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 $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$ the group ring of ${\rm Gal}(\mathbb{Q}_\infty/\mathbb{Q})$. For any elliptic curve defined over $\mathbb{Q}$ with odd conductor, the Mazur-Tate modular element associated with the curve is an element of $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$. 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 ${\rm Gal}(\mathbb{Q}_n/\mathbb{Q})$. 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 $\mathbb{Q}_\infty$.