Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part
Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part
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DOI：
10.1007/s11425-015-0742-7
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
WANG Zhangjie
中科院分区：
文献类型：
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作者：
WANG Zhangjie
Given a large positive number x and a positive integer k, we denote by Q_k(x) the set of congruent elliptic curves E (n): y^2 = z^3 − n^2 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) ∈ Q_k(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) ∈ Q_k(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.
