On a family of elliptic curves of rank at least 2

2022/01/02 by Chakraborty, Kalyan, Sharma, Richa
#FOS: Mathematics #Number Theory (math.NT) #Primary: 11G05 #Secondary: 14G05

paper · doi:10.48550/arxiv.2201.00310

Abstract

Let C_m : y² = x³ - m²x +p²q² be a family of elliptic curves over ℚ, where m is a positive integer and p, q are distinct odd primes. We study the torsion part and the rank of C_m(ℚ). More specifically, we prove that the torsion subgroup of C_m(ℚ) is trivial and the ℚ-rank of this family is at least 2, whenever m \not ≡ 0 \pmod 4 and m ≡ 2 \pmod 64 with neither p nor q divides m.

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