On invariants of elliptic curves on average

Amir Akbary ; Adam Tyler Felix

Acta Arithmetica, Tome 168 (2015), p. 31-70 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $AB > x(logx)^δ$.

Détail

Publié le : 2015-01-01

Zbl 1335.11039

EUDML-ID : urn:eudml:doc:279142