Finiteness Results in Descent Theory

It is shown that a -curve of genus g and with stable reduction (in some generalized sense)at every finite place outside a finite set S can be definedover a finite extension L of its field of moduli K dependingonly on g, S and K. Furthermore, there exist L-models that inheritall places of good and stable reduction of the original curve(except possibly for finitely many exceptional places dependingon g, K and S). This descent result yields this moduli formof the Shafarevich conjecture: given g, K and S as above, onlyfinitely many K-points on the moduli space M_g correspond to-curves of genus g and with good reduction outside S. Other applications to arithmetic geometry,like a modular generalization of the Mordell conjecture, aregiven.