UC San Diego

In this dissertation, we present a collection of results regarding the arithmetic of algebraic curves. More specifically, the curves involved are modular curves and elliptic curves. In the first part, we present an algorithm that computes a $p$-adic integration called Coleman integration on modular curves. Different from other methods, our algorithm does not require the knowledge of a model of the curve. The ability of computing such integrals aids the problem of finding rational points on modular curves. In the second part, we consider elliptic curves defined over finite fields of characteristic $p$. In particular, we are interested in supersingular elliptic curves. We first present a work on path-finding on supersingular $\ell$-isogeny graphs using the theory of orientations. We then present a work on counting the number of $\FF_p$-roots of the Hilbert class polynomial $\mathcal{H}_\mathcal{O}(x)$ modulo $p$, in the case when the $\FF_{p^2}$-roots are supersingular $j$-invariants.