Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000399 ◽

2020 ◽

pp. 1-17

Author(s):

Tongmu He

Keyword(s):

Abelian Varieties ◽

Residue Field ◽

Local Fields ◽

Field Case ◽

Residue Fields ◽

Valuation Field

Abstract Let K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

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Splitting properties of the reduction of semi-abelian varieties

International Journal of Number Theory ◽

10.1142/s1793042116501359 ◽

2016 ◽

Vol 12 (08) ◽

pp. 2241-2264

Author(s):

Alan Hertgen

Keyword(s):

Abelian Variety ◽

Abelian Varieties ◽

Residue Field ◽

Characteristic Exponent ◽

Jacobian Variety ◽

Ring Of Integers ◽

Identity Component ◽

Genus Formula ◽

Valuation Field

Let [Formula: see text] be a complete discrete valuation field. Let [Formula: see text] be its ring of integers. Let [Formula: see text] be its residue field which we assume to be algebraically closed of characteristic exponent [Formula: see text]. Let [Formula: see text] be a semi-abelian variety. Let [Formula: see text] be its Néron model. The special fiber [Formula: see text] is an extension of the identity component [Formula: see text] by the group of components [Formula: see text]. We say that [Formula: see text] has split reduction if this extension is split. Whereas [Formula: see text] has always split reduction if [Formula: see text] we prove that it is no longer the case if [Formula: see text] even if [Formula: see text] is tamely ramified. If [Formula: see text] is the Jacobian variety of a smooth proper and geometrically connected curve [Formula: see text] of genus [Formula: see text], we prove that for any tamely ramified extension [Formula: see text] of degree greater than a constant, depending on [Formula: see text] only, [Formula: see text] has split reduction. This answers some questions of Liu and Lorenzini.

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The different and differentials of local fields with imperfect residue fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500023798 ◽

1997 ◽

Vol 40 (2) ◽

pp. 353-365 ◽

Cited By ~ 3

Author(s):

Bart de Smit

Keyword(s):

Galois Extension ◽

Correction Term ◽

Residue Field ◽

Field Extension ◽

Local Fields ◽

Valuation Rings ◽

Complete Field ◽

Residue Fields ◽

Residue Characteristic ◽

Module Of Differentials

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

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A SIMPLIFIED PROOF OF HESSELHOLT’S CONJECTURE ON GALOIS COHOMOLOGY OF WITT VECTORS OF ALGEBRAIC INTEGERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711003315 ◽

2012 ◽

Vol 86 (3) ◽

pp. 456-460

Author(s):

WILSON ONG

Keyword(s):

Number Theory ◽

Abelian Group ◽

Residue Field ◽

Galois Cohomology ◽

Algebraic Integers ◽

Original Proof ◽

Cambridge Philos ◽

Witt Vectors ◽

Residue Fields ◽

Valuation Field

AbstractLet K be a complete discrete valuation field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G=Gal(L/K) and suppose that the induced extension of residue fields kL/kK is separable. Let 𝕎n(⋅) denote the ring of p-typical Witt vectors of length n. Hesselholt [‘Galois cohomology of Witt vectors of algebraic integers’, Math. Proc. Cambridge Philos. Soc.137(3) (2004), 551–557] conjectured that the pro-abelian group {H1 (G,𝕎n (𝒪L))}n≥1 is isomorphic to zero. Hogadi and Pisolkar [‘On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt’, J. Number Theory131(10) (2011), 1797–1807] have recently provided a proof of this conjecture. In this paper, we provide a simplified version of the original proof which avoids many of the calculations present in that version.

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A note on Sen’s theory in the imperfect residue field case

Mathematische Zeitschrift ◽

10.1007/s00209-010-0726-1 ◽

2010 ◽

Vol 269 (1-2) ◽

pp. 261-280

Author(s):

Shun Ohkubo

Keyword(s):

Residue Field ◽

Field Case

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Local Bounds for Torsion Points on Abelian Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-026-x ◽

2008 ◽

Vol 60 (3) ◽

pp. 532-555 ◽

Cited By ~ 6

Author(s):

Pete L. Clark ◽

Xavier Xarles

Keyword(s):

Abelian Varieties ◽

Residue Field ◽

Rational Numbers ◽

Abelian Surface ◽

Torsion Subgroup ◽

Good Reduction ◽

Torsion Points ◽

Ramification Index ◽

Numerical Invariants ◽

Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

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On value groups and residue fields of some valued function fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018897 ◽

1994 ◽

Vol 37 (3) ◽

pp. 445-454

Author(s):

Sudesh K. Khanduja

Keyword(s):

Cyclic Group ◽

Function Field ◽

Function Fields ◽

Residue Field ◽

Torsion Group ◽

Transcendence Degree ◽

Manuscripta Math ◽

Direct Sum ◽

Residue Fields ◽

The One

Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).

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NOTES ON EXTREMAL AND TAME VALUED FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2016.6 ◽

2016 ◽

Vol 81 (2) ◽

pp. 400-416

Author(s):

SYLVY ANSCOMBE ◽

FRANZ-VIKTOR KUHLMANN

Keyword(s):

Approximation Property ◽

Elementary Theory ◽

Residue Field ◽

Axiom System ◽

Complete Characterization ◽

Valued Fields ◽

Mixed Characteristic ◽

Valued Field ◽

Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

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