Number Theory Seminar: Alan Hertgen

Title: Splitting properties of the reduction of semi-abelian varieties

Abstract:

Let G/K be a semi-abelian variety over a discrete valuation field. The special fiber of the Neron model of G/K is an extension of the connected component of 0 by the group of components. We say that G/K has split reduction if this extension is split. We will recall some results obtained by Liu and Lorenzini and talk about some generalizations of these. For instance, whereas G/K has always split reduction if the residue characteristic is 0, we prove that it is no longer the case if the residue characteristic is> 0 even if G/K is tamely ramified. If J/K is the Jacobian variety of a smooth proper and geometrically connected curve C/K of genus g, we prove that for any tamely ramified extension M/K of degree greater than a constant, depending on g only, J_M/M has split reduction.