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Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

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Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita

Number Theory · Submitted 19 Apr 2018

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

The authors' main claim We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8.

how it is stated · asserted 1.00 read the record →

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arXiv ID1804.07109 (version 4)
Pith IDN4BQZWUI, Pith's permanent record of this paper and its review
CitationTatsuya Ohshita, Tetsushi Ito, Yasuhiro Ishitsuka. "Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic." arXiv:1804.07109 (2018). https://pith.science/paper/N4BQZWUI
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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.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,891 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.88 · compares with prior results 0.68 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic field. Thus we complete Faddeev's work in 1960.

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 use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. contribution 0.87 0.95 0.05 asserted 0.93
s1 With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. method 0.74 0.97 0.04 asserted 1.00
s2 We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. result 0.97 0.97 0.72 asserted 1.00
s3 As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. result 0.64 0.96 0.07 asserted 1.00
s4 We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic field. result 0.93 0.95 0.08 asserted 1.00
s5 Thus we complete Faddeev's work in 1960. interpretation 0.88 0.74 0.28 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from simulation or computation 0.59.

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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 use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic field. Thus we complete Faddeev's work in 1960.

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