Rational points on algebraic curves and its application to number theory

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In this research, I have studied rational points on algebraic curves. The main targets are Fermat curves and elliptic curves.As for Fermat curves, I treated the purity problem on Gauss sums and Jacobi sums which appear in the zeta functions of those curves. I was able to generalize certain known results, and proposed a new conjecture on the purity of Gauss sums. I showed that the conjecture is valid in some cases.As for elliptic curves, I treated special elliptic curves'which are closely related to the congruent number problem. The main result is an explicit formula for the size of the 2-Selmer groups. As far as I know, such an explicit formula has never been obtained in the literature. The key point in the proof is an explicit calculation of the Cassels pairing restricted to a subgroup of the Tate-Shafarevich group.