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 (Ep) of E over F for an odd prime number p. More precisely, we investigate the non-vanishing of the Ep-component in the semi-simplification of Cl (FE) /pCl (FE) as an FₚGal (FE/F) -module when Ep 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 Ep-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.