Espaces Homogenes Principaux et Points de 2-Division de Courbes Elliptiques

Dans cet article, nous nous intéressons aux problèmesde structure galoisienne associés aux couples (E/L, a),où E désigne une courbe elliptique définiesur le corps de nombres L et a un endomorphisme de E. Si E està multiplication complexe, nous notons k le corps quadratiqueimaginaire dont l'anneau des entiers décrit End(E) etL une extension de k. Ainsi a est-il selon les cas un entieralgébrique de k ou un élément de Z.     

The Denef-Loeser Zeta Function is not a Topological Invariant

An example is given which shows that the Denef–Loeserzeta function (usually called the topological zeta function)associated to a germ of a complex hypersurface singularity isnot a topological invariant of the singularity. The idea isthe following. Consider two germs of plane curves singularitieswith the same integral Seifert form but with different topologicaltype and which have different topological zeta functions. Makea double suspension of these singularities (consider them ina 4-dimensional complex space). A theorem of M. Kervaire andJ. Levine states that the topological type of these new hypersurfacesingularities is characterized by their integral Seifert form.Moreover the Seifert form of a suspension is equal (up to sign)to the original Seifert form. Hence these new singularitieshave the same topological type. By means of a double suspensionformula the Denef–Loeser zeta functions are computed forthe two 3-dimensional singularities and it is verified thatthey are not equal.