Pith. Sign in

RECORD

Reducible Galois representations and the homology of GL(3,Z)

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Avner Ash, Darrin Doud

Number Theory · Submitted 14 May 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1205.3086 (version 1)
Pith IDEDC6QH3X, Pith's permanent record of this paper and its review
CitationAvner Ash, Darrin Doud. "Reducible Galois representations and the homology of GL(3,Z)." arXiv:1205.3086 (2012). https://pith.science/paper/EDC6QH3X
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,740 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.75 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.14 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. background 0.98 0.19 0.02 asserted 0.55
s1 Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. background 0.81 0.11 0.03 asserted 0.52
s2 Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$. result 0.97 0.96 0.84 asserted 1.00
s3 In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott. result 0.92 0.86 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott.

Human review

How Pith checks papers → · Full audit of this paper →

References

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.