| This dissertation is mainly concerned with two topics, explicit descents via certain isogenies and quadratic sections on semistable elliptic K3 surfaces.;First, we consider an elliptic curve E with a rational 2- or 3-torsion point T, the isogeny &phis; defined as the quotient map by the group generated with T, and &phis;' the dual of &phis;. Then, we construct matrices M&phis;, M&phis;' calculating the &phis;, &phis;'-Selmer group as their column nullspaces respectively. This approach gives an effective algorithm of performing descent via these isogenies, working considerably faster than the existing methods. We also prove the existence of a duality between M&phis; and M &phis;' via the reciprocity laws, and explore the connections between the matrices and the local and global root numbers of E.;Second, we fix an elliptic fibration psi of a semistable elliptic K3 surface S/K, and construct a one-to-one correspondence between isomorphism classes of quadratic sections and a certain subset of essential lattice modulo an equivalence relation. This lattice theoretic description provides a systematic way of searching for all such quadratic sections of psi : S → P1 . Since such a quadratic section is a smooth rational curve admitting a degree 2 map to the base curve, quadratic base change via quadratic sections immediately gives a way to construct a number of non-isomorphic elliptic curves over K(t) of rank greater than the rank of (S, psi). |
