Title: Arithmetic on hyperelliptic Jacobians and singular curves Abstract: Mumford representation encodes each divisor class in the Jacobian of a nonsingular hyperelliptic curve as a unique pair of polynomials. And Cantor’s Algorithm provides an implementation of the addition of two divisor classes in this setting. In my thesis, I extended these ideas to study the arithmetic on Jacobians of…

Title: Admmissible sets for Erdos sieves Abstract: When considering a set of pairwise coprime numbers b_1,b_2,... the admissible sets, those which don't contain every congruence class mod b_i for any i, correspond to the closure of the shift {F+n} with n an integer and F those numbers not divisible by any b_i. When considering a generalization for Erdos sieves, this does not hold anymore. In this…