The Lang-Trotter Conjecture on Average
| dc.creator | Baier, Stephan | |
| dc.date | 2006-09-04 | |
| dc.date.accessioned | 2026-07-07T07:24:29Z | |
| dc.date.available | 2026-07-07T07:24:29Z | |
| dc.description | For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $π_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $π_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+ε}$ and $AB>x^{3/2+ε}$, then the arithmetic mean of $π_E^r(x)$ over all elliptic curves $E$ : $y^2=x^3+ax+b$ with $a,b\in \intz$, $|a|\le A$ and $|b|\le B$ is $\sim C_r\sqrt{x}/\log x$, where $C_r$ is some constant depending on $r$. This improves a result of C. David and F. Pappalardi. Moreover, we establish an ``almost-all'' result on $π_E^r(x)$. | |
| dc.description | 18 pages | |
| dc.identifier | https://arxiv.org/abs/math/0609095 | |
| dc.identifier | http://arxiv.org/abs/math/0609095 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/116370 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05 | |
| dc.title | The Lang-Trotter Conjecture on Average | |
| dc.type | text |