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.
Original language | English |
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Article number | 101725 |
Journal | Finite fields and their applications |
Volume | 67 |
DOIs | |
Publication status | Published - Oct 2020 |
Keywords
- Elliptic curves
- Recurrence sequence
- Subspace theorem
ASJC Scopus subject areas
- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics