Isogeny graphs, modular polynomials, and applications

This thesis has three main parts. The first part gives an algorithm to compute Hilbert modular polynomials for ordinary abelian varieties with maximal real multiplication. Hilbert modular polynomials of a given level b give a way of finding all of the abelian varieties that are b-isogeneous to any given abelian varieties satisfying the right conditions. The second part is the proof of a theorem giving the structure of an isogeny graph of simple ordinary abelian varieties with maximal real multiplication. The third part gives a new polynomial time algorithm to count points on genus 2 curves with maximal real multiplication. This algorithm is the fastest known for curves satisfying the right properties.ALGANT-docNumber theory, Algebra and Geometr