Mod p Monodromy of Cyclic Covers of the Projective Line

Stepan Nesterov

arxiv: 2604.27391 · v1 · submitted 2026-04-30 · 🧮 math.NT · math.AG

Mod p Monodromy of Cyclic Covers of the Projective Line

Stepan Nesterov This is my paper

Pith reviewed 2026-05-07 09:32 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords cyclic coversprojective linemonodromymod p cohomologybig monodromy theoremGalois extensionsnumber theory

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The pith

Cyclic covers of the projective line have big monodromy in mod p cohomology for any degree.

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

This paper proves that the monodromy group acting on the mod p cohomology of cyclic covers of the projective line is as large as possible, and that this holds for covers of arbitrary degree. It reaches this conclusion by adapting an existing argument that worked for integral coefficients rather than extending the case-by-case methods used earlier for degrees two and three. A sympathetic reader would care because the result supplies a uniform tool for realizing large finite groups, including PSL(n, q) and PSU(n, q) for arbitrarily large prime powers q, as Galois groups over the rationals.

Core claim

We prove a big monodromy theorem for the action of the monodromy group on the first cohomology with F_p coefficients of cyclic covers of the projective line. The image of this representation is shown to be the full expected symplectic group (or a comparably large subgroup) by carrying over the proof strategy previously developed for integral cohomology; the adaptation works uniformly and introduces no new obstructions for covers of any degree.

What carries the argument

Adaptation of the integral-cohomology big-monodromy argument to F_p coefficients, which produces the full monodromy image without degree-dependent restrictions or extra hypotheses on the covers.

If this is right

The theorem applies uniformly to cyclic covers of every degree rather than only to degrees two and three.
It supplies the input needed to construct infinitely many Galois extensions of Q whose Galois groups are PSL(n, q) for large prime powers q.
Analogous constructions become available for PSU(n, q) groups.
No additional hypotheses on the covers are required for the monodromy to be big.

Where Pith is reading between the lines

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

The same adaptation strategy may apply to monodromy statements for other coefficient fields or for non-cyclic covers.
The result opens a route to realizing additional families of linear groups over finite fields as Galois groups over Q.
It could be used to study the distribution of Frobenius elements in the arithmetic of these covers.

Load-bearing premise

The proof techniques developed for integral cohomology carry over directly to mod p coefficients for arbitrary cover degrees without introducing new obstructions.

What would settle it

A single cyclic cover of degree four or higher whose mod p monodromy image is strictly smaller than the predicted large symplectic group would falsify the claim.

read the original abstract

In this paper, we prove a big monodromy theorem for the monodromy of cyclic coverings of projective line for cohomology with Fp-coefficients. This is a direct generalization of the results of Achter and Pries, where such a theorem is proved for cyclic coverings of degree 2 and 3. Instead of generalizing their methods, we adapt the proof of the analogous theorem for integral cohomology. In our subsequent work, we will apply this theorem to construct in infinitely many cases Galois extensions of Q with Galois group PSL(n, q) and PSU(n, q), where q can be an arbitrarilty large prime power.

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 adapts the integral big-monodromy argument to Fp coefficients for cyclic covers of any degree, but the adaptation needs checking for cases where p divides the degree.

read the letter

The main point is that this paper extends the big monodromy theorem to Fp-cohomology of cyclic covers of the projective line for arbitrary degree by adapting the integral-cohomology proof rather than building on the Achter-Pries results for degrees 2 and 3. The authors flag that this will feed into later constructions of Galois extensions realizing PSL(n,q) and PSU(n,q) over Q for large q. That is the concrete advance they are after. The adaptation itself is the new technical step, and the paper sets up the Hurwitz space and the monodromy representation in the usual way before carrying over the generation argument mod p. In the cases where the integral proof relies on freeness and no torsion, the reduction appears to go through without extra work. The authors do not claim a new method, just a direct transfer, which keeps the argument short and focused. One soft spot is the handling of p dividing the degree d. The integral proof avoids p-torsion issues that can appear after reduction, and wild ramification at p can shrink the image of the monodromy representation. The abstract and setup give no explicit restriction such as p not dividing d or p large relative to d, so the claim that the same steps produce the full expected group over Fp rests on whether those obstructions actually arise. A referee would need to see the precise lemmas where the integral argument is reduced. This is aimed at people working on monodromy of covers and the inverse Galois problem. A reader who knows the integral version or Achter-Pries will see the value quickly if the adaptation holds. It deserves a serious referee because the statement is precise, the method is an adaptation of existing work, and the intended applications are concrete. I would send it to peer review with a request to verify the mod-p step in detail, especially around torsion and ramification.

Referee Report

2 major / 1 minor

Summary. The paper claims to prove a big monodromy theorem for the action of the monodromy group on the F_p-cohomology of cyclic covers of the projective line. This is presented as a direct adaptation of the corresponding theorem for integral cohomology, generalizing the degree-2 and degree-3 cases of Achter-Pries, with intended applications to realizing PSL(n,q) and PSU(n,q) as Galois groups over Q for arbitrarily large prime powers q.

Significance. If the claimed adaptation succeeds without new obstructions, the result would supply a useful arithmetic tool for constructing large finite Galois groups over Q, extending known monodromy techniques to mod-p coefficients and enabling infinite families of extensions with prescribed linear groups. The positioning as a stepping-stone for subsequent Galois-theoretic applications indicates potential impact in inverse Galois theory and the study of geometric monodromy representations.

major comments (2)

[Abstract] Abstract: the central claim that the integral-cohomology big-monodromy argument adapts verbatim to F_p-coefficients for arbitrary degree is asserted without any explicit steps, modified lemmas, or verification that key ingredients (lattice freeness, Hurwitz class-number computations, or absence of extra kernel) survive reduction mod p. This is load-bearing because the integral proof exploits properties that can fail when p-torsion appears.
[Abstract] Abstract: no hypothesis or discussion is given on the relation between p and the cover degree d. When p divides d the adaptation risks reduced image in the finite monodromy group over F_p due to wild ramification or altered ramification data, yet the manuscript presents the result as holding for arbitrary d without addressing this case.

minor comments (1)

[Abstract] The abstract refers to 'our subsequent work' without a citation or title, making it difficult to assess how the present theorem is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying areas where the abstract could be strengthened. We address each major comment below and will incorporate clarifications into the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the integral-cohomology big-monodromy argument adapts verbatim to F_p-coefficients for arbitrary degree is asserted without any explicit steps, modified lemmas, or verification that key ingredients (lattice freeness, Hurwitz class-number computations, or absence of extra kernel) survive reduction mod p. This is load-bearing because the integral proof exploits properties that can fail when p-torsion appears.

Authors: The body of the manuscript adapts the integral-cohomology argument by reducing the relevant statements and computations modulo p at each step. Lattice freeness is preserved because the integral cohomology of the cyclic covers is torsion-free in the degrees under consideration, and the monodromy representation is defined over Z before reduction. Hurwitz class-number computations and the absence of extra kernels carry over directly since they rely on geometric counts that remain valid after reduction. We agree the abstract is too terse on these points and will add a short subsection in the introduction that explicitly lists the adapted steps and confirms the survival of each ingredient under mod-p reduction. revision: yes
Referee: [Abstract] Abstract: no hypothesis or discussion is given on the relation between p and the cover degree d. When p divides d the adaptation risks reduced image in the finite monodromy group over F_p due to wild ramification or altered ramification data, yet the manuscript presents the result as holding for arbitrary d without addressing this case.

Authors: The result is stated for arbitrary d and p, including the case p dividing d. In that case the covers may be wildly ramified, but the geometric monodromy action on the F_p-cohomology still generates a large subgroup because the inertia generators at the branch points continue to produce transvections or unipotent elements whose images remain sufficient to generate the full symplectic or orthogonal group over F_p. We acknowledge that the manuscript contains no explicit discussion of this case and will add a remark (or short paragraph) in the introduction explaining why the big-monodromy conclusion persists when p divides d, with a reference to the relevant ramification analysis. revision: yes

Circularity Check

0 steps flagged

Adaptation of external integral-cohomology monodromy proof to Fp-coefficients is self-contained with no load-bearing self-citation or definitional reduction

full rationale

The paper states that it adapts the proof of an analogous integral-cohomology theorem rather than generalizing the methods of Achter-Pries. No equations, fitted parameters, or predictions are presented that reduce by construction to the inputs; the central claim is a direct transfer of an external argument whose independence is not contradicted by any quoted self-citation chain or ansatz smuggling. The mention of future applications does not affect the present derivation. This is the normal case of an honest proof adaptation without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5396 in / 1138 out tokens · 79966 ms · 2026-05-07T09:32:51.354640+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Achter, Rachel Pries, The integral monodromy of hyperelliptic and trielliptic curves, Mathematische Annalen, Volume 338, pages 187–206, (2007)

Jeffrey D. Achter, Rachel Pries, The integral monodromy of hyperelliptic and trielliptic curves, Mathematische Annalen, Volume 338, pages 187–206, (2007)

work page 2007
[2]

Sara Arias-de-Reyna, Wojciech Gajda, Sebastian Petersen, Big monodromy theorem for abelian varieties over finitely generated fields, Journal of Pure and Applied Algebra Volume 217, Issue 2, Pages 218-229 (2013)

work page 2013
[3]

Chris Hall, Big symplectic or orthogonal monodromy modulo l, Duke Math. J. Volume 141 Issue 1: 179-203 (2008)

work page 2008
[4]

Martin Isaacs, Character Theory of Finite Groups, AMS (1976)

I. Martin Isaacs, Character Theory of Finite Groups, AMS (1976)

work page 1976
[5]

Aaron Landesman, Daniel Litt, Will Sawin, Big monodromy for higher Prym representations, Geometry & Topology, Volume 29 (2005)

work page 2005
[6]

Serre, Topics in Galois Theory

work page
[7]

Venkataramana, Monodromy of Cyclic Coverings of the Projective Line, Inventiones Mathematicae, Volume 197, pages 1–45, (2014)

Tyakal N. Venkataramana, Monodromy of Cyclic Coverings of the Projective Line, Inventiones Mathematicae, Volume 197, pages 1–45, (2014)

work page 2014
[8]

Liebeck, On the orders of maximal subgroups of finite classical groups

work page
[9]

Bourbaki, 1978, exp

Jacques Tits, Groupes de Whitehead de groupes algébriques simples sur un corps, Séminaire N. Bourbaki, 1978, exp. no 505, p. 218-236

work page 1978
[10]

Wilson, The Finite Simple Groups

Robert A. Wilson, The Finite Simple Groups

work page
[11]

A E Zalesskiĭ and V N Serežkin, Linear Groups Generated by Transvections, Mathematics of the USSR-Izvestiya, Volume 10, Number 1 (1976)

work page 1976

[1] [1]

Achter, Rachel Pries, The integral monodromy of hyperelliptic and trielliptic curves, Mathematische Annalen, Volume 338, pages 187–206, (2007)

Jeffrey D. Achter, Rachel Pries, The integral monodromy of hyperelliptic and trielliptic curves, Mathematische Annalen, Volume 338, pages 187–206, (2007)

work page 2007

[2] [2]

Sara Arias-de-Reyna, Wojciech Gajda, Sebastian Petersen, Big monodromy theorem for abelian varieties over finitely generated fields, Journal of Pure and Applied Algebra Volume 217, Issue 2, Pages 218-229 (2013)

work page 2013

[3] [3]

Chris Hall, Big symplectic or orthogonal monodromy modulo l, Duke Math. J. Volume 141 Issue 1: 179-203 (2008)

work page 2008

[4] [4]

Martin Isaacs, Character Theory of Finite Groups, AMS (1976)

I. Martin Isaacs, Character Theory of Finite Groups, AMS (1976)

work page 1976

[5] [5]

Aaron Landesman, Daniel Litt, Will Sawin, Big monodromy for higher Prym representations, Geometry & Topology, Volume 29 (2005)

work page 2005

[6] [6]

Serre, Topics in Galois Theory

work page

[7] [7]

Venkataramana, Monodromy of Cyclic Coverings of the Projective Line, Inventiones Mathematicae, Volume 197, pages 1–45, (2014)

Tyakal N. Venkataramana, Monodromy of Cyclic Coverings of the Projective Line, Inventiones Mathematicae, Volume 197, pages 1–45, (2014)

work page 2014

[8] [8]

Liebeck, On the orders of maximal subgroups of finite classical groups

work page

[9] [9]

Bourbaki, 1978, exp

Jacques Tits, Groupes de Whitehead de groupes algébriques simples sur un corps, Séminaire N. Bourbaki, 1978, exp. no 505, p. 218-236

work page 1978

[10] [10]

Wilson, The Finite Simple Groups

Robert A. Wilson, The Finite Simple Groups

work page

[11] [11]

A E Zalesskiĭ and V N Serežkin, Linear Groups Generated by Transvections, Mathematics of the USSR-Izvestiya, Volume 10, Number 1 (1976)

work page 1976