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.
Shafarevich-Tate group distribution congruent elliptic curve multiplicative number theory number field independence property residue symbol
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This work was supported by National Natural Science Foundation of China (Grant No. 11501541). The author is greatly indebted to his advisor Professor Ye Tian for many instructions and suggestions. The author thanks Lvhao Yan for carefully reading the manuscript and giving valuable comments.
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