Galois representations, Mumford-Tate groups and good reduction of abelian varieties

2003/02/28 by Frederic Paugam · 1 citation
Mathematics · #math.NT #math.AG #msc:11Gxx #msc:14G35

paper · pdf

published as Math. Ann. 329, 119-160 (2004) · Minor modifications and corrections

arxiv created 2004/09/08 · arxiv updated 2009/11/30

Abstract

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without non trivial endomorphisms (Mumford's example, not of PEL type). We give new results for abelian varieties not of PEL type.

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• Arithmetic monodromy of hyper-Kähler varieties over p-adic fields

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