Torsion groups of elliptic curves that appear infinitely often over septic fields

Filip Najman

arxiv: 2602.03513 · v2 · submitted 2026-02-03 · 🧮 math.NT

Torsion groups of elliptic curves that appear infinitely often over septic fields

Filip Najman This is my paper

Pith reviewed 2026-05-16 07:30 UTC · model grok-4.3

classification 🧮 math.NT

keywords torsion groupselliptic curvesnumber fieldsdegree 7septic fieldsabelian groupsinfinite families

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The pith

The set of abelian groups appearing as torsion subgroups of infinitely many elliptic curves over degree-7 fields is completely determined.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper identifies the exact set Φ^∞(7) consisting of all abelian groups that arise as the torsion subgroup of infinitely many elliptic curves over number fields of degree 7, up to isomorphism over the algebraic closure. The determination proceeds by checking which torsion structures known from lower-degree fields can extend to degree 7 and by verifying whether any additional groups first become possible at degree 7. A reader cares because the result supplies a finite explicit list of allowable torsion types that can occur unboundedly often, thereby constraining the arithmetic geometry of elliptic curves in septic extensions.

Core claim

In this short note we determine the set Φ^∞(7) of Abelian groups that appear as torsion groups of infinitely many elliptic curves (up to Q-bar-isomorphism) over number fields of degree 7.

What carries the argument

The set Φ^∞(7) itself, obtained by exhaustive case analysis of possible torsion structures and reduction to known results for smaller degrees.

If this is right

Only the groups inside Φ^∞(7) can appear as torsion for infinitely many elliptic curves over septic fields.
Any torsion group outside this set occurs for only finitely many such curves up to isomorphism.
The result combines known torsion classifications over degrees 1 through 6 with direct checks for degree 7.
The classification is independent of any particular elliptic curve and applies uniformly to all curves over all degree-7 fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Any future search for elliptic curves with exotic torsion over septic fields is guaranteed to produce only finitely many examples.
The same reduction-to-lower-degree technique could be applied to determine Φ^∞(p) for other small primes p.
The finite list supplies concrete constraints that can be checked computationally for specific families of elliptic curves.

Load-bearing premise

The case analysis is exhaustive and misses no abelian group that could appear infinitely often over degree-7 fields.

What would settle it

An explicit elliptic curve over a degree-7 number field whose torsion subgroup lies outside the listed set Φ^∞(7) and that belongs to an infinite family of such curves up to algebraic closure isomorphism.

read the original abstract

In this short note we determine the set $\Phi^\infty(7)$ of Abelian groups that appear as torsion groups of infinitely many elliptic curves (up to $\overline \mathbb Q$-isomorphism) over number fields of degree 7.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Najman determines the precise set of torsion groups that occur infinitely often over septic fields by reducing to known lower-degree classifications and a finiteness argument.

read the letter

The main result here is a clean determination of Φ^∞(7), the exact list of abelian groups that arise as torsion subgroups for infinitely many elliptic curves over number fields of degree 7. This completes the picture for septic fields in the line of work that started with Mazur's theorem and has been extended degree by degree since then. The note achieves this by enumerating torsion structures possible in a degree-7 extension, then using finiteness results to exclude everything outside the list. It draws on the established classifications for smaller degrees without adding new technical machinery, which keeps the argument short and direct. The reduction looks standard and the case division appears to cover the possibilities without obvious omissions. The dependence on prior lists from Mazur, Kamienny-Merel, and Derickx-Sutherland is explicit and appropriate for a short note; those lists are the accepted foundation, so the main task is showing that degree 7 does not introduce new infinite families. No internal contradictions or hidden assumptions on the base field show up in the outline. The only real limitation is that the note is incremental rather than foundational, so its value is highest for readers already tracking the Φ^∞(d) sets. Specialists in elliptic curve torsion over number fields will find it useful as a reference for the degree-7 case. It is worth sending to peer review because the classification is definite and fills a natural gap in the literature.

Referee Report

0 major / 1 minor

Summary. The manuscript determines the set Φ^∞(7) of abelian groups that arise as the torsion subgroup of infinitely many elliptic curves (up to Q-bar isomorphism) over number fields of degree 7. The determination proceeds by reducing to known results on Φ^∞(d) for d < 7, combined with the classification of possible torsion structures for [K:Q]=7 and a finiteness argument excluding groups that occur only finitely often.

Significance. If the classification holds, the result completes the determination of Φ^∞(d) for d=7, extending the existing lists for smaller degrees obtained via Mazur, Kamienny–Merel, and Derickx–Sutherland. It supplies an explicit finite list of realizable infinite torsion groups over septic fields and confirms that all other groups appear only finitely often, which is a concrete contribution to the arithmetic geometry of elliptic curves over number fields of fixed degree.

minor comments (1)

The introduction would benefit from an explicit statement of the final list of groups in Φ^∞(7) rather than deferring the enumeration entirely to the proof section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes our determination of the set Φ^∞(7), and for recommending acceptance of the manuscript. We have no revisions to make in response to the report.

Circularity Check

0 steps flagged

No significant circularity; classification relies on external theorems

full rationale

The paper determines Φ^∞(7) through exhaustive enumeration of torsion structures compatible with degree-7 extensions, reducing via known results on Φ^∞(d) for d<7 and applying the classification of possible E(K)_tors for [K:Q]=7. These steps cite independent external results (Mazur, Kamienny–Merel, Derickx–Sutherland et al.) whose proofs are not internal to this work and do not depend on the target classification. The finiteness argument for omitted groups uses standard height or boundedness properties outside the paper's own definitions. No equation or step reduces to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on standard background results in the arithmetic of elliptic curves over number fields of bounded degree; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract statement.

axioms (1)

standard math Standard results on torsion subgroups of elliptic curves over number fields (e.g., generalizations of Mazur's theorem and known lists for small degrees)
The paper determines Φ^∞(7) by building directly on these established theorems.

pith-pipeline@v0.9.0 · 5314 in / 1125 out tokens · 37608 ms · 2026-05-16T07:30:22.699770+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. Φ∞(7) ={(1, n) : 1≤n≤30, n ∉ {25,29}} ∪ {(2,2n) : 1≤n≤10}.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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