Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Ding, Yiwen

doi:10.1093/imrn/rnx287

Cited by 3 publications

(1 citation statement)

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“…be the parabolic with component labelled by our fixed place v equal to the upper triangular Borel and other components equal to the whole group GL 2 . See, for example, [Din18] for another application of eigenvarieties defined with respect to a nonminimal parabolic. We also note that partial Hilbert modular eigenvarieties have been used recently by Barrera, Dimitrov and Jorza to study the exceptional zero conjecture for Hilbert modular forms [BDJ17].…”

Section: A Halo For the Partial Eigenvarietymentioning

confidence: 99%

Parallel Weight 2 Points on Hilbert Modular Eigenvarieties and the Parity Conjecture

Johansson

Newton

2019

Forum of Mathematics, Sigma

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Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

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Section: A Halo For the Partial Eigenvarietymentioning

confidence: 99%

Parallel Weight 2 Points on Hilbert Modular Eigenvarieties and the Parity Conjecture

Johansson

Newton

2019

Forum of Mathematics, Sigma

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A local model for the trianguline variety and applications

2019

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We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties. for discussions or answers to questions. E. H. is partially supported by SFB-TR 45 and SFB 878 of the D.F.G., B. S. and C. B. are supported by the C.N.R.S. 1 4.3. A locally analytic "Breuil-Mézard type" statement 58 5. Global applications 64 5.1. Classicality 64 5.2. Representation theoretic preliminaries 68 5.3. Companion constituents 71 5.4. Singularities on global Hecke eigenvarieties 78References 80are the Kazhdan-Lusztig polynomials and L(w ′ ) are certain finite type O Xp(ρ) wt(δ) ,x -modules such that:Formula (1.2) essentially comes from representation theory (in particular the structure of Verma modules) and doesn't use Theorem 1.5. By an argument analogous to the one for Theorem 1.8 (using Theorem 1.5), we have nonzeroMoreover we know that the cycle C(w 0 ) is irreducible and that [L(w 0 )] ∈ Z ≥0 C(w 0 ) (roughly because the support of the locally Q p -algebraic vectors lies in the locus of crystalline deformations). Consequently we can deduce Theorem 1.3 from the fact that P 1,w 0 w ′ (1) = 0, if we know that C(w ′ ) is contained in the support of L(w ′ ) for w x w ′ .We prove this last assertion by a descending induction on the length of the Weyl group element w x . Assume first that lg(w x ) = lg(w 0 ) − 1. In that case x is smooth on X p (ρ) and then M ∞ is locally free at x. Hence M ∞,wt(δ),x ≃ O r Xp(ρ) wt(δ) ,x for some r > 0 and we can combine (1.3) (multiplied by the integer r) with (1.2). Using C(w 0 ) = 0, C(w x ) = 0 and [L(w 0 )] ∈ Z ≥0 C(w 0 ), it is then not difficult to deduce [L(w x )] = 0, hence L(w x ) = 0 and then. By a Zariski-density argument analogous to the one in the proof of Theorem 1.7, we can then deduce [L(w ′ )] = 0 for any w ′ w x such that lg(w ′ ) ≥ lg(w 0 ) − 1 and any w x such that lg(w x ) ≤ lg(w 0 ) − 1. In particular we have the companion points x w ′ on Y (U p , ρ) for w ′ w x and lg(w ′ ) = lg(w 0 ) − 1 and formulas analogous to (1.2) and (1.3) localizing and completing atAssume now lg(w x ) = lg(w 0 ) − 2, we can repeat the argument of the case lg(w x ) = lg(w 0 ) − 1 but using the analogues of (1.2), (1.3) at x w ′ = (ρ, δ w ′ ) for w ′ w x and lg(w ′ ) = lg(w 0 ) − 1 = lg(w x ) + 1. The results on the local geometry o...

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Bernstein Eigenvarieties

Breuil

Ding

2023

Peking Math J

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Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Cited by 3 publications

References 32 publications

Parallel Weight 2 Points on Hilbert Modular Eigenvarieties and the Parity Conjecture

Parallel Weight 2 Points on Hilbert Modular Eigenvarieties and the Parity Conjecture

A local model for the trianguline variety and applications

Bernstein Eigenvarieties

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