Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$

Abstract

Let ${\mathbb{Q}}_{\infty }$ be the cyclotomic ${\mathbb{Z}}_2$-extension over $\mathbb{Q}$. For each integer $n\ge 1$, let ${\mathbb{Q}}_n$ denote the unique subfield in ${\mathbb{Q}}_{\infty }$ such that $[{\mathbb{Q}}_{\infty }:\mathbb{Q}]=2^n$. Denote by ${\mathbb{Z}}_2[\mathrm{Gal}({\mathbb{Q}}_n/\mathbb{Q})]$ the group ring of $\mathrm{Gal}({\mathbb{Q}}_{\infty }/\mathbb{Q})$. For any elliptic curve defined over $\mathbb{Q}$ with odd conductor, the Mazur-Tate modular element associated with the curve is an element of ${\mathbb{Z}}_2[\mathrm{Gal}({\mathbb{Q}}_n/\mathbb{Q})]$. In this paper, for each $n$, we study the $2$-adic properties of Mazur-Tate modular elements associated with quadratic twists of elliptic curves, under specializations by finite order characters of $\mathrm{Gal}({\mathbb{Q}}_n/\mathbb{Q})$. Using the congruence properties of Heegner points and an equivariant version of the Coates-Wiles theorem, we construct an elliptic curve $E/\mathbb{Q}$ and a family of quadratic twists $E^{(m)}$ of $E$ such that each $E^{(m)}$ has both analytic and algebraic rank one over ${\mathbb{Q}}_{\infty }$, and whose Tate-Shafarevich group is infinite over ${\mathbb{Q}}_{\infty }$.