Picard curves over Q with good reduction away from 3

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.Comment: 27 pages; A previous lemma was incorrect and has been removed; Corrected computation has identified 18 new such curves (63 in total

Good reduction of Fano threefolds and sextic surfaces

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the Shafarevich conjecture for smooth sextic surfaces, which appears to be the first non-trivial result in the literature on the arithmetic of such surfaces. Moreover, we exhibit certain moduli stacks of Fano varieties which are not hyperbolic, which allows us to show that the analogue of the Shafarevich conjecture does not always hold for Fano varieties. Our results also provide new examples for which the conjectures of Campana and Lang-Vojta hold.Comment: 22 pages. Minor change

Rational points on K3 surfaces and derived equivalence

We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.Comment: 30 page