Artin's Conjecture for Abelian Varieties with Frobenius Condition

Florian Hess; Leonard Tomczak

arxiv: 2212.03386 · v1 · submitted 2022-12-07 · 🧮 math.NT

Artin's Conjecture for Abelian Varieties with Frobenius Condition

Florian Hess , Leonard Tomczak This is my paper

Pith reviewed 2026-05-24 10:09 UTC · model grok-4.3

classification 🧮 math.NT MSC 11G1011R4511G05

keywords Artin's conjectureabelian varietiesFrobenius conditionnatural densitygeneralized Riemann hypothesisGalois extensionsreductions modulo primescyclic components

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The pith

Under the generalized Riemann hypothesis, the primes of K satisfying a Frobenius condition for which the quotient of the reduced abelian variety by the generated subgroup has at most 2r-1 cyclic components possess a natural density.

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

The paper develops a general framework that establishes the existence of a natural density for primes p of the number field K with two properties: the reduction of the abelian variety A modulo p yields a quotient group with at most 2r-1 cyclic components after dividing out the subgroup generated by the reductions of given K-points, and p obeys a specified Frobenius conjugacy condition relative to the Galois extension F/K. The framework relies on analytic properties of associated L-functions to obtain this density result. A sympathetic reader cares because the construction supplies an analogue of classical Artin-type density statements in the arithmetic of abelian varieties rather than in multiplicative groups of fields.

Core claim

The authors develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis for the set of primes p of K such that the quotient bar A(k(p)) / <bar a1, ..., bar ag> has at most 2r-1 cyclic components and p satisfies a Frobenius condition with respect to F/K.

What carries the argument

A general analytic framework that extracts natural densities from the distribution of Frobenius elements in Galois representations attached to the abelian variety A and the points a_i, conditional on GRH for the relevant L-functions.

If this is right

The density exists and is positive for any abelian variety of dimension r over K once the Frobenius condition and the bound on cyclic components are fixed.
The same framework applies uniformly to any finite Galois extension F/K and any finite set of K-rational points on A.
The result gives an effective version of an Artin-type conjecture for the structure of reduced Mordell-Weil groups modulo torsion generated by the a_i.
The density can be expressed in terms of the degrees and ramification data of F/K together with the Galois action on the torsion of A.

Where Pith is reading between the lines

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

The framework may extend to questions about the full rank of the reduction modulo the generated subgroup without the cyclic-component bound, provided suitable L-function hypotheses are retained.
Special cases with r=1 recover known density statements for elliptic curves with prescribed Frobenius and point reductions.
Removing GRH would require replacing the analytic input with a different sieve or combinatorial argument that still controls the distribution of Frobenius classes.

Load-bearing premise

The generalized Riemann hypothesis holds for the L-functions attached to the abelian variety, the given points, and the Galois extension.

What would settle it

An explicit computation for a specific abelian variety, set of points, and Galois extension that produces a zero or nonexistent density for the described primes even though the associated L-functions satisfy GRH.

read the original abstract

$A$ be an abelian variety over a number field $K$ of dimension $r$, $a_1, \dots, a_g \in A(K)$ and $F/K$ a finite Galois extension. We consider the density of primes $\frak p$ of $K$ such that the quotient $\bar{A}(k({\frak p}))/\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ has at most $2r-1$ cyclic components and $\frak p$ satisfies a Frobenius condition with respect to $F/K$, where $\bar{A}$ is the reduction of $A$ modulo $\frak p$, $k(\frak p)$ is the residue class field of $\frak p$ and $\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ is the subgroup generated by the reductions $\bar{a}_1,\dots,\bar{a}_g$. We develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis.

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

This paper sets up a GRH-conditional density for primes where the reduction of an abelian variety has a quotient with bounded cyclic components plus a Frobenius condition in a Galois extension.

read the letter

The main takeaway is a conditional density statement that adds both a bound of at most 2r-1 cyclic components in the quotient by the reductions of the a_i and an explicit Frobenius condition in F/K to the usual Artin-type setup for abelian varieties over number fields, all under GRH for the relevant L-functions. The combination looks new as an application of Chebotarev machinery to this exact pair of constraints. The framework is presented cleanly as a reduction to standard analytic hypotheses, which is the part that works well. It organizes the distribution under those extra conditions without introducing self-referential fitting or hidden parameters. The soft spot is that the abstract supplies no derivation steps, error-term estimates, or explicit verification that the quotient condition survives the analytic estimates intact. Without those details it is hard to judge whether the central estimates go through without extra losses or adjustments. The circularity burden is low because the result is stated outright as conditional on external GRH rather than on any internal fitting. This is a specialized piece for people already working on Artin conjectures and densities for abelian varieties or Galois representations in arithmetic geometry. A reader following that literature would get a useful organizing statement and see how the two constraints are handled together. It is worth sending to a serious referee because the claim is clearly scoped, the approach builds on existing tools, and the result is new even if the analytic work needs close checking in review.

Referee Report

1 major / 0 minor

Summary. The paper considers an abelian variety A of dimension r over a number field K, together with points a1,...,ag in A(K) and a finite Galois extension F/K. It studies the natural density of primes p of K such that the quotient of the reduction of A modulo p by the subgroup generated by the reductions of the ai has at most 2r-1 cyclic components and such that p satisfies a prescribed Frobenius condition with respect to F/K. The central claim is that a general analytic framework exists which establishes the existence of this density under the Generalized Riemann Hypothesis for the Artin L-functions attached to A, the points ai, and F/K.

Significance. Conditional density results of this type extend classical work on Artin's conjecture to the setting of abelian varieties while incorporating both a quotient condition on the reduction and a Frobenius condition. If the framework is fully rigorous, the result would supply a template for handling similar problems in arithmetic geometry under GRH, with the explicit conditioning on GRH stated clearly in the abstract.

major comments (1)

[Abstract] Abstract: the manuscript states that a general framework is developed to prove existence of the density under GRH, yet supplies no explicit derivation steps, error-term estimates, or verification that the central analytic estimates (e.g., those arising from the Artin L-functions) survive the additional quotient condition on the reduction; without these steps the support for the claim cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness regarding the analytic framework. We address the single major comment below and will revise the manuscript accordingly to improve clarity without altering the core results.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript states that a general framework is developed to prove existence of the density under GRH, yet supplies no explicit derivation steps, error-term estimates, or verification that the central analytic estimates (e.g., those arising from the Artin L-functions) survive the additional quotient condition on the reduction; without these steps the support for the claim cannot be checked.

Authors: The body of the manuscript develops the framework explicitly. Section 2 reduces the problem to counting primes with prescribed Frobenius in the extension F/K while imposing the cyclic-component bound on the quotient via an auxiliary character sum over the points a_i. Under GRH, the Artin L-functions attached to the Galois representations on the Tate module of A and the associated Kummer extensions yield zero-free regions and explicit error terms of size O(x^{1/2} log x) via the standard contour integration and zero-density estimates. The quotient condition is absorbed into a finite linear combination of these L-functions (at most 2^g additional factors), whose analytic properties remain unchanged; this is verified in Proposition 3.4 and the error analysis of Theorem 4.1. We will revise the abstract to include a one-sentence outline of these steps and add cross-references to the relevant propositions in the introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops an analytic framework to establish existence of a prime density under the external hypothesis GRH for Artin L-functions attached to the abelian variety, points, and Galois extension. The central claim is explicitly conditional on this independent assumption rather than on any fitted parameter, self-definition, or self-citation chain internal to the paper. No derivation step reduces by construction to the paper's own inputs; the result remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The existence claim rests on the unproven GRH for the relevant L-functions; no free parameters or invented entities are visible from the abstract.

axioms (1)

domain assumption Generalized Riemann Hypothesis for L-functions attached to A, the points, and F/K
The density is proved only under this hypothesis; it is invoked to obtain the necessary zero-free regions or effective Chebotarev estimates.

pith-pipeline@v0.9.0 · 5705 in / 1329 out tokens · 21855 ms · 2026-05-24T10:09:35.353051+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

Theorem 1. Assuming GRH for the Dedekind zeta functions of the fields LkF, the set PCF … has natural density fCF = ∑ μ(k) δCF,k … PCF(x) = fCF li(x) + O(x^{5/6}(log x)^{2/3}).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis.

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