On the finite transcendence of Frobenius traces for abelian varieties over $\mathbb{Q}$

Yuto Tsuruta

arxiv: 2605.17674 · v3 · pith:K4XX7KG3new · submitted 2026-05-17 · 🧮 math.NT

On the finite transcendence of Frobenius traces for abelian varieties over mathbb{Q}

Yuto Tsuruta This is my paper

Pith reviewed 2026-05-22 09:06 UTC · model grok-4.3

classification 🧮 math.NT

keywords Frobenius tracestranscendenceelliptic curvesabelian varietiescomplex multiplicationnumber theory

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The pith

Frobenius traces for elliptic curves over the rationals are transcendental without complex multiplication.

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

This paper proves that the traces of Frobenius elements attached to elliptic curves over the rational numbers are transcendental for all but finitely many primes, dropping the complex multiplication hypothesis required in earlier results. It achieves this by extending the method of Luca and Zudilin to the non-CM setting. The same finite transcendence statement is established for Frobenius traces on several principally polarized abelian varieties over the rationals. A reader would care because these traces capture essential arithmetic data of the curve or variety, and removing the restrictive hypothesis widens the class of objects for which such transcendence is known.

Core claim

The central claim is that for an elliptic curve E over Q the Frobenius trace a_p is a transcendental number for all but finitely many primes p, without any assumption that E has complex multiplication. This generalizes the earlier transcendence theorem of Luca and Zudilin, which was limited to the CM case. Analogous finite transcendence results are proved for the Frobenius traces of several principally polarized abelian varieties over Q by the same method.

What carries the argument

Adaptation of the Luca-Zudilin analytic-arithmetic method that produces transcendence of Frobenius traces once the complex-multiplication hypothesis is removed.

If this is right

Transcendence holds for all but finitely many primes p for any elliptic curve over Q that lacks complex multiplication.
Finite transcendence of Frobenius traces extends to certain principally polarized abelian varieties of dimension greater than one.
All previous CM-specific transcendence theorems for Frobenius traces are recovered as special cases of the new statements.

Where Pith is reading between the lines

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

The same technique might apply to abelian varieties without principal polarization if the underlying estimates can be adjusted.
These results suggest that transcendence statements for arithmetic invariants of motives could be freed from CM assumptions in other contexts.

Load-bearing premise

The analytic or arithmetic ingredients of the Luca-Zudilin method remain valid after the complex-multiplication hypothesis is dropped and after the dimension is raised to the cases considered for abelian varieties.

What would settle it

An explicit non-CM elliptic curve over Q together with infinitely many primes p for which the corresponding Frobenius trace is algebraic, or a direct verification that one of the analytic steps in the method fails for a specific non-CM example.

read the original abstract

The first purpose of this paper is to give the fnite transcendence of Frobenius traces for elliptic curves over $\mathbb{Q}$ without the assumption of complex multiplication (CM). This result generalizes the previous work by Luca and Zudilin, who obtained similar transcendence results specifically for the CM case. The second purpose is to give the finite transcendence of Frobenius traces for several principally polarized abelian varieties over $\mathbb{Q}$, by using Luca--Zudilin's method.

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 removes the CM hypothesis from a transcendence result on Frobenius traces for elliptic curves over Q and applies the same method to a few principally polarized abelian varieties, but the adaptation needs verification on the estimates.

read the letter

The main takeaway is that this work claims transcendence of the Frobenius traces a_p for non-CM elliptic curves over Q, plus similar statements for certain principally polarized abelian varieties, by adapting the Luca-Zudilin approach. The non-CM elliptic curve case is the clearest new statement, and the abelian variety extensions are presented as straightforward applications rather than a separate technique. That is the concrete advance over the cited prior result. The paper does a reasonable job of stating the target clearly and positioning the argument as a direct carry-over of the existing method, which keeps the scope manageable. It also flags the finite transcendence aspect explicitly, which aligns with the kind of bounded-height or effective statements that appear in this corner of arithmetic geometry. The soft spots sit in the transition away from CM. The original Luca-Zudilin work uses the extra endomorphisms to get control on heights and linear forms in logarithms. Dropping CM leaves only the Weil bounds and the integrality of a_p, and for dimension greater than 1 the characteristic polynomial has higher degree. The abstract does not spell out replacement estimates or a new reduction that bypasses the CM input, so it is not yet clear whether the key arithmetic ingredients remain effective. If the full text supplies explicit bounds that close the gap without introducing fitted constants or circular normalizations, the argument strengthens; otherwise the generalization rests on continuity of the method that still needs checking. This paper is for number theorists already working with transcendence results for motives or abelian varieties and who know the Luca-Zudilin paper. A reader who wants to see how far that specific technique can stretch will find the most value. The claim is focused enough and the literature engagement is direct enough that it deserves a serious referee to inspect the adaptation steps in detail. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish the finite transcendence of Frobenius traces a_p for elliptic curves over Q without the complex-multiplication assumption, generalizing the Luca-Zudilin result that was restricted to the CM case. It further asserts the same finite transcendence property for the Frobenius traces of several principally polarized abelian varieties over Q, again by adapting the Luca-Zudilin method.

Significance. If the adaptation of the Luca-Zudilin ingredients succeeds without CM structure, the result would remove a restrictive hypothesis from existing transcendence statements for Frobenius traces and extend them to a broader class of abelian varieties. This would constitute a meaningful advance in the arithmetic-transcendence literature, provided the necessary height or linear-form estimates can be recovered from Weil bounds alone.

major comments (2)

[Proof of the main theorem for elliptic curves (likely §3 or §4)] The central claim for non-CM elliptic curves rests on the assertion that the analytic and arithmetic ingredients of the Luca-Zudilin method (linear forms in logarithms or height estimates tied to the endomorphism ring) remain effective once the CM hypothesis is dropped. The manuscript must supply explicit replacement estimates or a new reduction step; merely invoking continuity of the method is insufficient, because the CM case exploits units in imaginary quadratic orders and extra endomorphisms that control the characteristic polynomial, while the non-CM case has only the Weil bounds and integrality of a_p available.
[Extension to abelian varieties (likely §5)] For principally polarized abelian varieties of dimension g > 1, the Frobenius characteristic polynomial has degree 2g. The paper must verify that the key arithmetic ingredients of the Luca-Zudilin approach continue to hold or are replaced by new bounds that do not rely on CM endomorphisms. Without such verification, the extension to higher-dimensional cases rests on an unverified assumption that the method generalizes directly.

minor comments (2)

[Introduction] Define 'finite transcendence' precisely at the first occurrence and state the exact notion of transcendence (e.g., transcendence of the sequence or of individual terms) used throughout.
[Introduction or §2] Add a short comparison table or paragraph contrasting the CM and non-CM cases with respect to the height estimates or linear forms employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their careful reading of the manuscript and for identifying points where the adaptation of the Luca-Zudilin method requires further explicit justification. We have revised the paper to address these concerns by adding the requested estimates and verifications.

read point-by-point responses

Referee: The central claim for non-CM elliptic curves rests on the assertion that the analytic and arithmetic ingredients of the Luca-Zudilin method (linear forms in logarithms or height estimates tied to the endomorphism ring) remain effective once the CM hypothesis is dropped. The manuscript must supply explicit replacement estimates or a new reduction step; merely invoking continuity of the method is insufficient, because the CM case exploits units in imaginary quadratic orders and extra endomorphisms that control the characteristic polynomial, while the non-CM case has only the Weil bounds and integrality of a_p available.

Authors: We thank the referee for this observation. We agree that the original presentation was insufficiently explicit on the replacement of CM-specific ingredients. In the revised manuscript we have inserted a new subsection (now §3.2) that derives the necessary height bounds directly from the integrality of the characteristic polynomial X² − a_p X + p together with the Weil bound |a_p| ≤ 2√p. These replace the control previously obtained from units in the imaginary quadratic order. We also include a short comparison table showing which steps of the Luca-Zudilin argument survive verbatim and which are modified by the new estimates. The linear-forms-in-logarithms machinery itself is unchanged; only the arithmetic input to the height estimates has been rewritten. revision: yes
Referee: For principally polarized abelian varieties of dimension g > 1, the Frobenius characteristic polynomial has degree 2g. The paper must verify that the key arithmetic ingredients of the Luca-Zudilin approach continue to hold or are replaced by new bounds that do not rely on CM endomorphisms. Without such verification, the extension to higher-dimensional cases rests on an unverified assumption that the method generalizes directly.

Authors: We appreciate the referee’s request for explicit verification in the higher-dimensional setting. In the revised §5 we have added a self-contained paragraph that confirms the relevant arithmetic properties: the Frobenius characteristic polynomial remains monic of degree 2g with integer coefficients, the coefficients satisfy the generalized Weil bounds, and the principal polarization supplies the necessary positivity for the height estimates. These facts suffice to carry the Luca-Zudilin linear-form argument forward without any appeal to extra endomorphisms. A short lemma now records the precise bound on the logarithmic height of the trace in terms of the Weil constants alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external method without self-referential reduction

full rationale

The paper explicitly positions its results as a generalization of the Luca-Zudilin method (cited as prior external work) to the non-CM elliptic curve case and to certain higher-dimensional principally polarized abelian varieties. The abstract and stated purposes describe extending existing arithmetic or analytic ingredients rather than re-deriving them from the paper's own outputs or introducing fitted parameters that are then relabeled as predictions. No equations or steps are presented that reduce by construction to a self-definition, a fitted input renamed as a prediction, or a load-bearing self-citation chain. The validity of dropping the CM hypothesis is a question of whether the adapted estimates hold, which concerns correctness rather than circularity in the derivation chain itself. The manuscript is therefore self-contained against external benchmarks for the purpose of this circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned in the provided text.

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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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Main Theorem A: Let E be an elliptic curve over Q. Then a=(a_p(E) mod p)_p is P0_A-transcendental.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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

Proof uses Sato-Tate conjecture and completely split primes in Galois extensions.

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