Root numbers and ranks in positive characteristic
| dc.creator | Conrad, B. | |
| dc.creator | Conrad, K. | |
| dc.creator | Helfgott, H. | |
| dc.date | 2004-08-11 | |
| dc.date | 2005-06-07 | |
| dc.date.accessioned | 2026-07-07T05:11:12Z | |
| dc.date.available | 2026-07-07T05:11:12Z | |
| dc.description | For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = kappa(u) over any finite field kappa with odd characteristic, we construct an explicit 2-parameter family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d} has elevated rank. | |
| dc.description | 40 pages; last version; to appear in Adv. Math | |
| dc.identifier | https://arxiv.org/abs/math/0408153 | |
| dc.identifier | http://arxiv.org/abs/math/0408153 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72164 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G05, 11G40 | |
| dc.title | Root numbers and ranks in positive characteristic | |
| dc.type | text |