Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

Let $F$ be a function field of characteristic $p>0$, \F/F a Galois extension with Gal(\F/F)\simeq \Z_l^d (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of \F via appropriate versions of Mazur's Control Theorem. As a consequence we prove that Sel_E(\F)_r is a cofinitely generated (in some cases cotorsion) \Z_r[[Gal(\F/F)]]-module.Comment: Final version to appear in Annales de l'Institut Fourie

Global geometric deformations of current algebras as Krichever-Novikov type algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.Comment: 35 pages, AMS-Late