Torsion of elliptic curves over mathbb{Q}_p with good reduction in cyclotomic extensions
Pith reviewed 2026-05-18 06:53 UTC · model grok-4.3
The pith
Elliptic curves over Q_p with good reduction have their torsion subgroups over all cyclotomic extensions Q_p(μ_{p^n}) fully classified for every prime p and every n including infinity.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every prime p and every integer n with 0 ≤ n ≤ ∞, the authors determine all possible isomorphism types of the torsion subgroup of the group of Q_p(μ_{p^n})-rational points on an elliptic curve over Q_p that has good reduction.
What carries the argument
The reduction map to the special fiber together with the formal group law, which together bound and identify the possible torsion points in the cyclotomic tower.
Load-bearing premise
The elliptic curves are assumed to have good reduction over Q_p so that the reduction map and formal group law can be used to control the torsion.
What would settle it
An explicit elliptic curve over some Q_p with good reduction whose torsion subgroup over Q_p(μ_{p^n}) is a finite abelian group that does not appear in the authors' list for that p and n would falsify the classification.
read the original 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 $\mathbb{Q}_p(\mu_{p^n})$-rational points of elliptic curves over $\mathbb{Q}_p$ with good reduction, where $\mu_{p^n}$ is the set of the $p^n$-th roots of unity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper classifies the torsion subgroup of E(Q_p(μ_{p^n})) for every prime p and every 0 ≤ n ≤ ∞, where E is an elliptic curve over Q_p with good reduction. The classification proceeds by case analysis on ordinary versus supersingular reduction at p, using compatibility of the reduction map with torsion and the formal group law to control p-power torsion in the cyclotomic tower, with an explicit description of the direct limit as n → ∞.
Significance. If the classification holds, the result supplies a complete and explicit list of possible torsion structures in local cyclotomic extensions for good-reduction curves. This is a useful reference in arithmetic geometry for controlling rational points and Galois representations over local fields. The manuscript employs only standard tools (reduction maps and formal groups) with no free parameters or ad-hoc constructions, and the direct-limit case at n = ∞ is handled uniformly.
minor comments (3)
- [§2] §2 (Preliminaries): the compatibility statement between the reduction map and torsion points is stated without an explicit reference to the relevant lemma in Silverman's Arithmetic of Elliptic Curves; adding the citation would improve readability for readers outside the immediate subfield.
- [§4] §4 (Supersingular case): the description of the division fields of the formal group could include a short remark on the height-2 Lubin-Tate theory to make the enumeration of possible orders fully self-contained.
- The final section on n = ∞ would benefit from a single summarizing corollary that lists the possible groups in the direct limit, rather than leaving the reader to assemble the statement from the finite-n cases.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation of minor revision. The referee's summary accurately describes the classification of torsion subgroups E(Q_p(μ_{p^n})) for elliptic curves E over Q_p with good reduction, distinguishing ordinary and supersingular cases via reduction maps and formal groups, together with the direct limit at n=∞. No specific major comments were raised in the report.
Circularity Check
No significant circularity in derivation chain
full rationale
The paper's classification of torsion subgroups proceeds via standard case analysis on ordinary versus supersingular reduction, combined with the reduction map and formal group law to bound p-power torsion in the cyclotomic tower. Possible torsion structures are enumerated from the orders of reduced curves over finite fields and the division fields of the formal group, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation is self-contained against external benchmarks in local arithmetic geometry and does not invoke uniqueness theorems or ansatzes from the authors' prior work in a circular manner.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Elliptic curves with good reduction over Q_p admit a formal group law whose torsion points are controlled by the reduction map to the special fiber.
discussion (0)
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