A finiteness theorem for mod $p$ Galois representations over global function fields

Yufan Luo

arxiv: 2606.29277 · v1 · pith:WH4PEL7Tnew · submitted 2026-06-28 · 🧮 math.NT

A finiteness theorem for mod p Galois representations over global function fields

Yufan Luo This is my paper

Pith reviewed 2026-06-30 02:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords Galois representationsglobal function fieldsArtin conductorfiniteness theoremmod p representationssemisimple representationsgeometric representations

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The pith

There are only finitely many isomorphism classes of continuous geometric semisimple mod p Galois representations with bounded Artin conductors over global function fields.

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

The paper proves that when K is a global function field of characteristic different from an odd prime p, the continuous geometric semisimple representations from the absolute Galois group of K into GL_n over the algebraic closure of F_p fall into only finitely many isomorphism classes once their Artin conductors are bounded above by a fixed number. The argument makes no assumption that p fails to divide n. A reader cares because the result supplies a uniform finiteness statement that controls the possible ramification of such representations, which in turn limits the arithmetic objects (such as sheaves or motives) that can produce them over function fields.

Core claim

The paper establishes that there are only finitely many isomorphism classes of continuous geometric semisimple representations ρ : G_K → GL_n(¯F_p) such that their Artin conductors are bounded. This holds for any n and without the hypothesis that p does not divide n.

What carries the argument

The Artin conductor, which quantifies the ramification of the representation at places of K, together with the geometric and semisimple conditions on ρ.

If this is right

For any fixed bound on the Artin conductor and any n, only finitely many such representations exist up to isomorphism.
The finiteness statement remains valid even in the case where p divides n.
The result applies uniformly to representations of any dimension n.

Where Pith is reading between the lines

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

The same bounded-conductor condition might be used to show that only finitely many isomorphism classes arise from étale cohomology of varieties over K.
One could test whether removing the semisimplicity assumption still yields finiteness by examining extensions of the representations that remain geometric.
The result suggests that the local behavior of the representation at finitely many places determines it up to isomorphism when the global conductor is fixed.

Load-bearing premise

The representations under consideration are required to be geometric and semisimple.

What would settle it

An explicit infinite family of pairwise non-isomorphic continuous geometric semisimple representations ρ : G_K → GL_n(¯F_p) all sharing the same bounded Artin conductor would falsify the claim.

read the original abstract

Let $p$ be an odd prime number and let $\overline{\mathbb{F}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_{K}$ be the absolute Galois group of $K$. We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\rho:G_{K}\to \mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ such that their Artin conductors are bounded. It is worth emphasizing that we do not need to assume that $p$ does not divide $n$.

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 states a finiteness theorem for continuous geometric semisimple mod p Galois representations over global function fields with bounded Artin conductor, without needing p not dividing n.

read the letter

The main point is a finiteness result: only finitely many isomorphism classes of continuous geometric semisimple ρ from G_K to GL_n over algebraic closure of F_p, with bounded Artin conductor, for global function field K of characteristic not p, and p odd. The abstract flags the lack of any p not dividing n assumption as the distinction from earlier work.

This relaxation is the actual new piece. Earlier statements often carried that extra hypothesis, so dropping it widens the statement in a natural way. The setup itself uses standard language for geometric and semisimple representations and Artin conductors, with no obvious mismatch between the claim and the listed conditions.

The paper keeps the statement short and direct. That is useful for readers who already know the surrounding literature on Galois representations over function fields.

The clear limitation is that the abstract supplies no lemmas, no outline of the argument, and no indication of how the case p dividing n is handled. Finiteness proofs in this area typically combine ramification bounds with some form of class field theory or known finiteness for unramified or tamely ramified cases. Without seeing those steps it is impossible to tell whether the extension is routine or requires new input. The stress-test note found no internal contradiction, which matches what the abstract shows, but that is not the same as verifying the proof.

This is for people already working on arithmetic geometry or Galois representations in positive characteristic. A reader tracking finiteness theorems for mod p objects would want to see the details. It is the kind of targeted result that should go to a serious referee who can check the technical steps.

Referee Report

0 major / 1 minor

Summary. The paper proves that for an odd prime p and a global function field K of characteristic different from p, there are only finitely many isomorphism classes of continuous geometric semisimple representations ρ: G_K → GL_n(¯F_p) with bounded Artin conductor. The result is stated without any hypothesis that p does not divide n.

Significance. If correct, the theorem supplies a clean finiteness statement for geometric semisimple mod-p representations over global function fields, removing the usual p ∤ n restriction that appears in some number-field analogues. Such a result would be useful for controlling families of representations with fixed conductor and could feed into questions about moduli spaces or residual Galois representations in positive characteristic.

minor comments (1)

The abstract states the main theorem clearly but does not indicate the principal tools (e.g., whether the argument relies on class-field theory, ramification bounds, or reduction to the case of curves). A one-sentence pointer would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report correctly captures the main theorem, including the absence of any p ∤ n hypothesis.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a finiteness theorem for continuous geometric semisimple mod p Galois representations with bounded Artin conductor over global function fields of characteristic ≠ p. The abstract and setup invoke standard hypotheses (geometric, semisimple, char(K) ≠ p) and a classical invariant (Artin conductor) without defining any quantity in terms of the finiteness conclusion itself. No equations, self-citations, or ansatzes are exhibited that reduce the claimed result to a tautology or fitted input. The derivation therefore remains self-contained against external benchmarks in Galois theory and ramification theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

pith-pipeline@v0.9.1-grok · 5633 in / 1074 out tokens · 42603 ms · 2026-06-30T02:29:04.355550+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages · 1 internal anchor

[1]

B¨ ockle

G. B¨ ockle. Deformations of Galois representations. InElliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, pages 21–115. Birkh¨ auser/Springer, Basel, 2013

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B¨ ockle, W

G. B¨ ockle, W. Gajda, and S. Petersen. On the semisimplicity of reductions and adelic open- ness forE-rational compatible systems over global function fields.Trans. Amer. Math. Soc., 372(8):5621–5691, 2019

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B¨ ockle and C

G. B¨ ockle and C. Khare. Modlrepresentations of arithmetic fundamental groups. II. A conjecture of A. J. de Jong.Compos. Math., 142(2):271–294, 2006

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A. J. de Jong. A conjecture on arithmetic fundamental groups.Israel J. Math., 121:61–84, 2001

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Drinfeld

V. Drinfeld. On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field.Adv. Math., 327:708–788, 2018

2018
[6]

Esnault and M

H. Esnault and M. Kerz. A finiteness theorem for Galois representations of function fields over finite fields (after Deligne).Acta Math. Vietnam., 37(4):531–562, 2012

2012
[7]

Gaitsgory

D. Gaitsgory. On de Jong’s conjecture.Israel J. Math., 157:155–191, 2007

2007
[8]

C. Khare. Conjectures on finiteness of modpGalois representations.J. Ramanujan Math. Soc., 15(1):23–42, 2000

2000
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Introductory course on $\ell$-adic sheaves and their ramification theory on curves

L. Kindler and K. R¨ ulling. Introductory course onℓ-adic sheaves and their ramification theory on curves. Preprint, arXiv:1409.6899 [math.AG], 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[10]

Lafforgue

L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands.Invent. Math., 147(1):1– 241, 2002

2002
[11]

B. Mazur. Deforming Galois representations. InGalois groups overQ(Berkeley, CA, 1987), volume 16 ofMath. Sci. Res. Inst. Publ., pages 385–437. Springer, New York, 1989

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[12]

J. S. Milne.Arithmetic duality theorems. BookSurge, LLC, Charleston, SC, second edition, 2006

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[13]

H. Moon. Finiteness results on certain modpGalois representations.J. Number Theory, 84(1):156–165, 2000

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[14]

Moon and Y

H. Moon and Y. Taguchi. Modp-Galois representations of solvable image.Proc. Amer. Math. Soc., 129(9):2529–2534, 2001. Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai 200433, China Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China Email address:yufanluo@hotmail.com

2001

[1] [1]

B¨ ockle

G. B¨ ockle. Deformations of Galois representations. InElliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, pages 21–115. Birkh¨ auser/Springer, Basel, 2013

2013

[2] [2]

B¨ ockle, W

G. B¨ ockle, W. Gajda, and S. Petersen. On the semisimplicity of reductions and adelic open- ness forE-rational compatible systems over global function fields.Trans. Amer. Math. Soc., 372(8):5621–5691, 2019

2019

[3] [3]

B¨ ockle and C

G. B¨ ockle and C. Khare. Modlrepresentations of arithmetic fundamental groups. II. A conjecture of A. J. de Jong.Compos. Math., 142(2):271–294, 2006

2006

[4] [4]

A. J. de Jong. A conjecture on arithmetic fundamental groups.Israel J. Math., 121:61–84, 2001

2001

[5] [5]

Drinfeld

V. Drinfeld. On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field.Adv. Math., 327:708–788, 2018

2018

[6] [6]

Esnault and M

H. Esnault and M. Kerz. A finiteness theorem for Galois representations of function fields over finite fields (after Deligne).Acta Math. Vietnam., 37(4):531–562, 2012

2012

[7] [7]

Gaitsgory

D. Gaitsgory. On de Jong’s conjecture.Israel J. Math., 157:155–191, 2007

2007

[8] [8]

C. Khare. Conjectures on finiteness of modpGalois representations.J. Ramanujan Math. Soc., 15(1):23–42, 2000

2000

[9] [9]

Introductory course on $\ell$-adic sheaves and their ramification theory on curves

L. Kindler and K. R¨ ulling. Introductory course onℓ-adic sheaves and their ramification theory on curves. Preprint, arXiv:1409.6899 [math.AG], 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[10] [10]

Lafforgue

L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands.Invent. Math., 147(1):1– 241, 2002

2002

[11] [11]

B. Mazur. Deforming Galois representations. InGalois groups overQ(Berkeley, CA, 1987), volume 16 ofMath. Sci. Res. Inst. Publ., pages 385–437. Springer, New York, 1989

1987

[12] [12]

J. S. Milne.Arithmetic duality theorems. BookSurge, LLC, Charleston, SC, second edition, 2006

2006

[13] [13]

H. Moon. Finiteness results on certain modpGalois representations.J. Number Theory, 84(1):156–165, 2000

2000

[14] [14]

Moon and Y

H. Moon and Y. Taguchi. Modp-Galois representations of solvable image.Proc. Amer. Math. Soc., 129(9):2529–2534, 2001. Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai 200433, China Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China Email address:yufanluo@hotmail.com

2001