Control theorems for elliptic curves over function fields
| dc.creator | Bandini, A. | |
| dc.creator | Longhi, I. | |
| dc.date | 2006-04-11 | |
| dc.date | 2007-04-05 | |
| dc.date.accessioned | 2026-07-07T07:54:56Z | |
| dc.date.available | 2026-07-07T07:54:56Z | |
| dc.description | Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ 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 $\mathcal F$ via appropriate versions of Mazur's Control Theorem. In the case $l=p$ we let $\mathcal F=\bigcup \mathcal F_d$ where $\mathcal F_d/F$ is a $\mathbb{Z}_p^d$-extension. With a mild hypothesis on $Sel_E(F)_p$ (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that $Sel_E(\mathcal F_d)_p$ is a cofinitely generated (in some cases cotorsion) $\mathbb{Z}_p[[Gal(\mathcal F_d/F)]]$-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic $L$-function associated to $E$ in $\mathbb{Z}_p[[Gal(\mathcal F/F)]]$, providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves. | |
| dc.description | 28 pages. Corrects a number of mistakes in the previous version math.NT/0604249, and formulates a new conjecture | |
| dc.identifier | https://arxiv.org/abs/math/0604249 | |
| dc.identifier | http://arxiv.org/abs/math/0604249 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126803 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05; 11R23 | |
| dc.title | Control theorems for elliptic curves over function fields | |
| dc.type | text |