FINITENESS RESULTS IN DESCENT THEORY

Abstract

It is shown that a\s Q ¯ -curve of genus g and with stable reduction (in some generalized sense) at every finite place outside a finite set S can be defined over a finite extension L of its field of moduli K depending only on g, S and K. Furthermore, there exist L-models that inherit all places of good and stable reduction of the original curve (except possibly for finitely many exceptional places depending on g, K and S). This descent result yields this moduli form of the Shafarevich conjecture: given g, K and S as above, only finitely many K-points on the moduli space Mg correspond to Q ¯ -curves of genus g and with good reduction outside S. Other applications to arithmetic geometry, like a modular generalization of the Mordell conjecture, are given.