We adapt the theory of currents in metric spaces, as developed by the
first-mentioned author in collaboration with B. Kirchheim, to currents with
coefficients in Z_p. Building on S. Wenger's work in the orientable case, we
obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these
inequalities to provide a proof of Gromov's filling radius inequality (and
therefore also the systolic inequality) which applies to nonorientable
manifolds, as well. With this goal in mind, we use the Ekeland principle to
provide quasi-minimizers of the mass mod(p) in the homology class, and use the
isoperimetric inequality to give lower bounds on the growth of their mass in
balls.Comment: 31 pages, to appear in Commentarii Mathematici Helvetic