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Ordinary reduction of K3 surfaces

Fedor Bogomolov;  Yuri Zarhin — 2009

Open Mathematics

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin — 2004

Bulletin de la Société Mathématique de France

Let K be a field of odd characteristic p , let f ( x ) be an irreducible separable polynomial of degree n ≥ 5 with big Galois group (the symmetric group or the alternating group). Let C be the hyperelliptic curve y 2 = f ( x ) and J ( C ) its jacobian. We prove that J ( C ) does not have nontrivial endomorphisms over an algebraic closure of K if either n ≥ 7 or p ≠ 3 .

Semistable reduction and torsion subgroups of abelian varieties

Alice Silverberg;  Yuri G. Zarhin — 1995

Annales de l'institut Fourier

The main result of this paper implies that if an abelian variety over a field F has a maximal isotropic subgroup of n -torsion points all of which are defined over F , and n ≥ 5 , then the abelian variety has semistable reduction away from n . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its n -torsion points are defined over a field F and n ≥ 3 , then the abelian variety has semistable reduction away from n . We also give information about the Néron models...

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