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

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 [ p n ])) of the p n -division field Q (E [ p n ]) of E. We introduce a certain subgroup E (Q) ur , p n of E (Q) and study the p -adic valuation of the class number # Cl (Q (E [ p n ])). 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. [ABSTRACT FROM AUTHOR]