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Local symplectic algebra of quasi-homogeneous curves

Wojciech Domitrz (2009)

Fundamenta Mathematicae

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a 𝕂-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of...

Local-global principle for Witt equivalence of function fields over global fields

Przemyslaw Koprowski (2002)

Colloquium Mathematicae

We examine the conditions for two algebraic function fields over global fields to be Witt equivalent. We develop a criterion solving the problem which is analogous to the local-global principle for Witt equivalence of global fields obtained by R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland [12]. Subsequently, we derive some immediate consequences of this result. In particular we show that Witt equivalence of algebraic function fields (that have rational places) over global fields implies...

Local-to-global extensions of representations of fundamental groups

Nicholas M. Katz (1986)

Annales de l'institut Fourier

Let K be a field of characteristic p > 0 , C a proper, smooth, geometrically connected curve over K , and 0 and ∞ two K -rational points on C . We show that any representation of the local Galois group at ∞ extends to a representation of the fundamental group of C - { 0 , ∞ } which is tamely ramified at 0, provided either that K is separately closed or that C is P 1 . In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group...

Loi de réciprocité quadratique dans les corps quadratiques imaginaires

Abdelmejid Bayad (1995)

Annales de l'institut Fourier

À partir d’une courbe elliptique définie sur le corps des classes de Hilbert d’un corps quadratique imaginaire K et à multiplicité complexe par l’anneau des entiers de K , on construit des fonctions elliptiques. Nous établissons des formules produits relatives à ces fonctions. De ce fait, nous obtenons une formulation analytique du lemme de Gauss généralisé ainsi qu’une expression explicite pour le symbole quadratique de Legendre défini sur l’anneau des entiers du corps quadratique imaginaire. Comme...

Loop groups, elliptic singularities and principal bundles over elliptic curves

Stefan Helmke, Peter Slodowy (2003)

Banach Center Publications

There is a well known relation between simple algebraic groups and simple singularities, cf. [5], [28]. The simple singularities appear as the generic singularity in codimension two of the unipotent variety of simple algebraic groups. Furthermore, the semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group. The aim of these notes is to extend this kind of relation to loop groups and simple elliptic singularities. It is the...

Lower bounds on the class number of algebraic function fields defined over any finite field

Stéphane Ballet, Robert Rolland (2012)

Journal de Théorie des Nombres de Bordeaux

We give lower bounds on the number of effective divisors of degree ≤ g - 1 with respect to the number of places of certain degrees of an algebraic function field of genus g defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.

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