We study 2-primary parts Ш（E（n）/Q）[2 ∞] of Shafarevich-Tate groups of congruent elliptic curves E（n） ： y2 = x3-n2x, n ∈ Q×/Q×2. Previous results focused on finding sufficient conditions for Ш（E（...We study 2-primary parts Ш（E（n）/Q）[2 ∞] of Shafarevich-Tate groups of congruent elliptic curves E（n） ： y2 = x3-n2x, n ∈ Q×/Q×2. Previous results focused on finding sufficient conditions for Ш（E（n）/Q）[2∞] trivial or isomorphic to （Z/2Z）2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E（n） is isomorphic to （Z/2Z）2 and the Mordell-Weil rank of E（n） is zero, provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for Ш（E（n）/Q）[2∞] （Z/2Z）2k, where k ≥ 2.展开更多
Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime ...Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.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.
In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
文摘We study 2-primary parts Ш（E（n）/Q）[2 ∞] of Shafarevich-Tate groups of congruent elliptic curves E（n） ： y2 = x3-n2x, n ∈ Q×/Q×2. Previous results focused on finding sufficient conditions for Ш（E（n）/Q）[2∞] trivial or isomorphic to （Z/2Z）2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E（n） is isomorphic to （Z/2Z）2 and the Mordell-Weil rank of E（n） is zero, provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for Ш（E（n）/Q）[2∞] （Z/2Z）2k, where k ≥ 2.
基金supported by National Natural Science Foundation of China (Grant No. 11501541)
文摘Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.
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.
文摘In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
