Finiteness Results in Descent Theory

Abstract

It is shown that a 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 M g 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.

Néron models and the arithmetic Shafarevich conjecture for certain canonically polarized surfaces

“…Other applications include Ullmo's finiteness theorem for S-integral points on Shimura varieties of abelian type [35,Théorème 3.2] and the Mordell conjecture [9]. Also, in [5], Faltings's finiteness theorems are used to establish field of moduli analogues of the Shafarevich conjecture for curves and abelian varieties. Finally, the connection of the Shafarevich conjecture to the Tate conjectures is explained in [1,9,36].…”