UC Irvine

Inspired by recent papers of Mazur-Rubin [8] and Klagsbrun-Mazur-Rubin [6], this thesis

investigates Selmer ranks of twists of Jacobians of various algebraic curves over number field. For example, we find sufficient conditions on hyperelliptic curves C over a number field

such that for any nonnegative integer r, there exist infinitely many quadratic twists of C

whose Jacobians have 2-Selmer ranks equal to r. This theorem is even more generalized to

the superelliptic curve case in this dissertation. We also present some results on 2-Selmer

ranks of elliptic curves. In particular, we prove if the set of 2-Selmer ranks of quadratic

twists of an elliptic curve over a number field contains an integer c, it contains all integers

larger than c having the same parity as c.