Zig-zag for Galois Representations

Eknath Ghate

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Zig-zag for Galois Representations

Eknath Ghate

Number Theory · Submitted 22 Nov 2022

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

The authors' main claim We prove this conjecture in smoothly varying families of such representations for $p \geq 5$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,133 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.83 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.41 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating sequence of irreducible and reducible mod $p$ representations. We prove this conjecture in smoothly varying families of such representations for $p \geq 5$. The proof uses a limiting argument due to Chitrao-Ghate-Yasuda to reduce to the case of semi-stable representations of weights at most $p+1$, and then appeals to the work of Breuil-M\'ezard, Guerberoff-Park and Chitrao-Ghate.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating sequence of irreducible and reducible mod $p$ representations.	background 1.00	0.05	0.10	asserted 0.51
s1	We prove this conjecture in smoothly varying families of such representations for $p \geq 5$.	result 0.59	0.96	0.95	asserted 1.00
s2	The proof uses a limiting argument due to Chitrao-Ghate-Yasuda to reduce to the case of semi-stable representations of weights at most $p+1$, and then appeals to the work of Breuil-M\'ezard, Guerberoff-Park and Chitrao-Ghate.	method 1.00	0.69	0.04	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

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arXiv listing abstract v2

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figures and tables
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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

The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating sequence of irreducible and reducible mod $p$ representations. We prove this conjecture in smoothly varying families of such representations for $p \geq 5$. The proof uses a limiting argument due to Chitrao-Ghate-Yasuda to reduce to the case of semi-stable representations of weights at most $p+1$, and then appeals to the work of Breuil-M\'ezard, Guerberoff-Park and Chitrao-Ghate.

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4 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 4 sentences without a confident label

Builds on it 1 paper

* It allows the second author to extend his proof of the zig-zag conjecture on $I_$ for slopes $1/2 ≤v_p(a_p) ≤p - 3/2$ outlined in [ref] to all of $Gal(/)$ and all slopes $1/2 ≤v(a_p) ≤p - 1/2$.

Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence · Introduction · 0.99

Compares with it 1 paper

To compare with the zig-zag conjecture for odd primes, we recall that $τ= v(a_2^2-2r2a_2)$ [ref] .

Behaviour of Certain Crystalline Representations modulo $2$ · Numerical examples · 0.90

What it cites · 14 sentences without a confident label

Compares with 2 papers

The paper [ref] computes instead the reduction of Breuil's Banach space $B(k_0,)$ and then uses the Iwahori mod $p$ local Langlands correspondence [ref] .

An Iwahori theoretic mod $p$ Local Langlands Correspondence · Introduction · 0.86
The paper [ref] computes instead the reduction of Breuil's Banach space $B(k_0,)$ and then uses the Iwahori mod $p$ local Langlands correspondence [ref] .

Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence · Introduction · 0.86

Cites as background 1 paper

For background on blow-ups in the rigid setting see [ref] .

Semi-stable representations as limits of crystalline representations · Limits of crystalline representations · 1.00

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