Arithmetic and reduction of one-dimensional and higher-dimensional Abelian varieties over function fields

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In this project, we shall be concerned with the arithmetic of elliptic curves and higher-dimensional Abelian varieties. Those are geometric objects which have been studied for a long time, particularly since they provide a bridge between arithmetic and geometry. In the present project, all Abelian varieties will be defined over discretely valued fields or function fields in positive characteristic. The research proposal consists of three projects, each of which will shed new light on some aspect of the arithmetic and reduction behaviour of these objects. In the first project, we shall look at the following situation: Let C be a smooth, projective, and geometrically integral curve over a non-perfect field k. Let E be an elliptic curve defined over the function field of C. Then we can consider the minimal proper regular model X of E over C. Then X is a regular surface over k. The goal of the first project is to prove a conjecture which characterises smoothness of such surfaces. Another goal would be proving new results about the behaviour of Néron models under base change.In the second project, we shall also consider elliptic surfaces X, this time defined over a discretely valued field K with algebraically closed residue field. There already exists a criterion which guarantees that X has logarithmic good reduction up to modification; this criterion only uses information contained in the Galois representations on the étale cohomology spaces of X. Such generalisations of the Néron-Ogg-Shafarevich criterion to logarithmic geometry are relatively new, and it is not currently known whether they are sufficient. The goal of this project is the proof of a precise cohomological characterisation of elliptic surfaces over K which have logarithmic good reduction. For certain related surfaces (Kummer surfaces), there already is a precise conjecture, which will probably be accessible using existing methods.For the third project, consider the following situation: Let C be a smooth, projective, and geometrically integral curve, defined this time over a finite field k of characteristic p. Let A be an Abelian variety defined over the function field K of C. If K^sep denotes a separable closure of K, the behaviour of the group A(K^sep) has not been completely understood (as opposed to the group A(K^alg) for an algebraic closure K^alg of K). Recently, Rössler proved that the p-power-torsion subgroup of A(K^sep) can only be infinite if the Néron model of A over C has certain very special properties. In this project, we shall study these properties, with a view towards generalising his newly developed global methods.