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

TL;DR

This paper refines Yu's formula for elliptic curves using cohomology, Tate duality, and Kramer-Tunnell results, enabling explicit calculations of Shafarevich-Tate groups in quadratic fields.

Contribution

It introduces a refined version of Yu's formula for elliptic curves, facilitating explicit determination of Shafarevich-Tate group orders in quadratic fields.

Findings

Explicit orders of Shafarevich-Tate groups obtained

Refinement of Yu's formula for elliptic curves

Applications include unconditional cases

Abstract

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.