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arxiv: 2207.02913 · v2 · submitted 2022-07-06 · 🧮 math.NT

Bounds for the distribution of the Frobenius traces associated to a generic abelian variety

Alina Carmen Cojocaru , Tian Wang This is my paper

Pith reviewed 2026-05-24 12:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords abelian varietiesFrobenius tracesGalois representationsGRHChebotarev density theoremArtin L-functions
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The pith

Under GRH and surjective residual Galois representations, the number of primes p with fixed Frobenius trace t for a dimension-g abelian variety A over Q is at most x to a power strictly less than 1.

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

The paper establishes explicit upper bounds on the counting function π_A(x, t) that records how often the first Frobenius trace a_{1,p}(A) equals a fixed integer t. Under the standing assumption that the mod-ℓ Galois representation is surjective for all large ℓ, and assuming GRH for Dedekind zeta functions, the bounds are of the shape x^{1-1/(2g²+g+1)} (log x)^{-something} for t=0 and a slightly stronger exponent for t≠0. These counting bounds immediately imply that the set of primes for which |a_{1,p}(A)| is smaller than p to a positive power (depending on g) has density zero. A second, stronger set of bounds is derived once Artin’s holomorphy conjecture and a pair-correlation conjecture for Artin L-functions are added.

Core claim

Assuming surjectivity of the residual Galois representations for all sufficiently large ℓ and assuming GRH, one has π_A(x,0) ≪_A x^{1-1/(2g²+g+1)}/(log x)^{1-2/(2g²+g+1)} and π_A(x,t) ≪_A x^{1-1/(2g²+g+2)}/(log x)^{1-2/(2g²+g+2)} for t≠0; consequently the set of primes p with |a_{1,p}(A)| ≤ p^{1/(2g²+g+1)}/(log p)^{2/(2g²+g+1)+ε} has density zero. Under the additional hypotheses of Artin holomorphy and pair correlation the exponents improve to 1-1/(g+1) and 1-1/(g+2) respectively, yielding the stronger lower bound |a_{1,p}(A)| > p^{1/(g+2)-ε} for almost all p.

What carries the argument

The surjectivity assumption on the residual mod-ℓ Galois representations, which permits the application of effective forms of the Chebotarev density theorem inside the division fields of A to control the distribution of Frobenius traces.

If this is right

  • For any fixed t the primes with a_{1,p}(A)=t form a zero-density set.
  • The typical size of |a_{1,p}(A)| is at least p raised to a positive power that depends only on the dimension g.
  • The same counting arguments apply verbatim to the higher traces a_{i,p}(A) once the corresponding Galois representations are known to be surjective.

Where Pith is reading between the lines

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

  • The exponent 1/(2g²+g+1) is an artefact of the current effective Chebotarev error terms; any improvement in those error terms would immediately raise the lower bound on |a_{1,p}(A)|.
  • The result supplies a uniform lower bound on the height of the image of the Frobenius conjugacy class inside the adelic Galois representation, which may be useful for studying the distribution of point counts on A mod p.

Load-bearing premise

That the mod-ℓ Galois representation attached to A is surjective for every sufficiently large prime ℓ.

What would settle it

An explicit abelian variety A of dimension g together with a positive-density set of primes p for which |a_{1,p}(A)| stays below p to the power 1/(2g²+g+1) times a slowly growing log factor, contradicting the GRH-derived bound.

read the original abstract

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote by $\pi_A(x, t)$ the number of primes $p \leq x$ for which the Frobenius trace $a_{1, p}(A)$ associated to $A \pmod p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that $\pi_A(x, 0) \ll_A x^{1 - \frac{1}{2g^2+g+1}}/(\log x)^{1 - \frac{2}{2g^2+g+1}}$ and $\pi_A(x, t) \ll_A x^{1 - \frac{1}{2g^2+g+2}}/(\log x)^{1 - \frac{2}{2g^2+g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)| > p^{\frac{1}{2 g^2 + g + 1}}/ (\log p)^{\frac{2}{2g^2+g+1}+\varepsilon}$ for any $\varepsilon>0$. Assuming, in addition to GRH, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that $\pi_A(x, 0) \ll_A x^{1 - \frac{1}{g+1}}/(\log x)^{1 - \frac{4}{g+1}}$ and $\pi_A(x, t) \ll_A x^{1 - \frac{1}{g+2}}/(\log x)^{1 - \frac{4}{g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)|> p^{\frac{1}{g + 2} - \varepsilon }$ for any $\varepsilon>0$.

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.

Referee Report

2 major / 1 minor

Summary. The paper assumes surjectivity of the residual Galois representation modulo ℓ for large ℓ for an abelian variety A/Q of dimension g. Under GRH it proves π_A(x,0) ≪_A x^{1-1/(2g²+g+1)}/(log x)^{1-2/(2g²+g+1)} and π_A(x,t) ≪_A x^{1-1/(2g²+g+2)}/(log x)^{1-2/(2g²+g+2)} (t≠0), and deduces that almost all p satisfy |a_{1,p}(A)| > p^{1/(2g²+g+1)}/(log p)^{2/(2g²+g+1)+ε}. Under GRH plus Artin's holomorphy conjecture and a pair-correlation conjecture it obtains the stronger bounds π_A(x,0) ≪_A x^{1-1/(g+1)}/(log x)^{1-4/(g+1)} and π_A(x,t) ≪_A x^{1-1/(g+2)}/(log x)^{1-4/(g+2)}, yielding |a_{1,p}(A)| > p^{1/(g+2)-ε} for almost all p.

Significance. Conditional bounds of this type on the distribution of Frobenius traces a_{1,p}(A) away from small values would refine our understanding of the Lang-Trotter conjecture and the typical size of traces for generic abelian varieties. The derivations rely on standard applications of GRH (and stronger conjectures) to Artin L-functions attached to the Galois representations, which is a natural extension of existing techniques.

major comments (2)
  1. [Abstract] Abstract (GRH deduction paragraph): the claimed consequence that almost all primes satisfy |a_{1,p}(A)| > p^{1/D}/(log p)^{2/D + ε} with D = 2g² + g + 1 does not follow from the stated bounds. Summing the t ≠ 0 bound over |t| ≤ x^{1/D} produces an upper bound ≫ x^{1 + 1/(D(D+1))} (ignoring logs), which exceeds x and therefore supplies no information on the exceptional set. This is load-bearing for the 'almost all' statement under GRH alone.
  2. [Abstract] Abstract (Artin+pair-correlation deduction): the manuscript should explicitly confirm that inserting the -ε in the exponent 1/(g+2)-ε makes the corresponding sum over |t| ≤ x^{1/(g+2)-ε} o(x), as the text only notes that the stronger conjectures 'avoid the issue via an explicit -ε'.
minor comments (1)
  1. The surjectivity assumption on the residual representations is stated only in the first sentence of the abstract; a precise formulation (including the precise meaning of 'sufficiently large') should appear in the introduction or §1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these issues in the abstract. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (GRH deduction paragraph): the claimed consequence that almost all primes satisfy |a_{1,p}(A)| > p^{1/D}/(log p)^{2/D + ε} with D = 2g² + g + 1 does not follow from the stated bounds. Summing the t ≠ 0 bound over |t| ≤ x^{1/D} produces an upper bound ≫ x^{1 + 1/(D(D+1))} (ignoring logs), which exceeds x and therefore supplies no information on the exceptional set. This is load-bearing for the 'almost all' statement under GRH alone.

    Authors: We agree with the referee's calculation. Summing the stated bound for π_A(x,t) (t≠0) over O(x^{1/D}) values of t indeed produces a quantity ≫ x^{1 + 1/(D(D+1))}, which is larger than x and therefore does not control the size of the exceptional set. This is an error in the manuscript. We will revise the abstract to remove the 'almost all' claim under GRH alone. revision: yes

  2. Referee: [Abstract] Abstract (Artin+pair-correlation deduction): the manuscript should explicitly confirm that inserting the -ε in the exponent 1/(g+2)-ε makes the corresponding sum over |t| ≤ x^{1/(g+2)-ε} o(x), as the text only notes that the stronger conjectures 'avoid the issue via an explicit -ε'.

    Authors: We agree that an explicit verification is desirable. Let θ = 1/(g+2) − ε. The number of admissible t is O(x^θ). The bound π_A(x,t) ≪_A x^{1−1/(g+2)} (log x)^{−(1−4/(g+2))} then yields a total of O(x^θ ⋅ x^{1−1/(g+2)}) = O(x^{1−ε}), which is o(x). We will insert this short calculation into the revised text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from external conjectures via standard methods

full rationale

The paper assumes surjectivity of residual Galois representations for large ℓ and derives explicit upper bounds for π_A(x,0) and π_A(x,t) under GRH (and under the additional Artin holomorphy + pair correlation conjectures) by applying standard analytic estimates to the associated Dedekind zeta or Artin L-functions. The exponents 1/(2g²+g+1) etc. arise directly from the zero-free regions or density theorems supplied by those external hypotheses, not from any redefinition or fitting of the target quantities inside the paper. The subsequent claim that almost all primes satisfy a lower bound on |a_{1,p}(A)| is presented as a consequence of summing the π_A bounds; while the skeptic correctly notes that the summation argument as written does not close, this is a potential gap in the deduction step rather than a circular reduction of the claimed result to its own inputs. No self-definitional, fitted-input, or load-bearing self-citation patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the surjectivity assumption for residual Galois representations and on GRH for Dedekind zeta functions (plus Artin holomorphy and pair correlation for the stronger bounds). No free parameters or invented entities are introduced.

axioms (3)
  • domain assumption Generalized Riemann Hypothesis for Dedekind zeta functions
    Invoked to obtain the stated upper bounds on π_A(x,t)
  • domain assumption Surjective residual Galois representation modulo ℓ for all sufficiently large ℓ
    Stated as an assumption on A in the first sentence
  • domain assumption Artin's Holomorphy Conjecture and Pair Correlation Conjecture for Artin L-functions
    Additional assumptions for the stronger bounds with exponents 1-1/(g+1)

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