Let F be a global field of characteristic p>0, \mathcalF/F a Galois extension with Gal(\mathcalF/F) \simeq \Z_p^\mathbbN and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups Sel_E(L)_l ( l any prime) as L varies through the subextensions of \mathcalF via appropriate versions of Mazur's Control Theorem. In the case l=p we let \mathcalF=\bigcup \mathcalF_d where \mathcalF_d/F is a \mathbbZ_p^d-extension. We prove that Sel_E(\mathcalF_d)_p is a cofinitely generated \mathbbZ_p[[Gal(\mathcalF_d/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in \mathbbZ_p[[Gal(\mathcalF/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.
Control theorems for elliptic curves over function fields
A. Bandini;
2009-01-01
Abstract
Let F be a global field of characteristic p>0, \mathcalF/F a Galois extension with Gal(\mathcalF/F) \simeq \Z_p^\mathbbN and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups Sel_E(L)_l ( l any prime) as L varies through the subextensions of \mathcalF via appropriate versions of Mazur's Control Theorem. In the case l=p we let \mathcalF=\bigcup \mathcalF_d where \mathcalF_d/F is a \mathbbZ_p^d-extension. We prove that Sel_E(\mathcalF_d)_p is a cofinitely generated \mathbbZ_p[[Gal(\mathcalF_d/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in \mathbbZ_p[[Gal(\mathcalF/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
