On invariants of elliptic curves on average

Amir Akbary, and Adam Tyler Felix. "On invariants of elliptic curves on average." Acta Arithmetica 168.1 (2015): 31-70. <http://eudml.org/doc/279142>.

@article{AmirAkbary2015,
abstract = {We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E(p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_p$. Let be the family of elliptic curves $E_\{a,b\} : y^2 = x^3 + ax + b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/|| ∑_\{E∈\} ∑_\{p≤x\} e_E^k(p) = C_k li(x^\{k+1\}) + O((x^\{k+1\})/(logx)^c$)$ $as x → ∞, as long as $A,B > exp(c_1 (logx)^\{1/2\})$ and $AB > x(logx)^\{4+2c\}$, where $c_1$ is a suitable positive constant. Here $C_k$ is an explicit constant given in the paper which depends only on k, and $li(x) = ∫_\{2\}^x dt/log\{t\}$. We prove several similar results as corollaries to a general theorem. The method of the proof is capable of improving some of the known results with $A,B > x^ϵ$ and $AB > x(logx)^δ$ to $A,B > exp(c_1 (logx)^\{1/2\})$ and $AB > x(logx)^δ$.},
author = {Amir Akbary, Adam Tyler Felix},
journal = {Acta Arithmetica},
keywords = {reduction mod of elliptic curves; invariants of elliptic curves; average results},
language = {eng},
number = {1},
pages = {31-70},
title = {On invariants of elliptic curves on average},
url = {http://eudml.org/doc/279142},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Amir Akbary
AU - Adam Tyler Felix
TI - On invariants of elliptic curves on average
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 1
SP - 31
EP - 70
AB - We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E(p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_p$. Let be the family of elliptic curves $E_{a,b} : y^2 = x^3 + ax + b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/|| ∑_{E∈} ∑_{p≤x} e_E^k(p) = C_k li(x^{k+1}) + O((x^{k+1})/(logx)^c$)$ $as x → ∞, as long as $A,B > exp(c_1 (logx)^{1/2})$ and $AB > x(logx)^{4+2c}$, where $c_1$ is a suitable positive constant. Here $C_k$ is an explicit constant given in the paper which depends only on k, and $li(x) = ∫_{2}^x dt/log{t}$. We prove several similar results as corollaries to a general theorem. The method of the proof is capable of improving some of the known results with $A,B > x^ϵ$ and $AB > x(logx)^δ$ to $A,B > exp(c_1 (logx)^{1/2})$ and $AB > x(logx)^δ$.
LA - eng
KW - reduction mod of elliptic curves; invariants of elliptic curves; average results
UR - http://eudml.org/doc/279142
ER -