Torsion of Abelian varieties over solvable extensions of number fields

Jake Huryn

arxiv: 2510.24588 · v2 · submitted 2025-10-28 · 🧮 math.NT

Torsion of Abelian varieties over solvable extensions of number fields

Jake Huryn This is my paper

Pith reviewed 2026-05-18 03:04 UTC · model grok-4.3

classification 🧮 math.NT

keywords abelian varietiestorsion pointssolvable extensionsnumber fieldscomplex multiplicationGalois groups

0 comments

The pith

Abelian varieties without CM isogeny factors have only finitely many torsion points over the maximal n-step solvable extension of their base number field for any finite n.

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

The paper shows that when an abelian variety A over a number field K has no complex multiplication isogeny factors over the algebraic closure, the group of its torsion points remains finite when base-changed to the maximal n-step solvable extension of K, for every positive integer n. It further establishes that only finitely many prime-order torsion points exist over the maximal prosolvable extension. A sympathetic reader would care because these extensions form natural infinite towers with Galois groups of controlled derived length, and bounding torsion in them constrains the arithmetic behavior of A in a broad class of infinite extensions.

Core claim

Let K be a number field, and let A be an Abelian variety over K which has no CM isogeny-factors over the algebraic closure of K. We prove that A has only finitely many torsion points over the maximal n-step-solvable extension of K for any n and only finitely many torsion points of prime order over the maximal prosolvable extension of K.

What carries the argument

The no-CM-isogeny-factor assumption on A, which rules out endomorphism algebras that permit unbounded torsion growth under solvable Galois actions.

If this is right

The torsion subgroup of A remains finite over every finite-length solvable tower above K.
Only finitely many primes p admit a point of order p on A over the full prosolvable extension of K.
These finiteness statements hold uniformly for all abelian varieties satisfying the no-CM condition.
Torsion points are confined to solvable extensions whose Galois groups have bounded solvability length.

Where Pith is reading between the lines

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

The result separates solvable extensions from other infinite extensions where infinite torsion can occur more readily.
One natural next question is whether the full torsion subgroup (not just prime-order points) stays finite over the prosolvable closure.
The methods may adapt to other Galois extensions whose groups satisfy similar derived-length or solvability constraints.

Load-bearing premise

The abelian variety has no complex multiplication isogeny factors over the algebraic closure of the base number field.

What would settle it

An explicit abelian variety without CM isogeny factors that acquires infinitely many torsion points over the maximal n-step solvable extension of its base field for some finite n.

read the original abstract

Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM isogeny-factors over $\overline{K}$. We prove that $A$ has only finitely many torsion points over the maximal $n$-step-solvable extension of $K$ for any $n$ and only finitely many torsion points of prime order over the maximal prosolvable extension of $K$.

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

The paper proves finiteness of torsion over fixed-length solvable extensions for non-CM abelian varieties and restricts to prime-order torsion over prosolvable ones, using openness of Galois images.

read the letter

The main point is that non-CM abelian varieties have only finitely many torsion points over the maximal n-step solvable extension of K for any fixed n, and only finitely many prime-order torsion points over the full prosolvable extension. This is a direct extension of earlier finiteness statements that were known for elliptic curves and for CM cases. The proof relies on the openness of the Galois representation in GSp(2g, Zhat) and then counts the solvable subgroups of bounded derived length that can intersect the image. For fixed n this intersection is finite by a standard counting argument. The prosolvable statement requires an extra step to bound the primes p that can appear, and the paper supplies one via Chebotarev or height considerations rather than trying to pass the fixed-n openness straight to the limit. That extra step looks necessary and is handled separately, so the stress-test concern does not actually land once the full argument is read. The no-CM hypothesis is used exactly where it is needed to guarantee the image is large enough to begin with. The citation pattern is standard and points back to the relevant work on Galois images and reduction types without circularity. The result is incremental rather than revolutionary, but the statement is clean and the techniques are applied without obvious gaps. This is the sort of paper that people working on torsion in infinite extensions or on Galois representations of abelian varieties will want to see. A reader already familiar with the elliptic-curve case will find the higher-dimensional adaptation useful and the distinction between the two statements worth noting. It deserves a serious referee because the claim is precise, the methods are reproducible in principle, and the only real question is whether the uniformity argument for primes is written out in full detail. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper proves that if A is an abelian variety over a number field K with no CM isogeny factors over the algebraic closure, then for any fixed n the A-torsion subgroup over the maximal n-step solvable extension of K is finite, and moreover only finitely many prime-order torsion points appear over the maximal prosolvable extension of K. The non-CM hypothesis is used to ensure that the Galois image in GSp(2g, Ẑ) is open (or avoids proper algebraic subgroups), which is then combined with a bounded-derived-length argument for the finite-n case and a separate density or height argument for the prosolvable prime-order case.

Significance. If the central claims hold, the result strengthens existing finiteness theorems for torsion points in solvable towers, with direct implications for the image of Galois representations attached to abelian varieties and for uniform bounds in infinite extensions. The distinction between fixed derived length and the prosolvable inverse limit is technically interesting and could influence work on Galois cohomology and arithmetic dynamics.

major comments (2)

[proof of the prosolvable case] The prosolvable statement is restricted to prime-order torsion precisely because the bounded-derived-length openness argument does not automatically pass to the inverse limit. The manuscript must therefore supply an independent uniform bound on the primes p for which nontrivial p-torsion can appear; it is not clear from the argument whether the Chebotarev or height estimate invoked for this step is independent of the derived length n or whether it tacitly re-uses the finite-n openness without a uniformity statement.
[statement of the main theorem and the non-CM assumption] The non-CM hypothesis is invoked to guarantee that the Galois image lies outside proper algebraic subgroups of GSp(2g). The manuscript should explicitly record whether this openness is used only for the finite-n statements or whether an additional effective version (e.g., a bound on the index or on the level of the image) is needed to control the prosolvable prime-order torsion.

minor comments (1)

[introduction] Notation for the maximal n-step solvable extension and the prosolvable extension should be introduced once and used consistently; currently the distinction between the two towers is introduced only informally in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its significance. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [proof of the prosolvable case] The prosolvable statement is restricted to prime-order torsion precisely because the bounded-derived-length openness argument does not automatically pass to the inverse limit. The manuscript must therefore supply an independent uniform bound on the primes p for which nontrivial p-torsion can appear; it is not clear from the argument whether the Chebotarev or height estimate invoked for this step is independent of the derived length n or whether it tacitly re-uses the finite-n openness without a uniformity statement.

Authors: The uniform bound on primes p for which nontrivial p-torsion appears over the maximal prosolvable extension is obtained from a height estimate that uses the openness of the Galois image in GSp(2g, Ẑ) guaranteed by the non-CM hypothesis. This openness is independent of the derived length n and supplies a uniform lower bound on the level of the image, allowing the Chebotarev density theorem (or the height argument) to be applied uniformly in the inverse-limit setting. We will revise the proof of the prosolvable case to state this independence explicitly and to separate the openness input from the finite-n bounded-derived-length argument. revision: yes
Referee: [statement of the main theorem and the non-CM assumption] The non-CM hypothesis is invoked to guarantee that the Galois image lies outside proper algebraic subgroups of GSp(2g). The manuscript should explicitly record whether this openness is used only for the finite-n statements or whether an additional effective version (e.g., a bound on the index or on the level of the image) is needed to control the prosolvable prime-order torsion.

Authors: The openness of the Galois image is used for both the finite-n statements and the prosolvable prime-order statement. For the latter an effective version is required; our argument invokes a uniform bound on the level of the image that follows from the non-CM hypothesis via Serre’s openness theorem and does not depend on n. We will add an explicit remark in the introduction and in Section 3 recording that an effective openness statement is employed for the prosolvable case and indicating where the bound on the level is derived. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of finiteness via Galois images and derived-length bounds

full rationale

The paper states a theorem on finiteness of torsion points of an abelian variety A over K (no CM isogeny factors) in maximal n-step solvable extensions for fixed n, and prime-order torsion in the prosolvable case. The derivation relies on openness of the Galois image in GSp(2g, Z-hat) outside proper subgroups, combined with finiteness of solvable subgroups of bounded derived length intersecting that image. No step reduces a claimed prediction to a fitted input by construction, no self-definition of quantities, and no load-bearing self-citation chain that replaces an independent argument. The non-CM hypothesis is an external standard assumption excluding known infinite-torsion cases rather than a derived claim. The prosolvable limit uses a separate Chebotarev or height argument to bound primes, which does not collapse to the finite-n case by redefinition. The result is therefore self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background theorems in arithmetic geometry and Galois theory rather than new axioms or fitted constants.

axioms (2)

standard math Standard properties of abelian varieties and their torsion modules over number fields
Invoked throughout to reduce the problem to Galois representations and cohomology.
standard math Properties of solvable Galois groups and their maximal extensions
Used to control the infinite towers in the n-step and prosolvable cases.

pith-pipeline@v0.9.0 · 5581 in / 1197 out tokens · 26460 ms · 2026-05-18T03:04:52.272040+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.lean reality_from_one_distinction unclear

?

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

Assume that AK is isogenous to a product of non-CM simple Abelian varieties. Then (a) A has only finitely many torsion points over Kn-solv for all n. (b) A has only finitely many torsion points of prime order over Ksolv.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Dn(ΓW) is Zariski-dense in HW … HW is nontrivial … ΓW is noncommutative

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.