Elliptic analogue of irregular prime numbers for the pⁿ-division fields of the curves y² = x³-(s⁴+t²)x

2022/05/18 by Dainobu, Naoto, Hirakawa, Yoshinosuke, Matsumura, Hideki
#11G05 (Primary) 11R29 #11R34 #14G40 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.08946

Abstract

A prime number p is said to be irregular if it divides the class number of the p-th cyclotomic field ℚ(ζ_p) = ℚ(\mathbbG_m[p]). In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number p ≥ 5 and a positive integer n, we study the p-divisibility of the class number of the pⁿ-division field ℚ(E[pⁿ]) of an elliptic curve E of the form y² = x³-(s⁴+t²)x. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that they are of Mordell-Weil rank 1 and the class numbers of their pⁿ-division fields are divisible by p²ⁿ. Moreover, we can prove that these division fields are not isomorphic to each other. In our construction, we use recent results obtained by the first author.

Related

• The splitting of primes in division fields of elliptic curves
• Local torsion primes and the class numbers associated to an elliptic curve over ℚ
• On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility
• Ideal class groups of division fields of elliptic curves and everywhere unramified rational points
• Uniform bounds on the level of cyclotomic division fields of elliptic curves