ≡ . (2) For each prime number l | pqDd , C d ( Q l ) = ∅ ⇐⇒ (cid:0) dl (cid:1) = 1 . (3) For each prime number l | d, C d ( Q l ) = ∅ ⇐⇒ (cid:16) pdDl − l (cid:17) = (cid:16) qdDl − l (cid:17) = 1 . Proposition 2.2
We assume ε = 1 and the elliptic curve E ′ = E ′ + be as in(1.2).(A) (1) For any d ∈ Q ( S, , C ′ d ( R ) = ∅ . If 2 | d, then d / ∈ S ( b ϕ ) ( E ′ / Q ) . (2) { , pq, − pD, − qD } ⊂ S ( b ϕ ) ( E ′ / Q ) . (B) For each d ∈ Q ( S,
2) satisfying d | pD, we have(B1) (1) If m = 1 , then C ′ d ( Q ) = ∅ if and only if one of the followingconditions holds: (a) d ≡ , (b) ( d + pD )( d + qD ) ≡ , (c) pqD d ≡ m = 2 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d + pD ≡ , (d) d ≡ p + 2) D ≡ m = 3 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d + pD ≡ , (d) d ≡ , and d + pD ≡ , (e) d ≡ d + pD ≡ m = 4 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d + pD ≡ , d ≡ d + pD ≡ , (e) d ≡ d + pD ≡ m ≥ , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d + pD ≡ . (B2) C ′ d ( Q p ) = ∅ and C ′ d ( Q q ) = ∅ . (B3) For each prime l | D, l ∤ d, C ′ d ( Q l ) = ∅ ⇐⇒ (cid:0) − (cid:0) dl (cid:1)(cid:1) (cid:0) − (cid:0) pqdl (cid:1)(cid:1) = 0 . (B4) For each prime l | D, l | d, C ′ d ( Q l ) = ∅ ⇐⇒ (cid:16) − (cid:16) − pdDl − l (cid:17)(cid:17) (cid:16) − (cid:16) − qdDl − l (cid:17)(cid:17) =0 . Proposition 2.3
We assume ε = − E = E − be as in(1.1).(A) For d ∈ Q ( S, , if one of the following conditions holds:(1) p | d ; (2) q | d. Then d / ∈ S ( ϕ ) ( E/ Q ) . Moreover, C d ( R ) = ∅ . (B) For each 2 | d, d | D, d ∈ Q ( S, , we have(1) if m = 1 , then C d ( Q ) = ∅ ⇐⇒ d + 2 D ( p + 1) + D d ≡ m = 2 , then C d ( Q ) = ∅ ;if m = 3 , then C d ( Q ) = ∅ ⇐⇒ d + D ( p + 4) + D d ≡ m = 4 , then C d ( Q ) = ∅ ⇐⇒ d + Dp ≡ m ≥ , then C d ( Q ) = ∅ ⇐⇒ Dp ≡ d + Dp ≡ . (2) For each odd prime number l | pqDd , C d ( Q l ) = ∅ ⇐⇒ (cid:0) dl (cid:1) = 1 . (3) For each odd prime number l | d, C d ( Q l ) = ∅ ⇐⇒ (cid:16) − pdDl − l (cid:17) = (cid:16) − qdDl − l (cid:17) = 1 . (C) For d | D, d ∈ Q ( S, , we have(1) if m = 1 , then C d ( Q ) = ∅ ⇐⇒ d ≡ m = 2 , then C d ( Q ) = ∅ ⇐⇒ d ≡ d + D ( p + 2) ≡ m ≥ , then C d ( Q ) = ∅ ⇐⇒ d ≡ d + Dp ≡ . (2) For each prime number l | pqDd , C d ( Q l ) = ∅ ⇐⇒ (cid:0) dl (cid:1) = 1 . (3) For each prime number l | d, C d ( Q l ) = ∅ ⇐⇒ (cid:16) − pdDl − l (cid:17) = (cid:16) − qdDl − l (cid:17) = 1 . Proposition 2.4.
We assume ε = − E ′ = E ′− be as in(1.2).(A) (1) For any d ∈ Q ( S,
2) and d > , C ′ d ( R ) = ∅ . If 2 | d or d < , then d / ∈ S ( b ϕ ) ( E ′ / Q ) . (2) { , pq, pD, qD } ⊂ S ( b ϕ ) ( E ′ / Q ) . (B) For each d ∈ Q ( S, , d | pD, d > , we have(B1) (1) If m = 1 , then C ′ d ( Q ) = ∅ if and only if one of the following conditionsholds:(a) d ≡ , (b) ( d − pD )( d − qD ) ≡ , (c) pqD d ≡ m = 2 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d − pD ≡ , (d) d ≡ p + 2) D ≡ m = 3 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d − pD ≡ , (d) d ≡ d − pD ≡ , (e) d ≡ d − pD ≡ . (4) If m = 4 , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d − pD ≡ , (d) d ≡ d − pD ≡ , (e) d ≡ d − pD ≡ m ≥ , then C ′ d ( Q ) = ∅ if and only if one of the following conditions holds:(a) d ≡ , (b) pqD d ≡ , (c) d − pD ≡ . (B2) C ′ d ( Q p ) = ∅ and C ′ d ( Q q ) = ∅ . (B3) For each prime l | D, l ∤ d, C ′ d ( Q l ) = ∅ ⇐⇒ (cid:0) − (cid:0) dl (cid:1)(cid:1) (cid:0) − (cid:0) pqdl (cid:1)(cid:1) = 0 . (B4) For each prime l | D, l | d, C ′ d ( Q l ) = ∅ ⇐⇒ (cid:16) − (cid:16) pdDl − l (cid:17)(cid:17) (cid:16) − (cid:16) qdDl − l (cid:17)(cid:17) =0 . Now let G = ( V, E ) be a directed graph. Recall that a partition ( V , V ) of V is called even if for any vertex, P ∈ V ( V ) , ♯ { P → V ( V ) } is even. In this case,we shall write ( V , V ) e V. The partition ( V , V ) is called quasi-even if for anyvertex, P ∈ V ( V ) ,♯ { P → V ( V ) } ≡ (cid:26) (cid:0) P (cid:1) = 1 , (cid:0) P (cid:1) = − . In this case, we shall write ( V , V ) qe V (see [F2] and [FJ] for these definitionsand related facts). Throughout this paper, for convenience, we write empty productas 1 . Definition 2.5
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. A directed graphG(+D) is defined as follows:Case 1. If m, p and D satisfy one of the following conditions:(1) m = 1; (2) m = 2 , ( p + 2) D m ≥ , pD ≡ , thendefine the directed graph G (+ D ) = G (+ D ) by defining the vertex V ( G (+ D )) tobe V ( G (+ D )) = {− , p, q, D , D , · · · , D n } and the edges E ( G (+ D )) as E ( G (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i :8 D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S {−−−→− D i : (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } . Case 2. If m, p and D satisfy one of the following conditions:(1) m = 2 , ( p +2) D ≡ m ≥ , pD ≡ , then define the directedgraph G (+ D ) = G (+ D ) by defining the vertex V ( G (+ D )) to be V ( G (+ D )) = { p, q, D , D , · · · , D n } and the edges E ( G (+ D )) as E ( G (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } . Here we define (cid:0) − (cid:1) = 1 , if m, p and D satisfy one of the following conditions:(1) m = 1 , pD ≡ D ≡ m = 1 , pD ≡ D ≡ m ≥ , pD ≡ . And we define (cid:0) − (cid:1) = − , if m, p and D satisfy one of the following conditions:(1) m = 1 , pD ≡ D ≡ m = 1 , pD ≡ D ≡ m ≥ , pD ≡ . Lemma 2.6.
For every even partition ( V , V ) of G (+ D ) such that V containsno − , p, q or D k ( s < k ≤ n ) , we have d ∈ S ( ϕ ) ( E/ Q ) where d = Q P ∈ V P . Conversely, suppose d is odd and d ∈ S ( ϕ ) ( E/ Q ) , we may write d = P P · · · P t with1 ≤ t ≤ s for distinct P j ∈ V ( G (+ D )) (1 ≤ j ≤ t ) , then ( V , V ) is even, where V = { P , P , · · · , P t } . Proof.
Suppose ( V , V ) is a nontrivial even partition of G (+ D ) such that − , p, q, D k / ∈ V ( s < k ≤ n ) . Let V = { D , D , · · · , D t } for some 1 ≤ t ≤ s. Consider d = D D · · · D t . For any 1 ≤ i ≤ t, we have (cid:16) pdDD − i D i (cid:17) = ( − ♯ {−−→ D i P : P ∈ V } =9 since ( V , V ) is even. Therefore, C d ( Q D i ) = ∅ by Proposition 2.1(C)(3). Also, for P ∈ V , P = − , (cid:0) dP (cid:1) = ( − ♯ {−−→ P D i : D i ∈ V } = 1 since ( V , V ) is even. Therefore, C d ( Q P ) = ∅ by Proposition 2.1(C)(2). We claim that C d ( Q ) = ∅ since ( V , V )is even. For an example in case 1, m = 1 : because ♯ {−−−→− D i : D i ∈ V } is even, d ≡ . Therefore, C d ( Q ) = ∅ by Proposition 2.1(C)(1). The remaining casescan be done similarly. And by Proposition 2.1(A), we have d in S ( ϕ ) ( E/ Q ) . Conversely, suppose d = P P · · · P t ∈ S ( ϕ ) ( E/ Q ) and d is odd. By Proposition2.1(C), P i ∈ { D , D , · · · , D s } and (cid:16) pdDP − i P i (cid:17) = 1 for each 1 ≤ i ≤ t. Let V = { P , P , · · · , P t } . Therefore, 1 = (cid:16) pdDP − i P i (cid:17) = ( − ♯ {−−→ P i P : P ∈ V } for 1 ≤ i ≤ t. So weget ♯ {−−→ P i P : P ∈ V } is even. For prime P | pqDd − , we have P ∈ V and (cid:0) dP (cid:1) = 1 . Therefore, 1 = (cid:0) dP (cid:1) = ( − ♯ {−−→ P P i : P i ∈ V } , which shows that ♯ {−−→ P P i : P i ∈ V } is even.If − ∈ V in case 1, then d ≡ C d ( Q ) = ∅ . Hence ♯ {−−−→− P i : 1 ≤ i ≤ t } iseven. To sum up, ( V , V ) is even. The proof of lemma 2.6 is completed. (cid:3) Lemma 2.7.
For every quasi-even partition ( V , V ) of G (+ D ) such that V contains no − , p, q or D k ( s < k ≤ n ) , we have 2 d ∈ S ( ϕ ) ( E/ Q ) , where d = Q P ∈ V P . Conversely, If d is even and d ∈ S ( ϕ ) ( E/ Q ) , we may write d =2 P P · · · P t with 1 ≤ t ≤ s for distinct P j ∈ V ( G (+ D )) (1 ≤ j ≤ t ) , then ( V , V ) isquasi-even, where V = { P , P , · · · , P t } . Proof.
Suppose ( V , V ) is a nontrivial quasi-even partition of G (+ D ) suchthat − , p, q, D k / ∈ V ( s < k ≤ n ) . Let V = { D , D , · · · , D t } for some 1 ≤ t ≤ s. Consider 2 d = 2 D D · · · D t . For any 1 ≤ i ≤ t, we have (cid:16) pdDD − i D i (cid:17) = (cid:16) D i (cid:17) ( − ♯ {−−→ D i P : P ∈ V } = 1 since ( V , V ) is quasi-even. Therefore, C d ( Q D i ) = ∅ byProposition 2.1(B)(3). Also, for P ∈ V and P = − , (cid:0) dP (cid:1) = (cid:0) P (cid:1) ( − ♯ {−−→ P D i : D i ∈ V } =10 since ( V , V ) is quasi-even. Therefore, C d ( Q P ) = ∅ by Proposition 2.1(B)(2). Weassert that C d ( Q ) = ∅ . To see this, we only need to prove case 1 with m = 1 , D ≡ pD ≡ , the other cases can be similarly done. Firstly, since (cid:0) − (cid:1) = 1 , we have ♯ {−−−→− D i : 1 ≤ i ≤ t } is even. So d ≡ D (2 d ) − ≡ . Next, since pD ≡ , we have d (1 − D (2 d ) − ) − pD ≡ , i.e., d − D ( p + 1) + D d ≡ , which shows that C d ( Q ) = ∅ by Proposition2.1(B)(1). Furthermore by Proposition 2.1(A), we get 2 d ∈ S ( ϕ ) ( E/ Q ) . Conversely, suppose d = 2 P P · · · P t ∈ S ( ϕ ) ( E/ Q ) . By Proposition 2.1(B), P i ∈{ D , D , · · · , D s } and (cid:16) pdDP − i P i (cid:17) = 1 for each 1 ≤ i ≤ t. Let V = { P , P , · · · , P t } . Therefore, 1 = (cid:16) pdDP − i P i (cid:17) = (cid:16) P i (cid:17) ( − ♯ {−−→ P i P : P ∈ V } for 1 ≤ i ≤ t. So ♯ { P i → V } =0(mod2) , if (cid:16) P i (cid:17) = 1 or 1(mod2) , if (cid:16) P i (cid:17) = − . For prime P | pqDd − , we have P ∈ V and (cid:0) dP (cid:1) = 1 . Therefore, 1 = (cid:0) dP (cid:1) = (cid:16) P i (cid:17) ( − ♯ {−−→ P P i : P i ∈ V } , which shows that ♯ { P → V } = 0(mod2) , if (cid:0) P (cid:1) = 1 or 1(mod2) , if (cid:0) P (cid:1) = − . If − ∈ V in case 1,e.g., m = 1 , D ≡ pD ≡ C d ( Q ) = ∅ , by Proposition2.1(A) we have d ≡ . Hence ♯ {−−−→− P i : 1 ≤ i ≤ t } is even (Here noticethat (cid:0) − (cid:1) = 1). The remaining cases can be done similarly. To sum up, ( V , V ) isquasi-even. The proof of lemma 2.7 is completed. (cid:3) Proof of Theorem 1.1.
By Proposition 2.1, S ( ϕ ) ( E/ Q ) ⊂ { , D , D , · · · , D n } . Furthermore, by lemma 2.6 and lemma 2.7, it is easy to obtain all the correspondingresults for different m, p, D.
The proof is completed. (cid:3)
Definition 2.8.
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s ) and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. A graph directedg(+D) is defined as follows : 11ase 1. If m, p and D satisfy one of the following conditions:(1) m = 1 , p ≡ p − D ≡ , m = 1 , p ≡ p − D ≡ , m = 2 , pD ≡ m = 2 , D ≡ pD ≡ m = 2 , D ≡ pD ≡ m = 3 , pD ≡ , then define the directed graph g (+ D ) = g (+ D ) by defining the vertex V ( g (+ D ))to be V ( g (+ D )) = {− , p, D , D , · · · , D n } and the edges E ( g (+ D )) as E ( g (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−−→ D i − (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } . Case 2. If m, p and D satisfy one of the following conditions:(1) m = 1 , p ≡ p − D ≡ , m ≥ , pD ≡ , then define the directed graph g (+ D ) = g (+ D ) by defining the vertex V ( g (+ D ))to be V ( g (+ D )) = {− , − , p, D , D , · · · , D n } and the edges E ( g (+ D )) as E ( g (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−−→ D i − (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − S {−−−−→− − } . Case 3. If m, p and D satisfy one of the following conditions:(1) m = 1 , p ≡ p − D ≡ , m ≥ , pD ≡ , then define the directed graph g (+ D ) = g (+ D ) by defining the vertex V ( g (+ D ))to be V ( g (+ D )) = {− , p, , D , D , · · · , D n } and the edges E ( g (+ D )) as E ( g (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−−→ D i − (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−→ D k : (cid:16) D k (cid:17) = − , ≤ k ≤ n } S {−→ p : (cid:16) p (cid:17) = − } . m, p and D satisfy one of the following conditions:(1) m = 2 , D ≡ pD ≡ m = 2 , D ≡ pD ≡ m = 3 , pD ≡ m = 4 , pD ≡ m ≥ , pD ≡ , then define the directed graph g (+ D ) = g (+ D ) by defining the vertex V ( g (+ D ))to be V ( g (+ D )) = {− , p, D , D , · · · , D n } and the edges E ( g (+ D )) as E ( g (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−−→ D i − (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − } . Case 5. If m, p and D satisfy one of the following conditions:(1) m = 3 , pD ≡ m ≥ , pD ≡ , then define the directed graph g (+ D ) = g (+ D ) by defining the vertex V ( g (+ D ))to be V ( g (+ D )) = {− , p, − , , D , D , · · · , D n } and the edges E ( g (+ D )) as E ( g (+ D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−−→ D i − (cid:16) − D i (cid:17) = − , ≤ i ≤ s } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − } S {−−→ D k : (cid:16) D k (cid:17) = − , ≤ k ≤ n } S {−→ p : (cid:16) p (cid:17) = − } S {−−−−→− − } . Lemma 2.9.
For every even partition ( V , V ) of g (+ D ) such that V containsno ± , we have d ∈ S ( b ϕ ) ( E ′ / Q ) , where d = Q P ∈ V P . Conversely, if d is odd and d ∈ S ( b ϕ ) ( E ′ / Q ) , we may write d = P P · · · P t for distinct P j ∈ V ( g (+ D )) (1 ≤ j ≤ t ) , then ( V , V ) is even, where V = { P , P , · · · , P t } . Proof.
Suppose ( V , V ) is a nontrivial even partition of g (+ D ) such that ± / ∈ V . Let V = { P , P , · · · , P t } , P i ∈ {− , p, D , D , · · · , D n } for each 1 ≤ i ≤ s. d = P P · · · P t . For each prime l | gcd( D, d ) , if l ∈ { D j : s < j ≤ n } , ( (cid:16) − pdDl − l (cid:17) − (cid:16) − qdDl − l (cid:17) −
1) = 0 because (cid:16) − pdDl − l (cid:17) (cid:16) − qdDl − l (cid:17) = (cid:0) pql (cid:1) = − l ∈ { D i : 1 ≤ i ≤ s } , then (cid:16) − pdDl − l (cid:17) = ( − ♯ {−→ lP : P ∈ V } = 1 because ( V , V ) iseven. Therefore, by Proposition2.2(B)(B4), we have C ′ d ( Q l ) = ∅ . Also for each prime l such that l | D and l ∤ d, if l ∈ { D j : s < j ≤ n } , then ( (cid:0) dl (cid:1) − (cid:0) pqdl (cid:1) −
1) = 0 be-cause (cid:0) dl (cid:1) (cid:0) pqdl (cid:1) = (cid:0) pql (cid:1) = −
1; if l ∈ { D i : 1 ≤ i ≤ s } , then (cid:0) dl (cid:1) = ( − ♯ {−→ lP : P ∈ V } = 1because ( V , V ) is even. So by Proposition2.2(B)(B3), we have C ′ d ( Q l ) = ∅ . We as-sert that C ′ d ( Q ) = ∅ . To see this, we only need to prove the case 3 with m = 1 , p ≡ p − D ≡ , , the other cases can be similarly done. In fact,since 2 ∈ V and ♯ {−→ P : P ∈ V } is even, we have d ≡ ± . So by Proposi-tion2.2(B)(B2), C ′ d ( Q ) = ∅ . This proves our assertion. So by Proposition2.2(B)(B3)and (A)(2), we obtain that d ∈ S ( b ϕ ) ( E ′ / Q ) . Conversely, suppose d = P P · · · P t ∈ S ( b ϕ ) ( E ′ / Q ) with distinct P , · · · , P t ∈{− , p, D , D , · · · , D n } . Let V = { P , P , · · · , P t } . For each prime l satisfying l | gcd( D, d ) , if l ∈ { D j : s < j ≤ n } , ♯ {−→ lP : P ∈ V } is even because ♯ {−→ lP : P ∈ V } = 0 . If l ∈ { D i : 1 ≤ i ≤ s } , by Proposition2.2(B)(B4) and (cid:0) pql (cid:1) = 1 , we have (cid:16) − pdDl − l (cid:17) − , and so 1 = (cid:16) − pdDl − l (cid:17) = ( − ♯ {−→ lP : P ∈ V } , whichshows that ♯ {−→ lP : P ∈ V } is even. Also, for each prime l satisfying l | D and l ∤ d, if l ∈ { D j : s < j ≤ n } , then ♯ {−→ lP : P ∈ V } is even because ♯ {−→ lP : P ∈ V } = 0; if l ∈ { D i : 1 ≤ i ≤ s } , by Proposition2.2(B)(B3) and (cid:0) pql (cid:1) = 1 , we have (cid:0) dl (cid:1) − . So 1 = (cid:0) dl (cid:1) = ( − ♯ {−→ lP : P ∈ V } , which shows that ♯ {−→ lP : P ∈ V } is even. As forthe vertex l = p, − , by the definition of g (+ D ) , we have ♯ {−→ lP : P ∈ V } = 0or ♯ {−→ lP : P ∈ V } = 0 . Now firstly, in case 2, − ∈ V . By C ′ d ( Q ) = ∅ and14he conditions for m, p, D, we have d ≡ , . So ♯ {−−→− P : P ∈ V } is even.Secondly, in case 3, 2 ∈ V . By C ′ d ( Q ) = ∅ and the conditions for m, p, D, wehave d ≡ , . So ♯ {−→ P : P ∈ V } is even. Lastly, in case 5, ± ∈ V . By C ′ d ( Q ) = ∅ and the conditions for m, p, D, we have d ≡ . So both ♯ {−−→− P , P ∈ V } and ♯ {−→ P : P ∈ V } are even. To sum up, ( V , V ) is even. TheProof is completed. (cid:3) Proof of Theorem 1.2.
By Proposition 2.2, we have { , pq, − pD, − qD } ⊂ S ( b ϕ ) ( E ′ / Q ) . Then the conclusion follows easily by Lemma2.9. The proof is com-pleted. (cid:3)
Definition 2.10.
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s )and (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. A directed graph G ( − D ) is defined as follows:Case 1. If m = 1 , D ≡ , then define the directed graph G ( − D ) = G ( − D )by defining the vertex V ( G ( − D )) to be V ( G ( − D )) = {− , p, q, D , D , · · · , D n } and the edges E ( G ( − D )) as E ( G ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S { D k − (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {− l : (cid:0) − l (cid:1) = − , l = p, q } . Case 2. If m, p and D satisfy one of the following conditions:(1) m = 1 , D ≡ . (2) m = 2 , D ≡ p + 2) D , thendefine the directed graph G ( − D ) = G ( − D ) by defining the vertex V ( G ( − D )) tobe V ( G ( − D )) = {− , p, q, D , D , · · · , D n } and the edges E ( G ( − D )) as E ( G ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) =15 , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S {− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−→ l − (cid:0) − l (cid:1) = − , l = p, q } . Case 3. If m, p and D satisfy one of the following conditions:(1) m = 2 , ( p + 2) D ≡ m ≥ , pD ≡ , then define the directed graph G ( − D ) = G ( − D ) by defining the vertex V ( G ( − D ))to be V ( G ( − D )) = {− , p, q, D , D , · · · , D n } and the edges E ( G ( − D )) as E ( G ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−−→ D k − (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ l − (cid:0) − l (cid:1) = − , l = p, q } . Case 4. If m = 2 , ( p + 2) D D ≡ , define the directedgraph G ( − D ) = G ( − D ) by defining the vertex V ( G ( − D )) to be V ( G ( − D )) = {− , p, q, D , D , · · · , D n } and the edges E ( G ( − D )) as E ( G ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−→ lD i : (cid:0) D i l (cid:1) = − , ≤ i ≤ s, l = p, q } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S { D k − (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ l − (cid:0) − l (cid:1) = − , l = p, q } S {−−→− p } . Case 5. If m ≥ , pD ≡ , define the directed graph G ( − D ) = G ( − D ) bydefining the vertex V ( G ( − D )) to be V ( G ( − D )) = {− , p, q, D , D , · · · , D n } andthe edges E ( G ( − D )) as E ( G ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−−→ D j D i : (cid:16) D i D j (cid:17) = − , ≤ i ≤ s, s < j ≤ n } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−→ lD i : (cid:0) D i l (cid:1) =16 , ≤ i ≤ s, l = p, q } S { D k − (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ l − (cid:0) − l (cid:1) = − , l = p, q } S {−−→− p : (cid:16) − p (cid:17) = − } . Here we define (cid:0) − (cid:1) = 1 , if m, p and D satisfy one of the following conditions:(1) m = 1 , pD ≡ D ≡ m = 1 , pD ≡ D ≡ m = 3 , pD ≡ m ≥ , pD ≡ m ≥ , pD ≡ . And we define (cid:0) − (cid:1) = − , if m, p and D satisfy one of the following conditions:(1) m = 1 , pD ≡ D ≡ m = 1 , pD ≡ D ≡ m ≥ , pD ≡ . Lemma 2.11.
For every even partition ( V , V ) of G ( − D ) such that V con-tains no p, q or D k ( s < k ≤ n ) , we have d ∈ S ( ϕ ) ( E/ Q ) , where d = Q P ∈ V P . Conversely, if d is odd and d ∈ S ( ϕ ) ( E/ Q ) , we may write d = δP P · · · P t for δ = ± P j ∈ V ( G ( − D )) (1 ≤ j ≤ t ) , then ( V , V ) is even. Here V = (cid:26) { P , P , · · · , P t } if δ = 1 , {− , P , P , · · · , P t } if δ = − . Proof.
Similar to the proof of Lemma 2.6.
Lemma 2.12.
For every quasi-even partition ( V , V ) of G ( − D ) such that V contains no p, q or D k ( s < k ≤ n ) , we have 2 d ∈ S ( ϕ ) ( E/ Q ) , where d = Q P ∈ V P . Conversely, if d is even and d ∈ S ( ϕ ) ( E/ Q ) , we may write d = 2 δP P · · · P t for δ = ± P j ∈ V ( G ( − D )) (1 ≤ j ≤ t ) , then ( V , V ) is quasi-even. Here V = (cid:26) { P , P , · · · , P t } if δ = 1 , {− , P , P , · · · , P t } if δ = − . Proof.
Similar to the proof of Lemma 2.7.
Definition 2.13.
Let D = D D · · · D s D s +1 · · · D n with (cid:16) pqD i (cid:17) = 1 ( i ≤ s )17nd (cid:16) pqD j (cid:17) = − s < j ≤ n ) for some non-negative integer s ≤ n. A graph directed g ( − D ) is defined as follows:Case 1. If m, p and D satisfy one of the following conditions:(1) m = 1 , p − D ≡ , m = 2 , pD ≡ m = 2 , D ≡ pD ≡ m = 2 , D ≡ pD ≡ m = 3 , pD ≡ , then define the directed graph g ( − D ) = g ( − D ) by defining the vertex V ( g ( − D ))to be V ( g ( − D )) = { p, D , D , · · · , D n } and the edges E ( g ( − D )) as E ( g ( − D ))= {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } . Case 2. If m, p and D satisfy one of the following conditions:(1) m = 1 , p ≡ p − D ≡ , m ≥ , pD ≡ , thendefine the directed graph g ( − D ) = g ( − D ) by defining the vertex V ( g ( − D )) to be V ( g ( − D )) = { p, − , D , D , · · · , D n } and the edges E ( g ( − D )) as E ( g ( − D ))= {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } Case 3. If m, p and D satisfy one of the following conditions:(1) m = 1 , p ≡ p − D ≡ , m ≥ , pD ≡ , thendefine the directed graph g ( − D ) = g ( − D ) by defining the vertex V ( g ( − D )) to be V ( g ( − D )) = { p, , D , D , · · · , D n } and the edges E ( g ( − D )) as E ( g ( − D ))= {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−→ p : (cid:16) p (cid:17) = − } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−→ D k : (cid:16) D k (cid:17) = − , ≤ k ≤ n } . Case 4. If m, p and D satisfy one of the following conditions:(1) m = 2 , D ≡ pD ≡ m = 2 , D ≡ D ≡ m ≥ , pD ≡ m = 4 , pD ≡ , thendefine the directed graph g ( − D ) = g ( − D ) by defining the vertex V ( g ( − D )) to be V ( g ( − D )) = { p, − , D , D , · · · , D n } and the edges E ( g ( − D )) as E ( g ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , ≤ i ≤ s, ≤ j ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } . Case 5. If m, p and D satisfy one of the following conditions:(1) m = 3 , pD ≡ m ≥ , pD ≡ , then define the directedgraph g ( − D ) = g ( − D ) by defining the vertex V ( g ( − D )) to be V ( g ( − D )) = { p, − , , D , D , · · · , D n } and the edges E ( g ( − D )) as E ( g ( − D )) = {−−−→ D i D j : (cid:16) D j D i (cid:17) = − , l ≤ i ≤ s, ≤ j ≤ n } S {−−→− p : (cid:16) − p (cid:17) = − } S {−−→ D i p : (cid:16) pD i (cid:17) = − , ≤ i ≤ s } S {−−−→− D k : (cid:16) − D k (cid:17) = − , ≤ k ≤ n } S {−−→ D k : (cid:16) D k (cid:17) = − , ≤ k ≤ n } S {−→ p : (cid:16) p (cid:17) = − } . Lemma 2.14.
For every even partition ( V , V ) of g ( − D ) such that V containsno − , ± , we have d ∈ S ( b ϕ ) ( E ′ / Q ) , where d = Q P ∈ V P . Conversely, if d is oddand d ∈ S ( b ϕ ) ( E ′ / Q ) , we may write d = P P · · · P t for distinct P j ∈ V ( g ( − D )) (1 ≤ j ≤ t ) , then ( V , V ) is even, where V = { P , P , · · · , P t } . Proof.
Similar to the proof of Lemma 2.9.
Proofs of Theorem 1.3 and 1.4.
By using Proposition 2.3, 2.4 and Lemma2.11, 2.12, 2.14, the proofs are similar to that of Theorem 1.1 and 1.2. (cid:3)
Appendix
In this appendix, by descent method, we obtain the following results aboutSelmer group of the elliptic curve (1.1) for all integers m ≥ , which generalize the19nes in [LQ] for the case m = 1 . The method is the same as in [LQ] (see also [QZ]and [DW]), so we omit the details.
Theorem A.1.
Let E = E + be the elliptic curve in (1.1) with ε = +1 and l be an odd prime number. For each i ∈ { , · · · , n } , denoteΠ + i ( D ) = δ i + (cid:16) − (cid:16) q c D i D i (cid:17)(cid:17) (cid:16) − (cid:16) p c D i D i (cid:17)(cid:17) + P l | pq c D i (cid:0) − (cid:0) D i l (cid:1)(cid:1) , where δ i = 0 if D i , m, p and D satisfy one of the following conditions:(1) D i ≡ , (2) m = 2 , p − D ≡ , (3) m ≥ , p + D ≡ δ i = 1 . And denoteΠ + n +1 ( D ) = δ n +1 + X l | pqD (cid:18) − (cid:18) l (cid:19)(cid:19) , where δ n +1 = 0 , if m, p and D satisfy one of the following conditions:(1) m = 3 , pD ≡ − , (2) m = 4 , pD ≡ , (3) m ≥
5; otherwise, δ n +1 = 1 . Here ( − ) is the ( Legendre ) quadratic residue symbol. And define afunction ρ + ( D ) by ρ + ( D ) = n +1 X i =1 (cid:20)
11 + Π + i ( D ) (cid:21) , where [x] is the greatest integer ≤ x. Then there exists a subset T ⊂ { , D , · · · , D n } with cardinal ♯T = ρ + ( D ) such that S ( ϕ ) ( E/ Q ) ⊃ < T mod( Q ⋆ ) > ∼ = ( Z / Z ) ρ + ( D ) . In particular, dim S ( ϕ ) ( E/ Q ) ≥ ρ + ( D ) . Theorem A.2.
Let E ′ = E ′ + be the elliptic curve in (1.2) with ε = +1 . Foreach i ∈ Z ( n ) = { , · · · , n } , denoteΠ + i ( D ′ ) = (cid:16) − (cid:16) − q c D i D i (cid:17)(cid:17) (cid:16) − (cid:16) − p c D i D i (cid:17)(cid:17) + P nj =1 , j = i (cid:16) − (cid:16) D i D j (cid:17)(cid:17) (cid:16) − (cid:16) pqD i D j (cid:17)(cid:17) andΠ + n +1 ( D ′ ) = δ ′ n +1 + P ni =1 (cid:16) − (cid:16) − D i (cid:17)(cid:17) (cid:16) − (cid:16) − pqD i (cid:17)(cid:17) , where δ ′ n +1 = 0 , if m, p and D satisfy one of the following conditions: (1) m = 2 , p − D , m = 3 , p − D ≡ , (3) m ≥ , p − D ≡ δ ′ n +1 = 1 . Here ( − ) is the ( Legendre ) quadratic residue symbol.Take a subset I of Z ( n ) as follows:if m = 2 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i + pD ≡ } S { i ∈ Z ( n ) : D i ≡ p − D ≡ } ;if m = 3 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i + pD ≡ } S { i ∈ Z ( n ) : D i ≡ pD + D i ≡ } S { i ∈ Z ( n ) : D i ≡ pD − D i ≡ } ;if m = 4 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i + pD ≡ } S { i ∈ Z ( n ) : D i ≡ pD + D i ≡ } ;if m ≥ , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i + pD ≡ } . Define a function ρ + ( D ′ ) by ρ + ( D ′ ) = X i ∈ I S { n +1 } (cid:20)
11 + Π + i ( D ′ ) (cid:21) , where [x] is the greatest integer ≤ x. Then there exists a subset T ⊂ {− , D , · · · , D n } with cardinal ♯T = ρ + ( D ′ ) such that S ( ϕ ) ( E/ Q ) ⊃ < T mod( Q ⋆ ) > ∼ = ( Z / Z ) ρ + ( D ′ ) . In particular, dim S ( ϕ ) ( E/ Q ) ≥ ρ + ( D ′ ) . Theorem A.3.
Let E = E − be the elliptic curve in (1.1) with ε = − l be an odd prime number. For each i ∈ { , · · · , n } , denoteΠ − i ( D ) = δ i + (cid:16) − (cid:16) − q c D i D i (cid:17)(cid:17) (cid:16) − (cid:16) − p c D i D i (cid:17)(cid:17) + P l | pq c D i (cid:0) − (cid:0) D i l (cid:1)(cid:1) , where δ i = 0 , if D i , m, p and D satisfy one of the following conditions:(1) D i ≡ , (2) m = 2 , p + D ≡ , (3) m ≥ , p − D ≡ δ i = 1 . And denoteΠ − n +1 ( D ) = δ n +1 + X l | pqD (cid:18) − (cid:18) l (cid:19)(cid:19) , where δ n +1 = 0 , if m, p and D satisfy one of the following conditions:(1) m = 3 , pD ≡ , (2) m = 4 , pD ≡ − , (3) m ≥
5; otherwise, δ n +1 = 1; and denote Π − n +2 ( D ) = δ n +2 + X l | pqD (cid:18) − (cid:18) − l (cid:19)(cid:19) , where δ n +2 = 0 , if m, p and D satisfy one of the following conditions:(1) pD ≡ , (2) m ≥ , pD ≡ δ n +2 = 1 . And define a function ρ − ( D ) by ρ − ( D ) = n +2 X i =1 (cid:20)
11 + Π − i ( D ) (cid:21) , where [x] is the greatest integer ≤ x. Then there exists a subset T ⊂ {− , , D , · · · , D n } with cardinal ♯T = ρ − ( D ) such that S ( ϕ ) ( E/ Q ) ⊃ < { D i : D i ∈ T } mod ( Q ⋆ ) > ∼ = ( Z / Z ) ρ − ( D ) . In particular, dim S ( ϕ ) ( E/ Q ) ≥ ρ − ( D ) . Theorem A.4.
Let E ′ = E ′− be the elliptic curve in (1.2) with ε = − . Foreach i ∈ Z ( n ) = { , · · · , n } , denote Π − i ( D ′ ) = (cid:16) − (cid:16) q c D i D i (cid:17)(cid:17) (cid:16) − (cid:16) p c D i D i (cid:17)(cid:17) + P nj =1 , j = i (cid:16) − (cid:16) D i D j (cid:17)(cid:17) (cid:16) − (cid:16) pqD i D j (cid:17)(cid:17) . Here ( − ) is the(Legendre) quadratic residue symbol. Take a subset I of Z ( n ) as follows:if m = 2 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i − pD ≡ } S { i ∈ Z ( n ) : D i ≡ p + D ≡ } ;if m = 3 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i − pD ≡ } S { i ∈ Z ( n ) : D i ≡ pD − D i ≡ } { i ∈ Z ( n ) : D i ≡ pD + D i ≡ } ;if m = 4 , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i − pD ≡ S { i ∈ Z ( n ) : D i ≡ pD − D i ≡ } ;if m ≥ , set I = { i ∈ Z ( n ) : D i ≡ } S { i ∈ Z ( n ) : D i − pD ≡ } . Define a function ρ − ( D ′ ) by ρ − ( D ′ ) = X i ∈ I (cid:20)
11 + Π − i ( D ′ ) (cid:21) , where [x] is the greatest integer ≤ x. Then there exists a subset T ⊂ { D , · · · , D n } with cardinal ♯T = ρ − ( D ′ ) such that S ( ϕ ) ( E/ Q ) ⊃ < T mod ( Q ⋆ ) > ∼ = ( Z / Z ) ρ − ( D ′ ) . In particular, dim S ( ϕ ) ( E/ Q ) ≥ ρ − ( D ′ ) . Acknowledgement
We are grateful to Prof. Keqin Feng for sending us hispapers [F1], [F2], [FX] and other materials which are helpful for this work.
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