scholarly journals On the ramification of étale cohomology groups

2019 ◽  
Vol 2019 (749) ◽  
pp. 295-304 ◽  
Author(s):  
Isabel Leal
Keyword(s):  
Positive Characteristic ◽  
Residue Field ◽  
Cohomology Groups ◽  
Generic Fiber ◽  
Codimension One ◽  
Etale Cohomology ◽  
Rank One ◽  
Étale Cohomology ◽  
Valuation Field ◽  
Dimension One

Abstract Let K be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let X be a connected, proper scheme over \mathcal{O}_{K} , and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair (X,U) is strictly semi-stable over \mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth \ell -adic sheaf \mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of \mathcal{G} is bounded by t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X, then the ramification of the étale cohomology groups with compact support of \mathcal{G} is bounded by t+ in the same sense.

scholarly journals Étale cohomology of rank one $$\ell $$-adic local systems in positive characteristic

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Hélène Esnault ◽  
Moritz Kerz
Keyword(s):  
Positive Characteristic ◽  
Vanishing Theorem ◽  
Deformation Spaces ◽  
Local Systems ◽  
Etale Cohomology ◽  
Lefschetz Theorem ◽  
Rank One ◽  
Étale Cohomology ◽  
Generic Vanishing ◽  
Hard Lefschetz Theorem

AbstractWe show that in positive characteristic special loci of deformation spaces of rank one $$\ell $$ ℓ -adic local systems are quasi-linear. From this we deduce the Hard Lefschetz theorem for rank one $$\ell $$ ℓ -adic local systems and a generic vanishing theorem.

Genus of abstract modular curves with level-ℓ structures

2019 ◽  
Vol 2019 (752) ◽  
pp. 25-61 ◽  
Author(s):  
Anna Cadoret ◽  
Akio Tamagawa
Keyword(s):  
Galois Cohomology ◽  
Fundamental Groups ◽  
Modular Curves ◽  
Cohomology Groups ◽  
Arbitrary Characteristic ◽  
Linear Representations ◽  
Etale Cohomology ◽  
Étale Cohomology

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

scholarly journals Representability of Chow groups of codimension three cycles

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Kalyan Banerjee
Keyword(s):  
Chow Groups ◽  
Generic Fiber ◽  
Projective Varieties ◽  
Rational Equivalence ◽  
Finite Dimensional ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Smooth Projective Varieties

Abstract Assume that we have a fibration of smooth projective varieties X → S over a surface S such that X is of dimension four and that the geometric generic fiber has finite-dimensional motive and the first étale cohomology of the geometric generic fiber with respect to ℚ l coefficients is zero and the second étale cohomology is spanned by divisors. We prove that then A 3(X), the group of codimension three algebraically trivial cycles modulo rational equivalence, is dominated by finitely many copies of A 0(S); this means that there exist finitely many correspondences Γi on S × X such that Σ i Γi is surjective from A 2(S) to A 3(X).

scholarly journals Reduced norms of division algebras over complete discrete valuation fields of local-global type

2019 ◽  
Vol 19 (11) ◽  
pp. 2050217 ◽  
Author(s):  
Yong Hu
Keyword(s):  
Positive Characteristic ◽  
Residue Field ◽  
Division Algebras ◽  
Central Division ◽  
Cup Product ◽  
Recent Theorem ◽  
Valuation Field ◽  
Global Type ◽  
Product Map ◽  
Field Of Positive Characteristic

Let [Formula: see text] be a complete discrete valuation field whose residue field [Formula: see text] is a global field of positive characteristic [Formula: see text]. Let [Formula: see text] be a central division [Formula: see text]-algebra of [Formula: see text]-power degree. We prove that the subgroup of [Formula: see text] consisting of reduced norms of [Formula: see text] is exactly the kernel of the cup product map [Formula: see text], if either [Formula: see text] is tamely ramified or of period [Formula: see text]. This gives a [Formula: see text]-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-[Formula: see text] degree.

scholarly journals Facteurs epsilon p-adiques

2008 ◽  
Vol 144 (2) ◽  
pp. 439-483 ◽  
Author(s):  
Adriano Marmora
Keyword(s):  
Residue Field ◽  
Smooth Curve ◽  
Local System ◽  
Product Formula ◽  
Mixed Characteristic ◽  
Characteristic Case ◽  
Rank One ◽  
Epsilon Factors ◽  
Valuation Field ◽  
Field Of Norms

AbstractWe develop and study the epsilon factor of a ‘local system’ of p-adic coefficients over the spectrum of a complete discrete valuation field K with finite residue field of characteristic p>0. In the equal characteristic case, we define the epsilon factor of an overconvergent F-isocrystal over Spec(K), using the p-adic monodromy theorem. We conjecture a global formula, the p-adic product formula, analogous to Deligne’s formula for étale ℓ-adic sheaves proved by Laumon, which explains the importance of this local invariant. Namely, for an overconvergent F-isocrystal over an open subset of a projective smooth curve X, the constant of the functional equation of the L-series is expressed as a product of the local epsilon factors at the points of X. We prove the conjecture for rank-one overconvergent F-isocrystals and for finite unit-root overconvergent F-isocrystals. In the mixed characteristic case, we study the behavior of the epsilon factor by deformation to the field of norms.

scholarly journals Loop Schrödinger–Virasoro Lie conformal algebra

2016 ◽  
Vol 27 (06) ◽  
pp. 1650057 ◽  
Author(s):  
Haibo Chen ◽  
Jianzhi Han ◽  
Yucai Su ◽  
Ying Xu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Cohomology Groups ◽  
Conformal Algebras ◽  
Rank One ◽  
Intermediate Series

In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger–Virasoro Lie algebra and the extended loop Schrödinger–Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and [Formula: see text]-graded free intermediate series modules over these two conformal algebras are also classified in the present paper.

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