Stable reduction of curves and tame ramification

Halle, Lars Halvard

doi:10.1007/s00209-009-0528-5

Cited by 18 publications

(45 citation statements)

References 11 publications

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“…-This is straightforward from the universal properties of the normalization and of the minimal desingularization. For a detailed proof, we refer to [10].…”

Section: )) That Is the Exceptional Locus Is Mapped Into Itself Undmentioning

confidence: 99%

“…When Assumptions 1 and 2 are valid, the following facts can be proved using the computations in [10]:…”

Section: With This Assumption We Can Find An Isomorphismmentioning

confidence: 99%

“…Let n be the degree of R /R, which by assumption is relatively prime to m 1 and m 2 . In the discussion that follows we will use some properties that were proved in [10].…”

Section: 1mentioning

confidence: 99%

“…In the case where a tamely ramified extension suffices one can do this by considering the geometry of suitable regular models for X over S (cf. [10]). In this paper we study, among other things, how the geometry of the Néron model contains information that is relevant for obtaining semistable reduction for X. extension of degree n, one can consider the filtration…”

Section: Introductionmentioning

confidence: 99%

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Galois actions on Néron models of Jacobians

Halle¹

2010

Ann. inst. Fourier

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We study group actions on regular models of curves. If X is a smooth curve defined over the fraction field K of a complete discrete valuation ring R, every tamely ramified extension K ′ /K with Galois group G induces a G-action on the extension X K ′ of X to K ′ . In this paper we study the extension of this G-action to certain regular models of X K ′ . In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups.We apply these results to study a natural filtration of the special fiber of the Néron model of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over Spec(R), and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2.

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“…-This is straightforward from the universal properties of the normalization and of the minimal desingularization. For a detailed proof, we refer to [10].…”

Section: )) That Is the Exceptional Locus Is Mapped Into Itself Undmentioning

confidence: 99%

“…When Assumptions 1 and 2 are valid, the following facts can be proved using the computations in [10]:…”

Section: With This Assumption We Can Find An Isomorphismmentioning

confidence: 99%

“…Let n be the degree of R /R, which by assumption is relatively prime to m 1 and m 2 . In the discussion that follows we will use some properties that were proved in [10].…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Galois actions on Néron models of Jacobians

Halle¹

2010

Ann. inst. Fourier

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“…Our approach is similar in spirit to the one in Saito's paper [Sa87], but our proof substantially simplifies the combinatorial analysis of C s . For different proofs of Saito's criterion, see [Sa04,St05] (using logarithmic geometry) and [Ha10] (using a geometric analysis of the behaviour of sncd-models under base change). For a survey on the semi-stable reduction theorem for curves, see [Ab00].…”

Section: Introductionmentioning

confidence: 99%

Geometric criteria for tame ramification

Nicaise

2012

Math. Z.

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Abstract. We prove an A'Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the ℓ-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito's criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito's criterion to arbitrary dimension.

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Néron Models and Base Change

Halle

Nicaise²

2016

Lecture Notes in Mathematics

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Abstract. We study various aspects of the behaviour of Néron models of semi-abelian varieties under finite extensions of the base field, with a special emphasis on wildly ramified Jacobians. In Part 1, we analyze the behaviour of the component groups of the Néron models, and we prove rationality results for a certain generating series encoding their orders. In Part 2, we discuss Chai's base change conductor and Edixhoven's filtration, and their relation to the Artin conductor. All of these results are applied in Part 3 to the study of motivic zeta functions of semi-abelian varieties. Part 4 contains some intriguing open problems and directions for further research. The main tools in this work are non-archimedean uniformization and a detailed analysis of the behaviour of regular models of curves under base change.

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Stable reduction of curves and tame ramification

Cited by 18 publications

References 11 publications

Galois actions on Néron models of Jacobians

Galois actions on Néron models of Jacobians

Geometric criteria for tame ramification

Néron Models and Base Change

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