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Given a hyperelliptic curve C of genus g over a number field K and a Weierstrass model \mathscr {C} of C over the ring of integers \mathcal {O}_K (i.e. the hyperelliptic involution of C extends to \mathscr {C} and the quotient is a smooth model of \mathbb {P}^1_K over \mathcal {O}_K), we give necessary and sometimes sufficient conditions for \mathscr {C} to be defined by a global Weierstrass equation. In particular, if C has everywhere good reduction, we prove that it is defined by a global integral Weierstrass equation with invertible discriminant if the class number h_K is prime to 2(2g+1), confirming a conjecture of M. Sadek. [ABSTRACT FROM AUTHOR]

Let S be a Dedekind scheme with field of functions K. We show that if XK is a smooth connected proper curve of positive genus over K, then it admits a Néron model over S, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of XK over S, as in the case of elliptic curves. When S is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models. [ABSTRACT FROM AUTHOR]

Given a number field K and a finite set S of places of K, the first main result of this paper shows that the quadratic rational maps ... defined over K which have good reduction at all places outside S form a Zariskidense subset of the moduli space M2 parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a Zariski nondensity result for the set of such maps with good reduction outside S. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure. [ABSTRACT FROM AUTHOR]