2025/08/07 by Das, Shamik, Jha, Somnath
#11A15 #11D25 #11G05 #11R29 #11R34 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.05361
We consider certain families of integers n determined by some congruence condition, such that the global root number of the elliptic curve E-432n2: Y2=X3-432n2 is 1 for every n, however a given n may or may not be a sum of two rational cubes. We give explicit criteria in terms of the 2-parts and 3-parts of the ideal class groups of certain cubic number fields to determine whether such an n is a cube sum. In particular, we study integers n divisible by 3 such that the global root number of E-432n2 is 1. For example, for a prime ℓ ≡ 7 \pmod9, we show that for 3ℓ to be a sum of two rational cubes, it is necessary that the ideal class group of \Q(√[3]12ℓ) contains (\Z)/(6\Z)⊕ (\Z)/(3\Z) as a subgroup. Moreover, for a positive proportion of primes ℓ ≡ 7 \pmod9, 3ℓ can not be a sum of two rational cubes. A key ingredient in the proof is to explore the relation between the 2-Selmer group and the 3-isogeny Selmer group of E-432n2 with the ideal class groups of appropriate cubic number fields.