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.
We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX...We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX(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 ⅢX(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
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.
文摘We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX(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 ⅢX(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.
