On invariants of elliptic curves on average

• Autorzy: Amir Akbary , Adam Tyler Felix
• Języki publikacji: EN

Abstrakty

EN

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 $𝔽_p$. Let 𝓒 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,
$1/|𝓒| ∑_{E∈𝓒} ∑_{p≤x} e_E^k(p) = C_k li(x^{k+1}) + O((x^{k+1})/(logx)^c})$
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 $AB > x(logx)^δ$.

Kategorie tematyczne

• 11G20: Curves over finite and local fields
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 168
• Numer: 1
• Strony: 31-70

Daty

wydano

2015

Twórcy

autor

Amir Akbary

• Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB, T1K 3M4, Canada

autor

Adam Tyler Felix

• Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB, T1K 3M4, Canada

Identyfikatory

DOI

10.4064/aa168-1-3