Year: 2026
Type: preprint
Abstract: Let $E/\Q$ be an elliptic curve and let $\widetilde K_p$ be the compositum of all $\Z_p$-extensions of a quadratic field $K$. We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$ for $p\geq5$. 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 $\Z_3$-extensions.
Source: arXiv (Cornell University)
Authors: Xu, Shisong, Yuan, Haoyang
Institution: Nanjing University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: 同余数问题、整二次型和代数K理论相关问题的研究, G8842371449
OpenAlex ID: W7169860421