A refined version of the Lang-Trotter Conjecture
| dc.creator | Baier, Stephan | |
| dc.creator | Jones, Nathan | |
| dc.date | 2008-01-25 | |
| dc.date | 2008-10-26 | |
| dc.date.accessioned | 2026-07-07T10:12:48Z | |
| dc.date.available | 2026-07-07T10:12:48Z | |
| dc.description | Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where $r$ is a {\it fixed} integer. However, as shown in this note, this asymptotic estimate cannot hold for {\it all} $r$ in the interval $|r|\le 2\sqrt{x}$ with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the "semicircular law", to an asymptotic formula that conjecturally holds for arbitrary integers $r$ in the interval $|r|\le 2\sqrt{x}$, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture. | |
| dc.description | 10 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0801.3946 | |
| dc.identifier | http://arxiv.org/abs/0801.3946 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/172306 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05 | |
| dc.title | A refined version of the Lang-Trotter Conjecture | |
| dc.type | text |