A Local-global Summation Formula for Abelian Varieties
| dc.creator | Wazir, Rania | |
| dc.date | 2003-02-21 | |
| dc.date | 2003-02-28 | |
| dc.date.accessioned | 2026-07-07T04:55:29Z | |
| dc.date.available | 2026-07-07T04:55:29Z | |
| dc.description | Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil rank of an arbitrary family of Abelian varieties $A$ over a number field $k$; this is the Abelian fibration analogue of the Nagao formula for elliptic surfaces $E$, originally conjectured by Nagao, and proven by Rosen and Silverman to be equivalent to Tate's Conjecture for $E$. We also give a short exact sequence relating the Picard Varieties of the family $A$, the parameter space, and the generic fiber, and use this to obtain an isomorphism (modulo torsion) relating the Neron-Severi group of $A$ to the Mordell-Weil group of $A$. | |
| dc.description | 16 pages; Introduction rewritten, Theorem 3.1 sharpened, references included | |
| dc.identifier | https://arxiv.org/abs/math/0302266 | |
| dc.identifier | http://arxiv.org/abs/math/0302266 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66594 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G40, 14K15 | |
| dc.title | A Local-global Summation Formula for Abelian Varieties | |
| dc.type | text |