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 behavior 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.     

Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.     

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 with algebraically closed residue 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? in C, the value exp(2πic(A)) is an ?-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 behavior of the Néron model under tame base change.     

Grothendieck's pairing for Jacobians and base change

We explicitly describe Grothendieck's pairing on the component group of the Néron model of the Jacobian of a curve using the intersection matrix of a regular proper model of the curve in some new cases where the curve does not have a K-rational point.     

Finiteness theorems in the class field theory of varieties over local fields

We show that the geometrical part of the abelian étale fundamental group of a proper smooth variety over a local field is finitely generated over with finite torsion, and describe its rank by the special fiber of the Néron model of the Albanese variety. As an application, we complete the class field theory of curves over local fields developed by Bloch and Saito, in which the theorem concerning the p-primary part in the positive characteristic case has remained unproven.     

An explicit semi-factorial compactification of the Néron model

C. Pépin recently constructed a semi-factorial compactification of the Néron model of an Abelian variety using the flattening technique of Raynaud–Gruson. Here we prove that an explicit semi-factorial compactification is a certain moduli space of sheaves — the family of compactified Jacobians.     

Semi-stable models for curves with cusps

Let R be a complete discrete valuation ring with residue characteristic zero, and let X be an integral regular flat curve over R with smooth generic fiber. Assume that the special fiber of X is smooth outside a single point where it has a cusp as singularity. We explicitly determine the structure of the minimal semi-stable model of X. In particular, we give an algebraic proof for the fact that the special fiber of any semi-stable model of X is treelike. This is equivalent to the finiteness of the monodromy of X over R. These two results were obtained in the 1970's by Lê Dung Tráng and A. Durfee using analytic methods. Received: 23 April 2001 / Published online: 8 November 2002 RID="*" ID="*" This research was made possible by the support of the Deutsche Forschungsgemeinschaft in form of grant LU 224/5-1.     

Purity for Hodge-Tate representations

We prove that a kind of purity holds for Hodge-Tate representations of the fundamental group of the generic fiber of a semi-stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field. As an application, we see that the relative p-adic étale cohomology with proper support of a scheme separated of finite type over the generic fiber is Hodge-Tate if it is locally constant.     

Local heights on abelian varieties with split multiplicative reduction

We express Néron functions and Schneider's local p-adic height pairing on an abelian variety A with split multiplicative reduction with theta functions and their automorphy factors on the rigid analytic torus uniformizing A.Moreover, we show formulas for the -splittingsof the Poincaré biextension corresponding to Néron's and Schneider's local height pairings.     

The Igusa local zeta functions of elliptic curves

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the Kodaira-Néron classification of the special fibers of elliptic curves as determined by Tate's algorithm.

     

Links between associated additive Galois modules and computation of H for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates-Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.     

Algebraic solutions of the Lamé equation, revisited

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S₄, then the degree parameter of the equation may differ by ±1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.     

Extension of elliptic curves on Krasner hyperfields

The aim of the article is to initiate a new study in the framework of algebraic hyperfields, with an applicative impact in cryptography. First we define the notions of generalized Weierstrass equation and elliptic hypercurve on Krasner hyperfields, as a generalization of the notion of elliptic curve on fields. Then we investigate properties of the hypergroups derived from elliptic hypercurves and of the associated H_v-groups. Finally, we present a class of canonical hypergroups, which can be used as an alphabet in a special cryptographic system.     

The Discrete Relativistic Toda Molecule Equation and a Padé Approximation Algorithm

The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szegö polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Padé approximation algorithm for a given power series.     

Singular hyperelliptic curves

Singular curves with a morphism of degree two onto a projective line should be classified into two types according as the equipped morphism is separable or not; we call a curve with a separable one a hyperelliptic curve of separable type, and the other a hyperelliptic curve of inseparable type. We give concrete expressions of a hyperelliptic curve of separable type by means of its global “equation” and a hyperelliptic curve of inseparable type by means of its local rings. Furthermore, we discuss about Weierstrass points of a hyperelliptic curve of inseparable type. Received: 26 March 1997 / Revised version: 21 May 1998     

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Let be a hyperelliptic curve of genus over a discrete valuation field . In this article we study the models of over the ring of integers of . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of over dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

     

Algebraic convergence for anisotropic edge elements in polyhedral domains

We study approximation errors for the h-version of Nédélec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error estimates for functions in anisotropic weighted Sobolev spaces. Here we consider not only the H(curl)-conforming Nédélec elements, but also the H(div)-conforming Raviart-Thomas elements which appear naturally in the discrete version of the de Rham complex. Our technique is to transport error estimates from the reference element to the physical element via highly anisotropic coordinate transformations. Second, Galerkin error estimates for the standard H(curl) approximation of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the solution on domains with three-dimensional edges and corners. We also prove the discrete compactness property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those of [40] to the case of polyhedral corners and higher order elements.     

Generalised soft binomial American real option pricing model (fuzzy–stochastic approach)

The stochastic discrete binomial models and continuous models are usually applied in option valuation. Valuation of the real American options is solved usually by the numerical procedures. Therefore, binomial model is suitable approach for appraising the options of American type. However, there is not in several situations especially in real option methodology application at to disposal input data of required quality. Two aspects of input data uncertainty should be distinguished; risk (stochastic) and vagueness (fuzzy). Traditionally, input data are in a form of real (crisp) numbers or crisp-stochastic distribution function. Therefore, hybrid models, combination of risk and vagueness could be useful approach in option valuation. Generalised hybrid fuzzy–stochastic binomial American real option model under fuzzy numbers (T-numbers) and Decomposition principle is proposed and described. Input data (up index, down index, growth rate, initial underlying asset price, exercise price and risk-free rate) are in a form of fuzzy numbers and result, possibility-expected option value is also determined vaguely as a fuzzy set. Illustrative example of equity valuation as an American real call option is presented.     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

Weierstrass preparation and algebraic invariants

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period–index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan’s Lemma.