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Galois actions on Néron models of Jacobians

Lars H. Halle (2010)

Annales de l’institut Fourier

Let X be a smooth curve defined over the fraction field K of a complete discrete valuation ring R . We 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...

Galois covers between K 3 surfaces

Gang Xiao (1996)

Annales de l'institut Fourier

We give a classification of finite group actions on a K 3 surface giving rise to K 3 quotients, from the point of view of their fixed points. It is shown that except two cases, each such group gives rise to a unique type of fixed point set.

Galois orbits and equidistribution: Manin-Mumford and André-Oort.

Andrei Yafaev (2009)

Journal de Théorie des Nombres de Bordeaux

We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.

Galois theory and torsion points on curves

Matthew H. Baker, Kenneth A. Ribet (2003)

Journal de théorie des nombres de Bordeaux

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with “ordinary good” or “ordinary semistable” reduction at a given prime. We also give new proofs of : (1) the Manin-Mumford conjecture : there are only finitely many torsion points lying on a curve of genus at least 2 embedded in its jacobian by an Albanese map; and (2) the...

Galois towers over non-prime finite fields

Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth (2014)

Acta Arithmetica

We construct Galois towers with good asymptotic properties over any non-prime finite field ℓ ; that is, we construct sequences of function fields = (N₁ ⊂ N₂ ⊂ ⋯) over ℓ of increasing genus, such that all the extensions N i / N 1 are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties...

Generalised Hermite constants, Voronoi theory and heights on flag varieties

Bertrand Meyer (2009)

Bulletin de la Société Mathématique de France

This paper explores the study of the general Hermite constant associated with the general linear group and its irreducible representations, as defined by T. Watanabe. To that end, a height, which naturally applies to flag varieties, is built and notions of perfection and eutaxy characterising extremality are introduced. Finally we acquaint some relations (e.g., with Korkine–Zolotareff reduction), upper bounds and computation relative to these constants.

Generalised Weber functions

Andreas Enge, François Morain (2014)

Acta Arithmetica

A generalised Weber function is given by N ( z ) = η ( z / N ) / η ( z ) , where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power N e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating N ( z ) and j(z). Our ultimate goal is the use of these invariants in constructing...

Generalization of Vélu's formulae for isogenies between elliptic curves.

Josep M. Miret Biosca, Ramiro Moreno, Anna Rio (2007)

Publicacions Matemàtiques

Given an elliptic curve E and a finite subgroup G, Vélu's formulae concern to a separable isogeny IG: E → E' with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P+G as the difference between the abscissa of IG(P) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstrass coefficients of E' as polynomials in...

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