Point counts of abelian varieties over finite fields determining their zeta function

Shiva Chidambaram; Timo Keller

arxiv: 2606.28989 · v1 · pith:3CSBWAPAnew · submitted 2026-06-27 · 🧮 math.NT · math.AG

Point counts of abelian varieties over finite fields determining their zeta function

Shiva Chidambaram , Timo Keller This is my paper

Pith reviewed 2026-06-30 08:22 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords abelian varietiesfinite fieldszeta functionspoint countsFrobenius endomorphismL-polynomialNewton identitiesisogeny classes

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The pith

If q is large enough relative to g, the first g point counts of an abelian variety over F_q determine its zeta function.

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

The paper establishes that for an abelian variety of dimension g over a finite field F_q, when q is sufficiently large compared to g, the point counts at the first g extensions of the field determine the zeta function completely. This works by combining the functional equation with Newton's identities to recover the eigenvalues of Frobenius as integers through controlled rounding. A reader would care because it reduces the number of necessary point counts to identify the isogeny class to just g, rather than up to 2g. For special cases like g=3, even fewer suffice, showing the bound is not always tight.

Core claim

We show that if q is sufficiently large relative to g, the g point counts #A(F_{q^i}) for 1 ≤ i ≤ g determine the zeta function of A, equivalently the characteristic polynomial of its Frobenius endomorphism, and hence the isogeny class of A. The proof uses the functional equation of the L-polynomial, Newton's identities, and an inductive error analysis to control the power sums of the inverse Frobenius eigenvalues with enough precision to recover them as integers by rounding.

What carries the argument

The inductive error analysis that controls the power sums of the inverse Frobenius eigenvalues with enough precision to recover them as integers by rounding, using the functional equation and Newton's identities.

If this is right

The isogeny class of A is determined by these g point counts when q is sufficiently large relative to g.
This number of counts is best possible for g=2 and g=4.
For g=3, two point counts already determine the zeta function, but a single count never does.
The zeta function is equivalently determined by the characteristic polynomial of Frobenius.

Where Pith is reading between the lines

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

Similar methods could be used to determine the zeta function with even fewer counts in some cases beyond what is shown.
Efficient computation of isogeny classes could benefit from this reduction in required point counts for large q.
The technique of inductive error analysis for power sums might apply to other problems involving recovery of integer coefficients from approximate data.

Load-bearing premise

The assumption that q is sufficiently large relative to g, which is required for the inductive error analysis to control the power sums of the inverse Frobenius eigenvalues with enough precision to recover them as integers by rounding.

What would settle it

An explicit pair of non-isogenous abelian varieties of dimension g over some F_q with q large relative to g that have the same first g point counts but different zeta functions.

read the original abstract

Let $A$ be an abelian variety of dimension $g$ over a finite field $\mathbf{F}_q$. We show that if $q$ is sufficiently large relative to $g$, the $g$ point counts $\#A(\mathbf{F}_{q^i})$ for $1 \leq i \leq g$ determine the zeta function of $A$, equivalently the characteristic polynomial of its Frobenius endomorphism, and hence the isogeny class of $A$. This count is best possible for $g=2$ and $g=4$, but not in general: for $g=3$ two point counts already determine the zeta function, whereas a single count never does. The proof combines the functional equation of the $L$-polynomial with Newton's identities and an inductive error analysis that controls the power sums of the inverse Frobenius eigenvalues with enough precision to recover them, as integers, by rounding.

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

g point counts determine the zeta function for large q relative to g, with the bound sharp for g=2 and 4 but improvable to 2 for g=3.

read the letter

The main result is that for an abelian variety of dimension g over F_q, if q is large enough relative to g, then the point counts over the first g extensions determine the zeta function, hence the characteristic polynomial of Frobenius and the isogeny class. They also prove this count is best possible for g=2 and g=4, while for g=3 two counts already work and one never does.

The paper applies the functional equation of the L-polynomial and Newton's identities to relate the point counts to power sums of the inverse Frobenius eigenvalues. It then runs an inductive rounding argument that keeps the accumulated error small enough to recover the exact integers when q is sufficiently large. The optimality examples are a useful check that shows the bound cannot be lowered in general.

This is a clean, concrete determination theorem that reduces the data needed to identify isogeny classes. The approach uses only standard external facts and avoids circularity.

The soft spots are the non-effective nature of the result and the details of the error control. The paper gives no explicit lower bound on q, only that it must be large enough, so the statement remains existential. The inductive step bounds errors using powers of sqrt(q) and previous coefficients, but without seeing the precise constants it is not immediate that the error stays below 1/2 at every step for all possible eigenvalue configurations. The stress-test concern about uniform closure of the induction is reasonable to check, though the abstract indicates the authors believe the growth is controlled.

This is for arithmetic geometers working on abelian varieties over finite fields or on computational classification of isogeny classes. It deserves serious peer review because the claim is precise, the method is transparent, and the optimality statements strengthen the contribution, even if explicit bounds would make it stronger.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that for an abelian variety A of dimension g over F_q, if q is sufficiently large relative to g, then the g point counts #A(F_{q^i}) for 1 ≤ i ≤ g uniquely determine the zeta function of A (equivalently, the characteristic polynomial of Frobenius) and hence the isogeny class. The argument combines the functional equation of the L-polynomial, Newton's identities relating the point counts to power sums p_k of the inverse eigenvalues, and an inductive rounding argument that recovers each p_k as the nearest integer provided the error is controlled below 1/2 at every step.

Significance. If the central claim holds, the result gives a sharp (or nearly sharp) bound on the number of point counts needed to determine the isogeny class when q ≫ g, with explicit optimality statements for g=2 and g=4 and an improvement for g=3. The proof relies only on standard external facts (functional equation, Newton identities) together with a self-contained inductive error analysis; no free parameters or ad-hoc fitted quantities are introduced.

major comments (1)

[Proof of Theorem 1.1] The inductive error analysis (described in the proof of the main theorem) must guarantee that the accumulated error remains strictly less than 1/2 at every step k=1 to g, uniformly over all possible configurations of the inverse eigenvalues on the circle of radius q^{-1/2}. The bounds grow with previous coefficients and with powers of sqrt(q); it is not immediate that a single 'sufficiently large' threshold for q closes the induction for all g and all eigenvalue placements (especially when some eigenvalues lie near the unit circle or when g is odd).

minor comments (2)

[Introduction] The introduction states that two point counts suffice for g=3; a short explicit statement or reference to the corresponding result would clarify the comparison with the general g-count bound.
Notation for the L-polynomial and the inverse eigenvalues α_i should be fixed consistently between the abstract and the body (currently the abstract uses 'inverse Frobenius eigenvalues' while the proof sketch uses p_k).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting a point that requires clarification in the proof of Theorem 1.1. We address the major comment below and will revise the manuscript to strengthen the presentation of the inductive error analysis.

read point-by-point responses

Referee: [Proof of Theorem 1.1] The inductive error analysis (described in the proof of the main theorem) must guarantee that the accumulated error remains strictly less than 1/2 at every step k=1 to g, uniformly over all possible configurations of the inverse eigenvalues on the circle of radius q^{-1/2}. The bounds grow with previous coefficients and with powers of sqrt(q); it is not immediate that a single 'sufficiently large' threshold for q closes the induction for all g and all eigenvalue placements (especially when some eigenvalues lie near the unit circle or when g is odd).

Authors: We agree that the current write-up of the induction, while correct in outline, does not make the uniform control of the accumulated error fully explicit. In the revised version we will expand the argument in the proof of Theorem 1.1 by deriving an explicit lower bound Q(g) on q (depending only on g) such that, whenever q > Q(g), the total propagated error at step k remains strictly less than 1/2 for every k = 1,…,g. The bound Q(g) is obtained by taking the maximum, over all admissible eigenvalue configurations, of the quantities that appear in the recursive error estimates; because each |p_j| is at most g q^{j/2} and the Newton identities involve only finitely many (g) steps, such a finite Q(g) exists. The same maximal-magnitude estimates automatically cover the case of eigenvalues near the unit circle. The induction proceeds identically for odd and even g; the middle coefficient (when g is odd) is recovered by the same rounding step once the preceding errors have been controlled. We will insert these explicit estimates and the resulting choice of Q(g) into the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard facts

full rationale

The paper proves its main result by combining the functional equation of the L-polynomial (a standard property of zeta functions of abelian varieties), Newton's identities (relating power sums of roots to coefficients), and an inductive error analysis to bound approximations of power sums sufficiently for integer rounding when q is large relative to g. These components are independent external mathematical tools; the induction controls errors using bounds on previous terms and powers of sqrt(q) without fitting parameters to the target data or redefining inputs in terms of outputs. No self-citations appear load-bearing, and the result is not obtained by renaming a known pattern or smuggling an ansatz. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background facts from algebraic geometry and number theory. No free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)

standard math The L-polynomial of an abelian variety satisfies a functional equation.
Invoked in the proof strategy described in the abstract.
standard math Newton's identities relate power sums of roots to the coefficients of the characteristic polynomial.
Used to recover the polynomial from power sums.

pith-pipeline@v0.9.1-grok · 5684 in / 1290 out tokens · 43143 ms · 2026-06-30T08:22:43.741103+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

[1]

Shiva Chidambaram and Timo Keller, Accompanying code for ``point counts of abelian varieties over finite fields determining their zeta function'', GitHub repository https://github.com/TimoKellerMath/PointCountsAbelianVarieties , 2026, Sage / Python scripts recover\_zeta.sage , verify\_g2\_family.sage , verify\_g3\_family.sage , verify\_g4\_family.sage , o...

2026
[2]

DiPippo and Everett W

Stephen A. DiPippo and Everett W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998), no. 2, 426--450. 1657992

1998
[3]

Kedlaya, Quantum computation of zeta functions of curves, Comput

Kiran S. Kedlaya, Quantum computation of zeta functions of curves, Comput. Complexity 15 (2006), no. 1, 1--19. 2226067

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[4]

The LMFDB Collaboration , The L -functions and modular forms database , https://www.lmfdb.org, 2026, [Online; accessed 25 June 2026]

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[5]

I. G. Macdonald, Symmetric functions and H all polynomials , second ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, With contribution by A. V. Zelevinsky and a foreword by Richard Stanley. 3443860

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[1] [1]

Shiva Chidambaram and Timo Keller, Accompanying code for ``point counts of abelian varieties over finite fields determining their zeta function'', GitHub repository https://github.com/TimoKellerMath/PointCountsAbelianVarieties , 2026, Sage / Python scripts recover\_zeta.sage , verify\_g2\_family.sage , verify\_g3\_family.sage , verify\_g4\_family.sage , o...

2026

[2] [2]

DiPippo and Everett W

Stephen A. DiPippo and Everett W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998), no. 2, 426--450. 1657992

1998

[3] [3]

Kedlaya, Quantum computation of zeta functions of curves, Comput

Kiran S. Kedlaya, Quantum computation of zeta functions of curves, Comput. Complexity 15 (2006), no. 1, 1--19. 2226067

2006

[4] [4]

The LMFDB Collaboration , The L -functions and modular forms database , https://www.lmfdb.org, 2026, [Online; accessed 25 June 2026]

2026

[5] [5]

I. G. Macdonald, Symmetric functions and H all polynomials , second ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, With contribution by A. V. Zelevinsky and a foreword by Richard Stanley. 3443860

2015