LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly de...LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly determined, e.g. S2(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii) Whenp ≡ 5 (3, 5 for σ= ?1) (mod 8), it is proved that the Mordell-Weil group E(Q) ? Z/2Z ⊕ Z/2Z, rankE(Q) = 0, and Shafarevich-Tate group III (E/Q)[2]= 0. (iii) In any case, the sum of rankE(Q) and dimension of III (E/Q)[2] is given, e.g. 0, 1, 2 whenp ≡ 5, 1 (or 3), 7 (mod 8) for σ= 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number fieldK, etc. are also obtained.展开更多
In this paper,we calculate the ()-Selmer groups S()(E/Q) and S()(E/Q) of elliptic curves y2 = x(x + εpD)(x + εqD) via the descent method.In particular,we show that the Selmer groups of several families of such ellip...In this paper,we calculate the ()-Selmer groups S()(E/Q) and S()(E/Q) of elliptic curves y2 = x(x + εpD)(x + εqD) via the descent method.In particular,we show that the Selmer groups of several families of such elliptic curves can be arbitrary large.展开更多
Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-idea...Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.展开更多
Let p be an odd prime number,and let E be an elliptic curve defined over a number field which has good reduction at every prime above p.Under suitable assumptions,we prove that the η-eigenspace and the ■-eigenspace ...Let p be an odd prime number,and let E be an elliptic curve defined over a number field which has good reduction at every prime above p.Under suitable assumptions,we prove that the η-eigenspace and the ■-eigenspace of mixed signed Selmer group of the elliptic curve are pseudoisomorphic.As a corollary,we show that the η-eigenspace is trivial if and only if the ■-eigenspace is trivial.Our results can be thought as a reflection principle which relates an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.展开更多
We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,...Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.展开更多
基金This work ws supported by the National Natural Science Foundation of China(Grant No.10071041).
文摘LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly determined, e.g. S2(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii) Whenp ≡ 5 (3, 5 for σ= ?1) (mod 8), it is proved that the Mordell-Weil group E(Q) ? Z/2Z ⊕ Z/2Z, rankE(Q) = 0, and Shafarevich-Tate group III (E/Q)[2]= 0. (iii) In any case, the sum of rankE(Q) and dimension of III (E/Q)[2] is given, e.g. 0, 1, 2 whenp ≡ 5, 1 (or 3), 7 (mod 8) for σ= 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number fieldK, etc. are also obtained.
文摘In this paper,we calculate the ()-Selmer groups S()(E/Q) and S()(E/Q) of elliptic curves y2 = x(x + εpD)(x + εqD) via the descent method.In particular,we show that the Selmer groups of several families of such elliptic curves can be arbitrary large.
文摘Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.
基金The second author is supported by National Natural Science Foundation of China(Grant Nos.11550110172 and 11771164)。
文摘Let p be an odd prime number,and let E be an elliptic curve defined over a number field which has good reduction at every prime above p.Under suitable assumptions,we prove that the η-eigenspace and the ■-eigenspace of mixed signed Selmer group of the elliptic curve are pseudoisomorphic.As a corollary,we show that the η-eigenspace is trivial if and only if the ■-eigenspace is trivial.Our results can be thought as a reflection principle which relates an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.
基金supported by NSFC(Nos.12231009,11971224,12071209).
文摘We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
基金supported by National Natural Science Foundation of China (Grant No. 11271212)
文摘Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.
