Notes on congruence zeta functions via a Berkovich approach

Yuto Yamada

arxiv: 2605.26488 · v1 · pith:RLEV7ULGnew · submitted 2026-05-26 · 🧮 math.NT

Notes on congruence zeta functions via a Berkovich approach

Yuto Yamada This is my paper

Pith reviewed 2026-06-29 16:19 UTC · model grok-4.3

classification 🧮 math.NT

keywords congruence zeta functionsBerkovich motivescategorical tracessmooth projective varietiesfinite fieldsarithmetic geometryzeta functions

0 comments

The pith

A zeta function built from Berkovich motives and categorical traces coincides with the classical congruence zeta function for smooth projective varieties over finite fields.

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

The paper re-examines congruence zeta functions of smooth projective varieties over finite fields inside Scholze's Berkovich motives. It uses this formalism together with categorical traces to define a new zeta function. The central result is that this new function agrees exactly with the classical congruence zeta function. A reader would care because the agreement supplies an alternative definition that lives inside a broader motivic framework.

Core claim

In the framework of Scholze's Berkovich motives, a new zeta function is constructed via categorical traces, and this function is shown to agree with the classical congruence zeta function when evaluated on smooth projective varieties over finite fields.

What carries the argument

Scholze's Berkovich motives framework combined with categorical traces, used to define the new zeta function and establish its agreement with the classical version.

If this is right

The new construction supplies an equivalent definition of the congruence zeta function inside the Berkovich motives setting.
Categorical traces can be used to recover the classical zeta values on the indicated class of varieties.
The agreement holds for all smooth projective varieties over finite fields.
The construction remains internal to the Berkovich motives formalism.

Where Pith is reading between the lines

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

The same trace-based construction might apply to other classes of varieties where a classical zeta function is not yet defined.
The approach could link Berkovich motives to existing trace formulas in arithmetic geometry.
It may allow zeta functions to be studied uniformly across different motivic categories.

Load-bearing premise

The combination of Scholze's Berkovich motives with categorical traces produces a well-defined zeta function whose values on smooth projective varieties over finite fields match the classical congruence zeta function.

What would settle it

Explicit computation of the new zeta function on a concrete example such as projective space over a finite field, followed by direct comparison to the known classical zeta polynomial for that variety.

read the original abstract

We revisit congruence zeta functions of smooth projective varieties over finite fields in the framework of Scholze's Berkovich motives. Via this formalism and categorical traces, we construct a new zeta function, and show that it agree with classical one

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 note shows a zeta function built from Berkovich motives and categorical traces matches the classical congruence zeta on smooth projective varieties over finite fields, but the work is mainly a verification step.

read the letter

The main takeaway is that Yamada defines a zeta function in Scholze's Berkovich motives using categorical traces and proves it agrees with the usual congruence zeta function for smooth projective varieties over finite fields.

The paper does a reasonable job of carrying the formalism through to this standard object. It makes the connection explicit and confirms consistency between the new setup and classical arithmetic geometry. That kind of check is useful when a framework is still being tested on known cases.

The soft spot is that the result is an equivalence rather than a new computation or prediction. The abstract gives no intermediate steps or explicit definitions, so it is difficult to tell how much of the agreement is forced by the way the zeta is constructed versus how much follows from independent properties. If the full text supplies clear, non-circular definitions and the trace calculations, then the claim holds up; otherwise it stays at the level of a sanity check. No load-bearing gaps appear in the stated claim, but the brevity limits what can be assessed.

This kind of note is mainly for people already working with Berkovich motives or categorical methods in arithmetic geometry. A reader outside that circle who wants new theorems about zeta functions or applications to open problems will not find them here.

I would send it to peer review. The claim is concrete, the framework is current, and referees can verify the details and decide whether the equivalence adds enough to the literature to be worth archiving.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to revisit the congruence zeta functions of smooth projective varieties over finite fields within Scholze's Berkovich motives framework. Using this formalism together with categorical traces, it constructs a new zeta function and asserts that the new function agrees with the classical congruence zeta function.

Significance. If the claimed agreement holds and is non-circular, the work would supply a categorical construction of the zeta function in the Berkovich-motives setting, potentially offering a new perspective on Weil conjectures or trace formulas. No machine-checked proofs, parameter-free derivations, or explicit computations are mentioned, so the significance remains difficult to gauge from the available text.

major comments (1)

Abstract: the central claim that the newly constructed zeta function 'agree[s] with classical one' is stated without any definition of the new zeta function, without the relevant categorical trace, and without a proof or even an outline of why the two functions coincide on smooth projective varieties over finite fields. This renders the main assertion unverifiable.

minor comments (1)

Abstract: grammatical error ('it agree' should read 'it agrees').

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. Below we address the single major comment point by point.

read point-by-point responses

Referee: Abstract: the central claim that the newly constructed zeta function 'agree[s] with classical one' is stated without any definition of the new zeta function, without the relevant categorical trace, and without a proof or even an outline of why the two functions coincide on smooth projective varieties over finite fields. This renders the main assertion unverifiable.

Authors: We agree that the abstract is extremely concise and omits any definition of the new zeta function, the categorical trace, or an outline of the agreement proof. While the body of the manuscript supplies these elements (definition via Berkovich motives and categorical traces, followed by the comparison theorem), the abstract itself does not. To address the concern we will revise the abstract to include a short indication of the construction and the statement that agreement is shown for smooth projective varieties over finite fields. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract states that a new zeta function is constructed via Scholze's Berkovich motives and categorical traces, then shown to agree with the classical congruence zeta on smooth projective varieties over finite fields. No equations, self-citations, fitted parameters, or ansatzes are provided that would reduce the agreement to a definitional identity or input fit. The claim is an equivalence result grounded in an external framework (Scholze's motives), with the classical zeta serving as an independent benchmark rather than a constructed output. No load-bearing step reduces by construction to the paper's own inputs. This is the expected honest non-finding for an abstract-level claim without visible internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The construction presumably relies on the existence and properties of Berkovich motives as developed by Scholze, but these are treated as background.

pith-pipeline@v0.9.1-grok · 5539 in / 1053 out tokens · 36725 ms · 2026-06-29T16:19:03.539919+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 5 canonical work pages · 1 internal anchor

[1]

Berkovich, Spectral theory and analytic geometry over non-Archimedean fields , Math

Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields , Math. Surv. Monogr. 33, AMS, 1990

1990
[2]

Dustin Clausen, Duality and linearization for p -adic lie groups , preprint (arXiv:2506.18174 https://arxiv.org/abs/2506.18174), 2025

work page arXiv 2025
[3]

Dustin Clausen and Peter Scholze, Condensed Mathematics and Complex Geometry , preprint (arXiv:2605.11731 https://arxiv.org/abs/2605.11731), 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[4]

Gelfand, Normierte Ringe , Sb

Israil M. Gelfand, Normierte Ringe , Sb. Math., 51 (1941) 1, 3-24

1941
[5]

Alexsander Grothendieck et al., S\' e minaire de G\' e om\' e trie Alg\' e brique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) , Lecture Notes in Mathematics, Vol. 589. Berlin; New York: Springer-Verlag, 1977

1965
[6]

Claudius Heyer and Lucas Mann, 6-Functor Formalisms and Smooth Representations , preprint (arXiv:2410.13038 https://arxiv.org/abs/2410.13038), 2024

work page arXiv 2024
[7]

Marc Hoyois, Sarah Scherotzke, and Nicolo Sibilla, Higher traces, noncommutative motives, and the categorified chern character , Adv. Math. 309 (2017), 97-154

2017
[8]

Bruno Kahn, Zeta and L functions of Voevodsky motives , preprint (arXiv:2412.08437 https://arxiv.org/abs/2412.08437), 2024

work page arXiv 2024
[9]

Peter Scholze, Berkovich motives , preprint (arXiv:2412.03382 https://arxiv.org/abs/2412.03382), 2024, to appear in J. Amer. Math. Soc

work page arXiv 2024
[10]

Andr\' e Weil, Numbers of solutions of equations in finite fields , Bull. Amer. Math. Soc. 55 (1949), 497–508

1949

[1] [1]

Berkovich, Spectral theory and analytic geometry over non-Archimedean fields , Math

Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields , Math. Surv. Monogr. 33, AMS, 1990

1990

[2] [2]

Dustin Clausen, Duality and linearization for p -adic lie groups , preprint (arXiv:2506.18174 https://arxiv.org/abs/2506.18174), 2025

work page arXiv 2025

[3] [3]

Dustin Clausen and Peter Scholze, Condensed Mathematics and Complex Geometry , preprint (arXiv:2605.11731 https://arxiv.org/abs/2605.11731), 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[4] [4]

Gelfand, Normierte Ringe , Sb

Israil M. Gelfand, Normierte Ringe , Sb. Math., 51 (1941) 1, 3-24

1941

[5] [5]

Alexsander Grothendieck et al., S\' e minaire de G\' e om\' e trie Alg\' e brique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) , Lecture Notes in Mathematics, Vol. 589. Berlin; New York: Springer-Verlag, 1977

1965

[6] [6]

Claudius Heyer and Lucas Mann, 6-Functor Formalisms and Smooth Representations , preprint (arXiv:2410.13038 https://arxiv.org/abs/2410.13038), 2024

work page arXiv 2024

[7] [7]

Marc Hoyois, Sarah Scherotzke, and Nicolo Sibilla, Higher traces, noncommutative motives, and the categorified chern character , Adv. Math. 309 (2017), 97-154

2017

[8] [8]

Bruno Kahn, Zeta and L functions of Voevodsky motives , preprint (arXiv:2412.08437 https://arxiv.org/abs/2412.08437), 2024

work page arXiv 2024

[9] [9]

Peter Scholze, Berkovich motives , preprint (arXiv:2412.03382 https://arxiv.org/abs/2412.03382), 2024, to appear in J. Amer. Math. Soc

work page arXiv 2024

[10] [10]

Andr\' e Weil, Numbers of solutions of equations in finite fields , Bull. Amer. Math. Soc. 55 (1949), 497–508

1949