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Lars Halvard Halle

Publications and source records attributed to Lars Halvard Halle.

13 recordsLinked to original sources
arXiv2024

Base change conductors through intersection theory and quotient singularities

We perform a systematic study of the base change conductor for Jacobians. Through the lens of intersection theory and Deligne's Riemann-Roch theorem, we present novel computational approaches for both the tame and wild parts of the base change conductor. Our key results include a general formula of the tame part, as well as a computation of the wild part in terms of Galois quotients of semistable models of the curves. We treat in detail the case of potential good reduction when the quotient only has weak wild quotient singularities, relying on recent advances by Obus and Wewers.

math.NT↗
arXiv2017

Motivic zeta functions of degenerating Calabi-Yau varieties

We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

math.AG↗
arXiv2016

Tropical Count of Curves on Abelian Varieties

We investigate the problem of counting tropical genus g curves in g-dimensional tropical abelian varieties. For g = 2, 3, we prove that the tropical count matches the count provided by Göttsche, Bryan-Leung, and Lange-Sernesi in the complex setting.

math.AG↗
arXiv2014

A logarithmic interpretation of Edixhoven's jumps for Jacobians

Let $A$ be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the Néron model of $A$ that measures the behaviour of the Néron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where $A$ is the Jacobian of a curve $C$, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of $C$.

math.AG↗
arXiv2012

Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties

This is a survey on motivic zeta functions associated to abelian varieties and Calabi-Yau varieties over a discretely valued field. We explain how they are related to Denef and Loeser's motivic zeta function associated to a complex hypersurface singularity and we investigate the relation between the poles of the zeta function and the eigenvalues of the monodromy action on the tame $\ell$-adic cohomology of the variety. The motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to Calabi-Yau varieties and to compute them explicitly on a model with strict normal crossings.

math.AG↗
arXiv2012

Néron models and base change

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.

math.AG↗
arXiv2011

Jumps and monodromy of abelian varieties

We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the size of the maximal Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.

math.AG↗
arXiv2009

Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety $A$ over a complete discretely valued field, its motivic zeta function has a unique pole at Chai's base change conductor $c(A)$ of $A$, and that the order of this pole equals one plus the potential toric rank of $A$. Moreover, we show that for every embedding of $\Q_\ell$ in $\C$, the value $\exp(2πi c(A))$ is an $\ell$-adic tame monodromy eigenvalue of $A$. The main tool in the paper is Edixhoven's filtration on the special fiber of the Néron model of $A$, which measures the behaviour of the Néron model under tame base change.

math.AG↗
arXiv2009

The Néron component series of an abelian variety

We introduce the Néron component series of an abelian variety $A$ over a complete discretely valued field. This is a power series in $\Z[[T]]$, which measures the behaviour of the number of components of the Néron model of $A$ under tame ramification of the base field. If $A$ is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T=1, whose order equals one plus the potential toric rank of $A$. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if $A$ is an elliptic curve, and if $A$ has potential purely multiplicative reduction.

math.AG↗
arXiv2008

Galois actions on Neron models of Jacobians

Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a 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 $R$, 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 types for curves of genus 1 and 2.

math.AG↗
arXiv2007

Galois actions on models of curves

We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider similar questions for Galois actions on the exceptional locus of a tame cyclic quotient singularity. 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.

math.AG↗
arXiv2007

Stable reduction of curves and tame ramification

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain purely geometric proof of Saito's criterion, avoiding the use of vanishing cycles.

math.AG↗