Curves D y^2 = x^3 - x of odd analytic rank
| dc.creator | Elkies, Noam D. | |
| dc.date | 2002-08-07 | |
| dc.date.accessioned | 2026-07-07T04:50:06Z | |
| dc.date.available | 2026-07-07T04:50:06Z | |
| dc.description | For nonzero rational D, which may be taken to be a squarefree integer, let E_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number" problem. It is known that the L-function of E_D has sign -1, and thus odd analytic rank, if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves E_D is odd, and therefore positive. We prove that E_D has positive rank for each D such that |D| is in one of the above congruence classes mod 8 and also satisfies |D|<10^6. Our proof is computational: we use the modular parametrization of E_1 or E_2 to construct a rational point P_D on each E_D from CM points on modular curves, and compute P_D to enough accuracy to usually distinguish it from any of the rational torsion points on E_D. In the 1375 cases in which we cannot numerically distinguish P_D from a torsion point of E_D, we surmise that P_D is in fact a torsion point but that E_D has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D|<10^7 of the list of curves E_D whose analytic rank is odd and greater than 1, which raises several new questions. | |
| dc.description | 8 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0208056 | |
| dc.identifier | http://arxiv.org/abs/math/0208056 | |
| dc.identifier | Pages 244-251 of the proceedings of ANTS-V (LNCS 2369 (2002), ed. C.Fieker and D.Kohel) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64676 | |
| dc.subject | Number Theory | |
| dc.subject | 14H52, 11G15 (Primary) 11Y50, 11Y70 (secondary) | |
| dc.title | Curves D y^2 = x^3 - x of odd analytic rank | |
| dc.type | text |