Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2017,（4）
• 页码范围: 593-612
• 页数: 20页
• 语种: 英文
• DOI: 10.1007/s11425-015-0742-7

摘要

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-n2z 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.