By Lars Halvard Halle, Johannes Nicaise

Presenting the 1st systematic therapy of the habit of Néron types below ramified base switch, this booklet will be learn as an creation to numerous sophisticated invariants and buildings regarding Néron types of semi-abelian kinds, inspired via concrete examine difficulties and complemented with particular examples.

Néron versions of abelian and semi-abelian forms became an integral device in algebraic and mathematics geometry considering Néron brought them in his seminal 1964 paper. functions diversity from the speculation of heights in Diophantine geometry to Hodge concept.

We concentration particularly on Néron part teams, Edixhoven’s filtration and the bottom switch conductor of Chai and Yu, and we learn those invariants utilizing quite a few recommendations equivalent to types of curves, sheaves on Grothendieck websites and non-archimedean uniformization. We then observe our effects to the examine of motivic zeta services of abelian kinds. the ultimate bankruptcy features a record of not easy open questions. This ebook is aimed in the direction of researchers with a historical past in algebraic and mathematics geometry.

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**Sample text**

G3 is surjective. 5 For every exact sequence 0 ! G1 ! G2 ! G3 ! G/ D every commutative algebraic F-group G. 5. 6 Let ` be a prime invertible in F. Let G1 ! G2 ! 3 Néron Models 27 be a complex of connected smooth commutative algebraic F-groups such that the sequence of Tate modules 0 ! T` G1 ! T` G2 ! T` G3 ! 0 is exact. 3, we may suppose that F is algebraically closed. If we denote by Ui the unipotent radical of Gi , then the morphism T` Gi ! Gi =Ui / is an isomorphism, since multiplication by ` defines an automorphism of Ui .

K/ ! K/ ! K/ ! 0 that we can identify with the sequence 0 ! G1 /ok ! G2 /ok ! G3 /ok ! 5]. 6. 1) Let G be a semi-abelian K-variety, with Néron lft-model G . Let K 0 be a finite extension of K, with valuation ring R0 , and denote by G 0 the Néron lft-model of G0 D G K K 0 . By the universal property of the Néron lft-model, there exists a unique morphism of R0 -group schemes hWG R R0 ! G 0 that extends the natural isomorphism between the generic fibers. It is not an isomorphism, in general. G/ ! 2) One of the principal aims of this monograph is to study the properties of ˛G .

Note that A i will in general not be local. 5 Let regular. NR be part of a regular refinement of i . 8], but the proof that is given there is not complete (one cannot directly carry over the methods from the classical theory of toric varieties over an algebraically closed field, since the equation ' is not homogeneous with respect to the torus action). 4]. 5); this is the only case we used in this chapter. 6) The natural maps Spec A desingularization Fmin i ! Spec A are birational, and glue to give a W XFmin D [i Spec A i !