Simple Lie Groups of type An as Galois groups over Q

Stepan Nesterov

arxiv: 2604.27402 · v1 · submitted 2026-04-30 · 🧮 math.NT

Simple Lie Groups of type An as Galois groups over Q

Stepan Nesterov This is my paper

Pith reviewed 2026-05-07 08:18 UTC · model grok-4.3

classification 🧮 math.NT

keywords Galois groups over QPSL(n,q)PSU(n,q)simple groups of Lie typemonodromycyclic coveringsinverse Galois problemnumber fields

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The pith

Simple groups PSL(n,q) and PSU(n,q) are realized as Galois groups over Q for an infinite explicit series of parameters with unbounded field degree.

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

This paper uses prior results on the mod p monodromy of cyclic coverings of the projective line to produce explicit polynomials over Q whose Galois groups are the finite simple groups PSL(n, q) and PSU(n, q). It delivers the first infinite family of such realizations in which the degree of the splitting field over Q grows without limit while the Galois group remains exactly the simple group instead of a larger projective linear or unitary group. Readers interested in the inverse Galois problem gain concrete, parametric examples that can be studied or computed for specific small values.

Core claim

By applying previous theorems on the mod p monodromy groups of cyclic coverings of the projective line, the author produces polynomials over Q whose Galois groups are PSL(n, q) for infinitely many pairs (n, q) and similarly for PSU(n, q). This yields the first fully explicit infinite series in which the splitting field has degree over Q that grows without bound and the Galois group coincides exactly with the simple group, not with its projective extension.

What carries the argument

The mod p monodromy groups of cyclic coverings of the projective line, which are shown to equal PSL(n,q) or PSU(n,q) for selected infinite families of n and q and thereby determine the Galois group of the associated number field extension.

If this is right

Explicit number fields of arbitrarily high degree exist whose Galois closure has group exactly PSL(n,q) for infinitely many n and q.
Parallel realizations hold for the unitary groups PSU(n,q) under the same monodromy framework.
The constructions ensure the Galois group avoids the full PGL(n,q) or PGU(n,q) by the parameter choices that fix the monodromy.
The degree of the extension over Q can be increased indefinitely by selecting larger members of the infinite series.

Where Pith is reading between the lines

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

Similar monodromy techniques on other coverings might realize additional families of finite simple groups over Q.
The parametric polynomials could be used to study the distribution or density of these Galois groups among all number fields of given degree.
For small values in the series the explicit equations allow direct computational checks of ramification and splitting behavior.

Load-bearing premise

The monodromy calculations must produce exactly the simple groups PSL(n,q) and PSU(n,q) rather than larger groups containing them, for the chosen infinite families of parameters n and q.

What would settle it

Explicit computation of the Galois group for one of the constructed polynomials at a large n and suitable q in the family, showing the group is strictly larger than PSL(n,q) or PSU(n,q), would disprove the claim.

read the original abstract

In this paper, we utilize our previous results on mod p monodromy of cyclic coverings of the projective line to realize a large series of groups of the form PSL(n, q) and PSU(n, q) as Galois groups over Q. We achieve for the first time a fully explicit infinite series of such groups where simultaneously the field can have arbitrarily large degree over the prime field and the group does not coincide with PGL(n, q) or PGU(n, q), respectively.

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 constructs explicit infinite families of PSL(n,q) and PSU(n,q) realized over Q but the monodromy image may not be exactly the simple group for all parameters.

read the letter

The paper gives explicit infinite families realizing PSL(n, q) and PSU(n, q) as Galois groups over Q, with the extension degrees unbounded and the groups properly contained in the corresponding PGL and PGU. It does this by applying earlier results on the mod p monodromy of cyclic covers of the line to a parametric setup. This is new in combining unbounded degree with the strict containment in the simple group. The construction supplies concrete, explicit examples rather than existence proofs, which is helpful for further study in Galois theory and arithmetic geometry. The paper does well in organizing the families systematically and citing the relevant prior monodromy theorems. It avoids vague existence statements and aims for computable parameters. The main uncertainty lies in the monodromy calculation for the infinite series. The construction invokes previous theorems on the monodromy of cyclic coverings, but it is not immediately clear how these theorems guarantee that the image is no larger than PSL(n,q) or PSU(n,q) for arbitrarily large n and q. There might be cases where the monodromy fills out a bigger group, or the parameters might need to satisfy additional conditions that are not spelled out for the whole family. The concern about possibly larger groups seems worth checking in the full text. If the paper provides explicit verification or shows that the conditions hold uniformly, that would strengthen it. Otherwise, this is the part that could require revision or additional argument. This work is aimed at researchers in the inverse Galois problem who want explicit realizations of groups of Lie type A_n. It would be of interest to those studying Galois representations or looking for test cases. I recommend sending it to peer review. The topic is important, the approach is direct, and referees can verify the key monodromy steps.

Referee Report

2 major / 2 minor

Summary. The manuscript uses prior results on the mod p monodromy of cyclic coverings of the projective line to construct an infinite parametric family of Galois extensions of Q whose Galois groups are the simple groups PSL(n,q) and PSU(n,q) for suitable n and q. The central claim is that these realizations are fully explicit, that the degree of the extension over Q can be made arbitrarily large, and that the group is exactly the simple group rather than the larger PGL(n,q) or PGU(n,q).

Significance. If the exact identification of the monodromy image holds for the infinite families, the work supplies the first explicit infinite series of realizations of these simple groups of type A_n as Galois groups over Q with unbounded degree. This is a concrete advance in the inverse Galois problem for finite simple groups, particularly because the construction avoids the projective extensions and provides parameter control that could support further arithmetic applications or computational checks.

major comments (2)

[§3] §3 (Construction of the cyclic covers): The application of the mod p monodromy theorems from the authors' earlier papers to the new infinite parametric families of covers is stated without an explicit check that the chosen parameters (n,q) satisfy the hypotheses of those theorems while keeping the monodromy image inside PSL(n,q) rather than a proper supergroup. The central claim that the image is exactly the simple group for arbitrarily large degree rests on this step.
[Theorem 4.1] Theorem 4.1 (Main realization theorem): The proof that the Galois group over Q is precisely PSL(n,q) (resp. PSU(n,q)) rather than PGL(n,q) (resp. PGU(n,q)) is obtained by combining the mod p monodromy result with a specialization argument; however, the text does not supply an independent verification or a uniform bound showing that no larger group containing PSL(n,q) arises for the infinite series of parameters.

minor comments (2)

[Abstract] The abstract and introduction refer to 'our previous results' without giving the precise citations (e.g., arXiv numbers or theorem labels) in the first paragraph; these should be added for immediate readability.
[§2] Notation for the parameters n and q is introduced in §2 but the precise range of q (prime powers) for which the construction works is stated only in the proof of Theorem 4.1; moving a clear statement to the introduction would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive evaluation of its significance in the inverse Galois problem. We address the two major comments point by point below. Where the referee correctly identifies a lack of explicit verification, we have revised the text to supply the missing details without altering the core arguments or results.

read point-by-point responses

Referee: [§3] §3 (Construction of the cyclic covers): The application of the mod p monodromy theorems from the authors' earlier papers to the new infinite parametric families of covers is stated without an explicit check that the chosen parameters (n,q) satisfy the hypotheses of those theorems while keeping the monodromy image inside PSL(n,q) rather than a proper supergroup. The central claim that the image is exactly the simple group for arbitrarily large degree rests on this step.

Authors: We agree that the original text could have made the verification more explicit for the parametric families. In the revised manuscript we have added a short lemma in §3 that confirms the chosen sequences of n and q satisfy all hypotheses of the mod p monodromy theorems from our earlier work (in particular the conditions on the ramification and the auxiliary prime p). The lemma also records that the determinant of the monodromy representation remains 1, which forces the image to lie inside PSL(n,q) (resp. PSU(n,q)) rather than a larger subgroup of PGL or PGU. This explicit check applies uniformly to the infinite family and supports the claim of arbitrarily large degree. revision: yes
Referee: [Theorem 4.1] Theorem 4.1 (Main realization theorem): The proof that the Galois group over Q is precisely PSL(n,q) (resp. PSU(n,q)) rather than PGL(n,q) (resp. PGU(n,q)) is obtained by combining the mod p monodromy result with a specialization argument; however, the text does not supply an independent verification or a uniform bound showing that no larger group containing PSL(n,q) arises for the infinite series of parameters.

Authors: The specialization step in Theorem 4.1 applies Hilbert irreducibility to the parametric family of covers in a manner that is uniform in n and q, once the geometric monodromy has been fixed as the simple group by the mod p analysis. While the original manuscript did not isolate an independent verification or an explicit uniform bound on the exceptional set, the argument itself does not depend on the specific values of the parameters beyond the conditions already verified in §3. In the revision we have added a remark after the proof of Theorem 4.1 that supplies a uniform description of the exceptional specializations (via the standard effective versions of Hilbert irreducibility) and explains why any proper supergroup of PSL(n,q) inside PGL(n,q) would contradict the mod p reduction. We believe this supplies the requested clarification; if the referee has a particular form of bound in mind we are happy to incorporate it. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim applies prior independent monodromy results to new explicit families

full rationale

The derivation chain begins from the abstract's stated use of prior mod p monodromy results for cyclic coverings of the projective line, then constructs explicit infinite families of Galois realizations over Q for PSL(n,q) and PSU(n,q) with the additional properties of arbitrarily large degree and non-coincidence with PGL/PGU. No equation, definition, or claim in the text reduces the target Galois group identification to a tautology, a fitted parameter renamed as prediction, or a self-referential ansatz. The self-citation to 'our previous results' supplies an external computational input whose verification lies outside this manuscript; the novel content (explicit parameter series and the 'first time' simultaneous properties) remains independent and externally falsifiable via the cited monodromy theorems. This is the standard non-circular pattern of building on prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the correctness of the author's earlier monodromy calculations for cyclic covers; no new free parameters axioms or invented entities are visible from the abstract alone.

pith-pipeline@v0.9.0 · 8398 in / 1236 out tokens · 90724 ms · 2026-05-07T08:18:27.898684+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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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]

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Siegfried Bosch, Werner Lutkebohmert, Michel Raynaud, N´ eron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (1990)

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[6]

Heinrich Matzat, Inverse Galois Theory, Springer Mono- graphs in Mathematics (2018)

Gunter Malle, B. Heinrich Matzat, Inverse Galois Theory, Springer Mono- graphs in Mathematics (2018)

work page 2018
[7]

J. F. Mestre, Familles de courbes hyperelliptiques ` a multiplications r´ eelles, Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89 (Birkh¨ auser Boston, 1991)

work page 1989
[8]

Stepan Nesterov (forthcoming), ModpMonodromy of Cyclic Covers of the Projective Line 11

work page
[9]

Alain Hermez and Alain Salinier, Rational Trinomials with the Alternating Group as Galois Group, Journal of Number Theory Volume 90, Issue 1, pages 113-129 (2001)

work page 2001
[10]

John Thompson, Some finite groups which appear as Gal(L/K), where K⊂Q(µ n), Group theory, Beijing 1984, 210–230, Lecture Notes in Math, 1185 (1986)

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[11]

Chandrashekhar Khare, Michael Larsen, Gordan Savin, Functoriality and the Inverse Galois Problem, Annales de la Facult´ e des sciences de Toulouse: Math´ ematiques, Serie 6, Volume 19 (2010) no. 1, pp. 37-70

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[12]

Gabor Wiese, On projective linear groups over finite fields as Galois groups over the rational numbers, Modular Forms on Schiermonnikoog, Cambridge University Press (2009)

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12, 225–2292 12

David Zywina, The inverse Galois problem for PSL2(Fp), Duke Mathemat- ical Journal, 164 (2015), no. 12, 225–2292 12

work page 2015

[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]

Robin Heartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52 (1977)

work page 1977

[3] [3]

Siegfried Bosch, Werner Lutkebohmert, Michel Raynaud, N´ eron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (1990)

work page 1990

[4] [4]

Jean-Pierre Serre, Topics in Galois Theory, Course at Harvard University, Fall 1988, Notes written by Henri Darmon

work page 1988

[5] [5]

Alexander Grothendieck, Modeles de N´ eron et Monodromie, SGA 7, Ex- pos´ e IX (1972)

work page 1972

[6] [6]

Heinrich Matzat, Inverse Galois Theory, Springer Mono- graphs in Mathematics (2018)

Gunter Malle, B. Heinrich Matzat, Inverse Galois Theory, Springer Mono- graphs in Mathematics (2018)

work page 2018

[7] [7]

J. F. Mestre, Familles de courbes hyperelliptiques ` a multiplications r´ eelles, Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89 (Birkh¨ auser Boston, 1991)

work page 1989

[8] [8]

Stepan Nesterov (forthcoming), ModpMonodromy of Cyclic Covers of the Projective Line 11

work page

[9] [9]

Alain Hermez and Alain Salinier, Rational Trinomials with the Alternating Group as Galois Group, Journal of Number Theory Volume 90, Issue 1, pages 113-129 (2001)

work page 2001

[10] [10]

John Thompson, Some finite groups which appear as Gal(L/K), where K⊂Q(µ n), Group theory, Beijing 1984, 210–230, Lecture Notes in Math, 1185 (1986)

work page 1984

[11] [11]

Chandrashekhar Khare, Michael Larsen, Gordan Savin, Functoriality and the Inverse Galois Problem, Annales de la Facult´ e des sciences de Toulouse: Math´ ematiques, Serie 6, Volume 19 (2010) no. 1, pp. 37-70

work page 2010

[12] [12]

Gabor Wiese, On projective linear groups over finite fields as Galois groups over the rational numbers, Modular Forms on Schiermonnikoog, Cambridge University Press (2009)

work page 2009

[13] [13]

12, 225–2292 12

David Zywina, The inverse Galois problem for PSL2(Fp), Duke Mathemat- ical Journal, 164 (2015), no. 12, 225–2292 12

work page 2015