Torsion of elliptic curves over $\mathbb{Q}_p$ with good reduction in cyclotomic extensions

TL;DR

This work provides a complete $p$-adic analogue of Mazur’s torsion classification for elliptic curves with good reduction, detailing the possible torsion subgroups over $\mathbb{Q}_p$ and over cyclotomic extensions $\mathbb{Q}_p(\mu_{p^{\infty}})$. The authors separate the analysis by odd primes and the prime $p=2$, using $p$-adic Hodge theory, Lubin-Tate theory, ramification bounds, and canonical lifts to tightly constrain torsion structures and realize all possibilities via explicit examples. For $p\ge 3$, they show the $p$-power torsion over the cyclotomic tower vanishes and classify the remaining torsion by the reduction type, with sharp bounds and exceptional small-$p$ phenomena. The $p=2$ case is handled with Fontaine ramification bounds and computational verification, yielding detailed lists and an explicit exceptional case, and the results extend to the maximal abelian cyclotomic extension with sharp, realizable bounds. Overall, the paper provides explicit, sharp classifications of torsion over $p$-adic fields and their cyclotomic extensions, illuminating how good reduction constrains torsion in $p$-adic and cyclotomic contexts.

Abstract

In this paper, for every prime $p$ and every $0\le n\le \infty$, we classify the structure of the torsion subgroup of the group of $\mathbb{Q}_p(μ_{p^n})$-rational points of elliptic curves over $\mathbb{Q}_p$ with good reduction, where $μ_{p^n}$ is the set of the $p^n$-th roots of unity.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Theorem 2.1
• Theorem 2.2: =Theorem \ref{['E(Qp):good']} (1)
• Theorem 2.3: =Theorem \ref{['E(Qp):good']} (2)
• proof : Proof of Theorem \ref{['E(Qp(mu)):good']} (1)
• Lemma 3.1