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

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 (?/2?)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 (?/2?)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.