On the class numbers of the fields of the pⁿ-torsion points of certain elliptic curves over Q

2013/07/29 by Fumio Sairaiji, Sairaiji, Fumio, Takuya Yamauchi +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1307.7691

Abstract

Let E be an elliptic curve over Q with prime conductor p. For each non-negative integer n we put K_n:=Q(E[pⁿ]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of K_n. We give a lower bounds in terms of the Mordell-Weil rank of E(\Q). As an application of our result, we give an example such that p²ⁿ divides the class number of the field K_n in the case of p=5077 for each positive integer n.

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