Issue of the Sum of Cubes, Class Groups of Prime, and Functions Based on Ellipses in Ternary Selmer groups

In this dissertation, the relation between Selmer groups over some number fields of certain elliptic curves induced by a rational 3-isogeny and the 3-part of the ideal class groups of some associated number fields is studied. The results obtained in this direction are then applied to study various classical problems in number theory. One of these is a Diophantine problem related to the rational cube sums: which integers can be written as a sum of cubes of two rational numbers? Let