On p-adic L-functions of elliptic curves and ideal class groups of the division fields

Let E be an elliptic curve defined over Q and F be Q or an imaginary quadratic field with certain conditions. Our research object in this article is the ideal class group Cl(FE) of the p-division field FE := F(E[p]) of E over F for an odd prime number p. More precisely, we investigate the non-vanishing of the $E[p]$-component in the semi-simplification of Cl(FE)/pCl(FE) as an Fₚ[Gal(FE/F)]-module when $E[p]$ is an irreducible Gal(FE /F)-module. When the analytic rank of E over F is 1, we establish a new relationship between the non-vanishing of the $E[p]$-component and the p-divisibility of a certain p-adic analytic quantity associated with E. The quantity is defined by the leading coefficient of the cyclotomic p-adic L-function of E when F = Q and by that of the anticyclotomic p-adic L-function of E when F is the imaginary quadratic field.