An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal(ℚ¯p/ℚp)

Rozensztajn, Sandra

doi:10.1142/s1793042118501117

Cited by 8 publications

(5 citation statements)

References 16 publications

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“…A conjecture. Based on these results for slopes 1 2 and 1, and some computations of Rozensztajn [Roz18] for some small half-integral slopes, one might guess that in the general exceptional case v ∈ 1 2 Z and b = 2v, there are b + 1 possibilities for Vk,ap , with various irreducible and reducible cases occurring alternately. More precisely, we make the following qualitative conjecture.…”

Section: Introductionmentioning

confidence: 86%

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: 86%

A zig-zag conjecture and local constancy for Galois representations

Ghate

2019

Preprint

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“…For other examples, for small values of k and p, we refer the reader to [Roz16], where an algorithm to computeV ss k,ap is described and implemented for all positive slopes.…”

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confidence: 99%

Reductions of Galois representations of slope 1

Bhattacharya

Ghate

Rozensztajn³

2018

Journal of Algebra

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We compute the reductions of irreducible crystalline two-dimensional representations of G Qp of slope 1, for primes p ≥ 5, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or très ramifiée, in the relevant reducible nonsemisimple cases. The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and nonstandard lattices in certain p-adic Banach spaces.• However, when v(a p ) = 1, the theorem shows that in each congruence class of weights mod (p − 1) there are further congruence classes of weights mod p whereV k,ap is irreducible.• A surprising trichotomy occurs for b = 2. This seems to be a more complicated manifestation of the dichotomy that occurs for weights r ≡ 1 mod (p − 1) when v(a p ) = 1 2 in [BG13].The proof of Theorem 1.1 uses the compatibility of the p-adic and mod p Local Langlands Correspondences, with respect to the process of reduction [Ber10]. This compatibility allows one to reduce the reduction problem to one on the 'automorphic side', namely, to computing the reduction of a lattice in a certain Banach space. Let G = GL 2 (Q p ) and let B(V k,ap ) be the unitary G-Banach space associated to V k,ap by the p-adic Local Langlands Correspondence. The reduction B(V k,ap ) ss of a lattice in this Banach space coincides with the image ofV ss k,ap under the (semisimple) mod p Local 1 Recent work by Arsovski [Ars15] includes a study of the reductionV ss k,ap when the slope is 1, for b = 3, p, though the reduction is not uniquely specified there. REDUCTIONS OF GALOIS REPRESENTATIONS OF SLOPE 1 3Langlands Correspondence defined in [Bre03b]. Since the mod p correspondence is by definition injective, it suffices to compute the reduction B(V k,ap ) ss .Let K = GL 2 (Z p ), a maximal compact open subgroup of G = GL 2 (Q p ), and let Z = Q × p be the center of G. Let X = KZ\G be the (vertices of the) Bruhat-Tits tree associated to G. The module Sym rQ2 p , for r = k − 2, carries a natural action of KZ, and the projectionconsists of all sections f : G → Sym k−2Q2 p of this local system which are compactly supported mod KZ. There is a G-equivariant Hecke operator T acting on this space of sections. Let Π k,ap be the locally algebraic representation of G defined by taking the cokernel of T − a p acting on the above space. Let Θ k,ap be the image of the integral sections ind G KZ Sym k−2Z2 p in Π k,ap . Then B(V k,ap ) is the completionΠ k,ap of Π k,ap with respect to the lattice Θ k,ap . The completionΘ k,ap , and sometimes by abuse of notation Θ k,ap itself, is called the standard lattice in B(V k,ap ). We have∼ =Θ ss k,ap . Thus, to computeV ss k,ap , it suffices to compute the reductionΘ ss k,ap of Θ k,ap . There are two main steps. First, we study a quotient P of Sym k−2F2 p whose Jordan-Hölder factors provide an upper bound on the possible Jordan-Hölder fac...

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“…With some exceptions, V k,a has been computed when v p (a) < 2 in the work of Buzzard, Gee, Ganguli, Ghate, Bhattacharya, Rozensztajn, and Rai in [BG15], [BGR18], [BG09], [BG13], [GG15], and [GR]. Finally, Rozensztajn's work [Roz18] gives an algorithm that computes V k,a for given p, k, a, and her work [Roz20] gives an algorithm that finds the locus of all a such that V k,a = ρ for given p, k, and a modulo p representation ρ.…”

Section: Known Resultsmentioning

confidence: 99%

On the reductions of certain two-dimensional crystalline representations

Arsovski

2021

Doc. Math.

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The question of computing the reductions modulo p of twodimensional crystalline p-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small slopes, or very large slopes. It was conjectured by Breuil, Buzzard, and Emerton that these reductions are irreducible if they have even weight and non-integer slope. We prove some instances of this conjecture for slopes up to p−1 2 .

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An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal(ℚ¯p/ℚp)

Cited by 8 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 Galois representations of slope 1

On the reductions of certain two-dimensional crystalline representations

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