Torsion growth of rational elliptic curves over ${\mathbb{Z}}_p$-extensions of quadratic fields

Abstract

Let $E/\Q$ be an elliptic curve and let ${\tilde{K}}_p$ be the compositum of all ${\mathbb{Z}}_p$-extensions of a quadratic field $K$. We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$ for $p\ge 5$. For $p=3$ and imaginary quadratic $K\neq\Q(\sqrt{-3})$, torsion on each extension is determined by its intersections with the cyclotomic extension and the $2$-division field. Over $\Q(\sqrt{-3})$, we construct infinitely many non-CM curves with full $3$-torsion in the first anticyclotomic layer and compute the $3$-primary torsion on every slope for eight CM curves. For $p=2$, we bound the odd-primary torsion and exclude all primes greater than $7$. We also give uniform bounds for non-CM primary torsion and correct two assertions in Li's preprint about noncyclotomic ${\mathbb{Z}}_3$-extensions.