Finiteness with Coefficients via a Local Monodromy Theorem

2016 ◽  
pp. 71-129
Author(s):  
Christopher Lazda ◽  
Ambrus Pál
Keyword(s):  
Local Monodromy

scholarly journals Deformation of ℓ -adic sheaves with undeformed local monodromy

2013 ◽  
Vol 133 (2) ◽  
pp. 675-691 ◽  
Author(s):  
Lei Fu
Keyword(s):  
Local Monodromy

scholarly journals LOCAL MONODROMY OF p-ADIC DIFFERENTIAL EQUATIONS: AN OVERVIEW

2005 ◽  
Vol 01 (01) ◽  
pp. 109-154 ◽  
Author(s):  
KIRAN S. KEDLAYA
Keyword(s):  
Differential Equations ◽  
Local Monodromy ◽  
Expository Article

This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent ∇-modules, the Christol–Mebkhout construction of the ramification filtration, and the Christol–Dwork Frobenius antecedent theorem. We also briefly discuss the p-adic local monodromy theorem without proof.

scholarly journals The p-adic local monodromy theorem

2012 ◽  
pp. 326-337
Author(s):  
Kiran S. Kedlaya
Keyword(s):  
Local Monodromy

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

scholarly journals THE HOLOMORPHY CONJECTURE FOR NONDEGENERATE SURFACE SINGULARITIES

2016 ◽  
Vol 227 ◽  
pp. 160-188
Author(s):  
WOUTER CASTRYCK ◽  
DENIS IBADULA ◽  
ANN LEMAHIEU
Keyword(s):  
Zeta Function ◽  
Arbitrary Dimension ◽  
Newton Polyhedron ◽  
Character Sums ◽  
Specific Character ◽  
Trivial Character ◽  
Surface Singularities ◽  
Local Monodromy

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on$\mathbb{C}$whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over$\mathbb{C}$with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by$B_{1}$-facets.

scholarly journals Ramification theory and perfectoid spaces

2014 ◽  
Vol 150 (5) ◽  
pp. 798-834 ◽  
Author(s):  
Shin Hattori
Keyword(s):  
Positive Integer ◽  
Rational Number ◽  
Positive Rational Number ◽  
Ramification Theory ◽  
Local Monodromy ◽  
Residue Fields ◽  
Perfectoid Spaces ◽  
Residue Characteristic ◽  
The Absolute ◽  
Separable Extensions

AbstractLet $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ whose ramifications are bounded by $j$ are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl’s theory of higher fields of norms with the ramification theory of Abbes–Saito, and the integrality of small Artin and Swan conductors of $p$-adic representations with finite local monodromy.

scholarly journals Arithmetic and differential Swan conductors of rank one representations with finite local monodromy

2009 ◽  
Vol 131 (6) ◽  
pp. 1743-1794 ◽  
Author(s):  
Bruno Chiarellotto ◽  
Andrea Pulita
Keyword(s):  
Rank One ◽  
Local Monodromy
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