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

2020/03/22 by Chim, Kwok Chi, Luca, Florian
#11B39 #11D45 #11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2003.09951

Abstract

Let q be a perfect power of a prime number p and E(\mathbb F_q) be an elliptic curve over \mathbb F_q given by the equation y²=x³+Ax+B. For a positive integer n we denote by # E(\mathbb F_qⁿ) the number of rational points on E (including infinity) over the extension \mathbb F_qⁿ. Under a mild technical condition, we show that the sequence \lbrace # E(\mathbb F_qⁿ) \rbrace_n>0 contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then #E(\mathbb F_qⁿ) is a perfect square for infinitely many n including all the multiples of 24. 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.

Related

• Perfect powers in the product of denominators of elliptic curves
• Elliptic curves maximal over extensions of finite base fields
• Rational points on Cubic, Quartic and Sextic Curves over Finite Fields
• The number of points on an elliptic curve with square x-coordinates
• A method for construction of rational points over elliptic curves II:\n Points over solvable extensions