Semistable deformation rings in even
Hodge–Tate weights

Guerberoff, Lucio; Park, Chol

doi:10.2140/pjm.2019.298.299

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…As far as we are aware, the reduction problem for slope 2 is still open, though partial results for small slopes larger than 2 have been announced by Arsovski [Ars18] and Nagel-Pande [NP18]. In related parallel developments on the reduction problem, we remark that certain crystabelian cases of weight 2 and slope at most 1 had been treated earlier by Savitt [Sav05], and that similarly, semistable (non-crystalline) cases of small even weights k ≤ p + 1 had earlier been treated by Breuil-Mézard [BM02], and more recently by Guerberoff-Park [GP18] for the remaining odd weights in this range, using the theory of strongly divisible modules.…”

Section: Introductionmentioning

confidence: 74%

A zig-zag conjecture and local constancy for Galois representations

Ghate

2019

Preprint

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We make a zig-zag conjecture describing the reductions of irreducible crystalline twodimensional representations of G Qp of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod (p − 1). We show that zig-zag holds for half-integral slopes at most 3 2 . We then explore the connection between zig-zag and local constancy results in the weight. First we show that known cases of zig-zag force local constancy to fail for small weights. Conversely, we explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least 2. However, we expect zig-zag to be qualitatively true in general.We end with some compatibility results between zig-zag and other results.

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Section: Introductionmentioning

confidence: 74%

A zig-zag conjecture and local constancy for Galois representations

Ghate

2019

Preprint

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“…The situation is more complicated when . In that case, for , Guerberoff and Park showed that exactly on [17, Theorem 5.0.5]. Thus, the bound from Theorem 4.1 produces too large a region of -invariants, whereas formula (4.10) produces a region too small.…”

Section: Descent and Reductionsmentioning

confidence: 99%

“…Twenty years ago, Breuil and Mézard determined for even and any [7, Théorème 4.2.4.7]. Guerberoff and Park recently studied odd [17, Theorem 5.0.5]. The reader who takes a moment to examine the cited theorems should be left with an impression of the complicated dependence of on , and that is just for .…”

Section: Introductionmentioning

confidence: 99%

Reductions of -Dimensional Semistable Representations With Large -Invariant

Bergdall

Левин

Liu

2022

J. Inst. Math. Jussieu

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We determine reductions of $2$ -dimensional, irreducible, semistable, and non-crystalline representations of $\mathrm {Gal}\left (\overline {\mathbb {Q}}_p/\mathbb {Q}_p\right )$ with Hodge–Tate weights $0 < k-1$ and with $\mathcal L$ -invariant whose p-adic norm is sufficiently large, depending on k. Our main result provides the first systematic examples of the reductions for $k \geq p$ .

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“…Let us illustrate this with some examples. The reductions of semi-stable representations have been computed completely for even weights in the range [2, p] by Breuil and Mézard [BM02], and for odd weights in the same range by Guerberoff and Park [GP19] at least on inertia. In [Gha21], the second author made a conjecture called the zig-zag conjecture describing the reductions of crystalline representations of exceptional weights and half-integral slopes in terms of an alternating sequence of reducible and irreducible mod p representations.…”

Section: Reductions Of Semi-stable Representationsmentioning

confidence: 99%

Semi-stable representations as limits of crystalline representations

Chitrao¹,

Ghate²,

Yasuda³

2021

Preprint

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We construct an explicit sequence of crystalline representations converging to a given irreducible two-dimensional semi-stable representation of Gal(Q p /Q p ). The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest.Our convergence result can be used to compute the reductions of any irreducible two-dimensional semi-stable representation in terms of the reductions of certain crystalline representations of exceptional weight. In particular, this provides an alternative approach to computing the reductions of irreducible two-dimensional semi-stable representations that circumvents the somewhat technical machinery of integral p-adic Hodge theory. For instance, using the zig-zag conjecture made in [Gha21] on the reductions of crystalline representations of exceptional weights, we recover completely the work of Breuil-Mézard and Guerberoff-Park on the reductions of irreducible semi-stable representations of weights at most p + 1, at least on the inertia subgroup. In the cases where the zig-zag conjecture is known, we are further able to obtain some new information about the reductions for small odd weights.Finally, we use the above ideas to explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight which were noticed in [Gha21].

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Semistable deformation rings in even Hodge–Tate weights

Cited by 6 publications

References 16 publications

A zig-zag conjecture and local constancy for Galois representations

A zig-zag conjecture and local constancy for Galois representations

Reductions of -Dimensional Semistable Representations With Large -Invariant

Semi-stable representations as limits of crystalline representations

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