Algorithms for p-adic cohomology and p-adic heights

n 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.