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We prove that a K3 surface with an automorphism acting on the global 2-forms by a primitive m-th root of unity, m ≠ 1, 2, 3, 4, 6, does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the l-adic cohomology groups of the surface. [ABSTRACT FROM AUTHOR]

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]

We give a criterion for the good reduction of semistable K3 surfaces over padic fields. We use neither p-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or K3 surfaces. We achieve our goal by realizing the special fiber Xs of a semistable model X of a K3 surface over the p-adic field K, as a special fiber of a log-family in characteristic p and use an arithmetic version of the Clemens-Schmid exact sequence in order to obtain a Kulikov-- Persson--Pinkham classification theorem in characteristic p. [ABSTRACT FROM AUTHOR]

Let K be a two-dimensional global field of characteristic ≠ 2 and let V be a divisorial set of places of K. We show that for a given n ≥ 5, the set of K-isomorphism classes of spinor groups G = Spinn(q) of nondegenerate n-dimensional quadratic forms over K that have good reduction at all v ∈ V is finite. This result yields some other finiteness properties, such as the finiteness of the genus genK(G) and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups Hi(K,μ2)V for i ≥ 1 established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type G2. [ABSTRACT FROM AUTHOR]

Let K be the field of fractions of a local Henselian discrete valuation ring OK of characteristic zero with perfect residue field k. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second l-adic cohomology group of a K3 surface over K implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties. [ABSTRACT FROM AUTHOR]

Let K be a number field and let ϕ in K ( z ) be a rational function of degree d ≥ 2 . Let S be the set of places of bad reduction for ϕ (including the archimedean places). Let Per ( ϕ , K ) , PrePer ( ϕ , K ) , and Tail ( ϕ , K ) be the set of K -rational periodic, preperiodic, and purely preperiodic points of ϕ , respectively. The present paper presents two main results. The first result is a bound for | PrePer ( ϕ , K ) | in terms of the number of places of bad reduction | S | and the degree d of the rational function ϕ . This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that | Per ( ϕ , K ) | ≥ 4 (resp. | Tail ( ϕ , K ) | ≥ 3 ), we prove bounds for | Tail ( ϕ , K ) | (resp. | Per ( ϕ , K ) | ) that depend only on the number of places of bad reduction | S | (and not on the degree d ). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when | Per ( ϕ , K ) | < 4 (resp. | Tail ( ϕ , K ) | < 3 ). [ABSTRACT FROM AUTHOR]

Fix a number field and a rational prime . We consider abelian varieties whose -power torsion generates a pro- extension of which is unramified away from . It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from . In the special case of , we demonstrate that for abelian surfaces , good reduction away from does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from . An explicit example is constructed to demonstrate that good reduction away from is not sufficient, at , for abelian varieties of sufficiently high dimension. [ABSTRACT FROM AUTHOR]

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results of Grothendieck and of Gao. [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]

We prove a Néron-Ogg-Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $$l$$ -adic cohomology group is unramified. We also prove a $$p$$ -adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero. Let K be the rational function field K = k(t). Let ϕ be a nonisotrivial rational function in K(z). We prove a bound for the cardinality of the set of K-rational preperiodic points for ϕ in terms of the number of places of bad reduction and the degree d of ϕ. [ABSTRACT FROM AUTHOR]

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials. [Copyright &y& Elsevier]

Let R be a complete discrete valuation Fq-algebra with fraction field K and perfect residue field k. For an irreducible smooth affine curve C, with field of constants Fq, let M denote a τ-sheaf over CK, endowed with a characteristic morphism ι : Spec K→C. Given a Tate module Tℓ(M) with trivial action of the inertia group IK, we construct a good model M for M over CR. This yields an analog for τ-sheaves of the classical Ne´ron–Ogg–Shafarevicˇ theorem on good reduction of abelian varieties.We can actually extend this result to a criterion for nondegenerate and semistable reduction. As an application, we show how the local L-factor of a τ-sheaf at a place of bad reduction is related to the action of Frobenius on the associated Galois representations. Finally, we discuss the implications of these results to Drinfeld modules and their associated t-motives. [Copyright &y& Elsevier]

We investigate the existence of distinct polynomialsF, G having roots of prescribed multiplicities and deg(F −G) as small as predicted by Mason’sabc theorem. The case of characteristic zero has been treated completely in a previous paper, but those methods do not apply in positive characteristic. Here we study this problem through reduction; it turns out that what we require amounts to proving good reduction for certain covers of the projective line, unramified except above 0, 1, ∞. We shall give sufficient conditions for good reduction of those covers, which sometimes go beyond known criteria due to Grothendieck, Fulton and Beckmann. The methods are completely different from those used by such authors and rely on results by Dwork and Robba onp-adic analytic continuation of Puiseux series. [ABSTRACT FROM AUTHOR]

We apply the ''almost good reduction'' (AGR) criterion, which has been introduced in our previous works, to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlevé III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR.