ABSTRACT: I will discuss p-adic methods for the construction of curves over number fields whose Jacobian has complex multiplication (CM) and for the computation of zeta functions of abelian varieties over finite fields. My focus is on higher dimensional methods based on the computation of theta constants. A p-adic CM algorithm typically has as input an ordinary abelian variety over a finite field of characteristic p and as output an arithmetic invariant of the canonical lift over a p-adic ring. My talk includes recent results by Gaudry, Houtman, Kohel, Lercier, Lubicz, Mestre, Ritzenthaler, Weng and myself

.