Volume 16, Issue 6

We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L -functions).

We consider a minimizing sequence for the conformal energy in a given homotopy class of maps between two compact Riemannian manifolds M and N . In general this sequence will fail to be (strongly) convergent in the natural Sobolev class, but will have a weak limit which is not a priori in the original homotopy class. We prove a topological decomposition theorem: the homotopy class of the original map is given as the composition (in an appropriate sense) of the homotopy class of the weak limit with a finite number of free homotopy classes of maps from the sphere (with dimension that of the manifold M ) into N . The method of proof shows that the weak limit is a minimizer in its homotopy class, and also shows that the homotopy classes of maps from the sphere occurring in the decomposition can be represented by minimizers in their respective classes.

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem.

Necessary and sufficient conditions for flatness of control systems are obtained by using geometry of differential equations and deformation theory. The necessary conditions coincide with sufficient ones if some regularity condition holds. An example is considered to illustrate the approach.

Recall that for a function field F over an algebraically closed field the gonality of F is defined as the minimal index of a rational subfield. For n ε IF q [ T ] we derive a lower bound for the gonality of the Drinfeld modular curve X 0 (n). Then for Drinfeld IF q [ T ]-modules ɸ of rank 2 on a function field F we discuss explicit uniform bounds (in terms of the gonality of F ) for the F -rational torsion of ɸ . We also complete the existing analogous results for elliptic curves over function fields by bounding the p -primary torsion in characteristic p .