Solving p-adic matrix differential equations Hendrik Hubrechts, Katholieke Universiteit Leuven, Belgium Let F be the finite field with p^n elements, p prime, and let C be a hyperelliptic (or C_(a,b)) curve defined over F. How can we efficiently compute the number of F-rational points on C? Using Dwork's deformation theory it is possible to reduce this question essentially to the problem of solving a p-adic matrix differential equation. In this talk we discuss some strategies to achieve this reduction, together with various approaches that allow us to solve the resulting equation in an efficient way. One such approach is a new result that leads to a solution of the original problem in time less than cubic in n, thereby improving Kedlaya's algorithm that had complexity Soft-Oh(n^3).