Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Kwok Chi Chim*, Florian Luca

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

Abstract

Let q be a perfect power of a prime number p and E(Fq) be an elliptic curve over Fq given by the equation y2=x3+Ax+B. For a positive integer n we denote by #E(Fqn) the number of rational points on E (including infinity) over the extension Fqn. Under a mild technical condition, we show that the sequence {#E(Fqn)}n>0 contains at most 10200 perfect squares. If the mild condition is not satisfied, then #E(Fqn) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q<50 and n≤1000.

Originalspracheenglisch
Aufsatznummer101725
FachzeitschriftFinite fields and their applications
Jahrgang67
DOIs
PublikationsstatusVeröffentlicht - Okt 2020

ASJC Scopus subject areas

  • !!Theoretical Computer Science
  • !!Algebra and Number Theory
  • !!Engineering(all)
  • Angewandte Mathematik

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