Ph.D. thesis (April 2008).
In Part I, we present a new algorithm for computing the zeta function of a hyperelliptic curve over a finite field, based on Kedlaya's approach via p-adic cohomology. It is the first known algorithm for this task whose time complexity is polynomial in the genus of the curve and quasilinear in the square root of the characteristic of the base field. In Part II, we study and improve the Mazur–Stein–Tate algorithm for computing the p-adic height of a rational point on an elliptic curve E/Q, where p ≥ 5 is a prime of good ordinary reduction for E.
(Note: this thesis has essentially the same content as the papers Kedlaya's algorithm in larger characterstic and Efficient computation of p-adic heights, plus some algorithmic improvements.)