FINITENESS RESULTS IN DESCENT THEORY

Abstract

It is shown that a $\scriptstyle \overline {\mathbb Q}$-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 ${\cal M}_g$ correspond to $\scriptstyle \overline {\mathbb Q}$-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.