On quadratic twists of elliptic curves and some applications of a refined version of Yu's formula

In this paper, we study some cohomology groups and quadratic twists of elliptic curves, and apply Tate local duality and the results of Kramer-Tunnell on local norm cokernel to give a refined version of Yu's formula in the case of elliptic curves. Then, by using this refinement formula, we obtain explicit orders of Shafarevich-Tate groups of some elliptic curves in quadratic number fields, including a few unconditional cases.