Hasse-Weil Zeta Functions Modulo a Prime

Chris Hall

arxiv: 2606.05340 · v1 · pith:PYRHOJKPnew · submitted 2026-06-03 · 🧮 math.NT

Hasse-Weil Zeta Functions Modulo a Prime

Chris Hall This is my paper

Pith reviewed 2026-06-28 04:04 UTC · model grok-4.3

classification 🧮 math.NT

keywords Hasse-Weil zeta functionmod r reductionGalois coverbranch locusfinite schemescurves over finite fieldshyperelliptic curves

0 comments

The pith

For a generically Galois cover of prime order r not equal to the characteristic, the mod-r reduction of the zeta function of Y is expressed using the zeta function of X and the branch locus Z.

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

The paper shows how to write the zeta function of the cover Y modulo r directly from the zeta function of the base X together with data about the points where the cover ramifies. The setup requires a finite morphism of schemes over a finite field that is Galois of prime degree r away from the characteristic. This relation simplifies arithmetic calculations because zeta functions count points over extensions of the base field. The result is applied to give concrete formulas for curves and for the numerators of zeta functions attached to hyperelliptic and superelliptic curves.

Core claim

We determine the mod-r reduction of the zeta function of Y in terms of the zeta function of X and the branch locus Z subset X of pi, where pi is a finite F_q-morphism of separated F_q-schemes of finite type that is generically Galois with group G of prime order r not equal to p.

What carries the argument

The mod-r reduction of the Hasse-Weil zeta function, obtained by combining the zeta function of the base with the contribution of the branch locus under the prime-order Galois action.

If this is right

The zeta function of Y modulo r is obtained from the zeta function of X and the geometry of Z without enumerating points on Y directly.
For curves over finite fields the relation supplies explicit comparisons between the mod-r zeta functions of the cover and the base.
The numerators of the zeta functions of hyperelliptic and superelliptic curves are determined modulo r by the same data.

Where Pith is reading between the lines

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

The formula could be checked by direct point counting on low-genus examples such as Artin-Schreier covers of the projective line.
Similar reductions might exist when the Galois group is not prime order, provided the characteristic does not divide the group order.

Load-bearing premise

The morphism pi is generically Galois with group of prime order r different from the characteristic p, and X and Y are separated schemes of finite type over the finite field F_q.

What would settle it

An explicit example of schemes X and Y over a finite field, together with a morphism pi satisfying the generically Galois prime-order condition, for which the mod-r zeta function of Y fails to equal the expression built from the zeta function of X and the branch locus Z.

read the original abstract

Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $\pi\colon Y\to X$ be a finite $\mathbb{F}_q$-morphism of separated $\mathbb{F}_q$-schemes of finite type. Suppose $\pi$ is generically Galois with group $G$ of prime order $r\neq p$. We determine the mod-$r$ reduction of the zeta function of $Y$ in terms of the zeta function of $X$ and the branch locus $Z\subset X$ of $\pi$. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.

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 claims an explicit mod-r formula for the zeta of a prime-order Galois cover from the base and branch locus, but the non-semisimple group algebra makes it unclear if that's sufficient.

read the letter

The main takeaway is that Hall claims to have an explicit way to write the mod-r reduction of the zeta function of Y using only the zeta function of X and the branch locus Z for a generically Galois cover with prime order group r not equal to the characteristic. This would be a useful relation if it holds up.

What is new is this specific determination, presented as not previously available in the literature for this setup. The paper does well in giving direct applications to curves and formulas for the numerators of hyperelliptic and superelliptic curves, which are the kind of explicit results that people in the field can test on examples.

The soft spot is whether the result actually follows from the given data. Because the group algebra over the field with r elements is non-semisimple, the usual factorization of the zeta function into L-functions does not work in the standard way. The nilpotent part of the algebra might affect the counts in ways that require more than the support of the branch locus. The abstract does not indicate how the proof gets around this, so the central claim rests on whether an alternative argument is supplied in the body. If it is, the result is fine; if not, that would be a problem.

The assumptions look standard for the area: separated schemes of finite type over a finite field, finite morphism that is generically Galois. No obvious circularity or extra parameters.

This is a paper for people working on zeta functions and point counts for varieties over finite fields, especially those using covers to compute things. A reader who wants mod-r information for such computations would get something out of the applications section.

It deserves a serious referee because the claim is precise and the topic is appropriate for the journal. I would send it out for review.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that if π: Y → X is a finite generically Galois F_q-morphism of separated finite-type schemes with Galois group G of prime order r ≠ p, then the reduction of the Hasse-Weil zeta function ζ_Y modulo r can be expressed explicitly in terms of ζ_X and the branch locus Z ⊂ X. Applications to the zeta functions of curves and to the numerators of hyperelliptic and superelliptic curves are indicated.

Significance. If the central claim holds, the result would supply a concrete relation between zeta functions of Galois covers modulo r that depends only on the base zeta function and the support of the branch locus. This could simplify certain point-counting problems and the study of L-functions modulo primes for covers of curves over finite fields.

major comments (1)

The central claim requires that the mod-r point counts on Y are controlled by those on X together with the geometry of Z, even though F_r[G] ≅ F_r[t]/(t-1)^r is local and non-semisimple. No indication is given in the abstract (or in the provided description of the argument) of how the nilpotent contributions arising from the action of σ-1 are determined solely by the branch locus; this is load-bearing for the determination asserted in the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need to clarify the treatment of nilpotent contributions in the group ring. We address the single major comment below.

read point-by-point responses

Referee: The central claim requires that the mod-r point counts on Y are controlled by those on X together with the geometry of Z, even though F_r[G] ≅ F_r[t]/(t-1)^r is local and non-semisimple. No indication is given in the abstract (or in the provided description of the argument) of how the nilpotent contributions arising from the action of σ-1 are determined solely by the branch locus; this is load-bearing for the determination asserted in the main theorem.

Authors: The argument in Sections 2–3 proceeds by expressing the Lefschetz trace formula for the twisted Frobenius action on Y in terms of the base X and the ramification data along Z. For the non-identity elements of G the fixed loci coincide with the support of the branch divisor; the nilpotent part of the operator (σ−1) then contributes only through the lengths of the ramification chains at those points, which are encoded in the geometry of Z. This reduces the mod-r zeta function to an explicit combination of ζ_X and a correction term supported on Z. We will insert a short paragraph in the introduction summarizing this reduction step to address the concern that the abstract provides insufficient indication. revision: partial

Circularity Check

0 steps flagged

No circularity; theorem stated without reduction to inputs or self-citations

full rationale

The paper states a determination of the mod-r reduction of ζ_Y in terms of ζ_X and branch locus Z for a generically Galois cover of prime order r ≠ p. No equations, derivations, or citations appear in the provided abstract or description that reduce the claimed result to its inputs by construction, fit parameters then rename them as predictions, or rely on load-bearing self-citations. The result is presented as a theorem derived from the given assumptions on schemes and morphisms. Absence of any quoted self-referential steps or ansatzes means the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5621 in / 1077 out tokens · 30609 ms · 2026-06-28T04:04:11.622339+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

[1]

Arithmetic geometry, number theory, and computation , SERIES =

Costa, Edgar and Donepudi, Ravi and Fernando, Ravi and Karemaker, Valentijn and Springer, Caleb and West, Mckenzie , TITLE =. Arithmetic geometry, number theory, and computation , SERIES =. 2021 , ISBN =. doi:10.1007/978-3-030-80914-0\_7 , URL =

work page doi:10.1007/978-3-030-80914-0 2021
[2]

American Journal of Mathematics , volume =

Dwork, Bernard , title =. American Journal of Mathematics , volume =. 1960 , publisher =

1960
[3]

2011 , publisher=

Etale Cohomology Theory , author=. 2011 , publisher=

2011
[4]

Gabber, Ofer , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1983 , NUMBER =

1983
[5]

Grothendieck, Alexander , title =. S. 1966 , publisher =

1966
[6]

Journal of Number Theory , volume =

Hall, Chris , title =. Journal of Number Theory , volume =. 2006 , publisher =

2006
[7]

1977 , publisher=

Algebraic Geometry , author=. 1977 , publisher=

1977
[8]

2002 , note =

Liu, Qing , title =. 2002 , note =

2002
[9]

, TITLE =

Milne, James S. , TITLE =. 1980 , PAGES =

1980
[10]

1977 , isbn =

Deligne, Pierre , title =. 1977 , isbn =

1977
[11]

Cohomologie -adique et Fonctions L , editor =

Grothendieck, Alexander , series =. Cohomologie -adique et Fonctions L , editor =. 1977 , doi =

1977
[12]

The Stacks project , howpublished =

The. The Stacks project , howpublished =

[1] [1]

Arithmetic geometry, number theory, and computation , SERIES =

Costa, Edgar and Donepudi, Ravi and Fernando, Ravi and Karemaker, Valentijn and Springer, Caleb and West, Mckenzie , TITLE =. Arithmetic geometry, number theory, and computation , SERIES =. 2021 , ISBN =. doi:10.1007/978-3-030-80914-0\_7 , URL =

work page doi:10.1007/978-3-030-80914-0 2021

[2] [2]

American Journal of Mathematics , volume =

Dwork, Bernard , title =. American Journal of Mathematics , volume =. 1960 , publisher =

1960

[3] [3]

2011 , publisher=

Etale Cohomology Theory , author=. 2011 , publisher=

2011

[4] [4]

Gabber, Ofer , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1983 , NUMBER =

1983

[5] [5]

Grothendieck, Alexander , title =. S. 1966 , publisher =

1966

[6] [6]

Journal of Number Theory , volume =

Hall, Chris , title =. Journal of Number Theory , volume =. 2006 , publisher =

2006

[7] [7]

1977 , publisher=

Algebraic Geometry , author=. 1977 , publisher=

1977

[8] [8]

2002 , note =

Liu, Qing , title =. 2002 , note =

2002

[9] [9]

, TITLE =

Milne, James S. , TITLE =. 1980 , PAGES =

1980

[10] [10]

1977 , isbn =

Deligne, Pierre , title =. 1977 , isbn =

1977

[11] [11]

Cohomologie -adique et Fonctions L , editor =

Grothendieck, Alexander , series =. Cohomologie -adique et Fonctions L , editor =. 1977 , doi =

1977

[12] [12]

The Stacks project , howpublished =

The. The Stacks project , howpublished =