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
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
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
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.
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.
