Ideal class groups of division fields of elliptic curves and everywhere unramified rational points

2023/04/11 by Dainobu, Naoto
#11G05 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R29 #Secondary 11R34

paper · doi:10.48550/arxiv.2304.05035

Abstract

Let E be an elliptic curve over ℚ, p an odd prime number and n a positive integer. In this article, we investigate the ideal class group Cl(ℚ(E[pⁿ])) of the pⁿ-division field ℚ(E[pⁿ]) of E. We introduce a certain subgroup E(ℚ)_ur,pⁿ of E(ℚ) and study the p-adic valuation of the class number #Cl(ℚ(E[pⁿ])). In addition, when n = 1, we further study Cl(ℚ(E[p])) as a Gal(ℚ(E[p])/ℚ)- module. More precisely, we study the semi-simplification (Cl(ℚ(E[p]))⊗ ℤ_p)^ss of Cl(ℚ(E[p]))⊗ ℤ_p as a ℤ_p[Gal(ℚ(E[p])/ℚ)]-module. We obtain a lower bound of the multiplicity of the E[p]-component in the semi-simplification when E[p] is an irreducible Gal(ℚ(E[p])/ℚ)-module.

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