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

A proof of a conjecture of Y. Morita

“…Since this construction is an equivalence of suitable additive categories, for each place λ|p, we get the 'λ-component' N λ,ψ : V λ (A)(1) → V λ (A). Then, for a place λ 0 as in our assumption, Paugam's arguments in [15,p. 130], which deduces N p,ψ = N = 0 from the assumption of the non-existence of the unipotent Q p -rational element of index 2, work without change to show that N λ0,ψ and thus N λ0 as well are the zero maps.…”

“…If G A has no Q-rational unipotent element other than 1, then A has potentially good reduction at every finite prime of E. This was first conjectured by Morita [12] in the PEL-type setting (that is, (G A , {h A }) defines a PEL-type Shimura datum in the sense of Kottwitz [9]), and proved by himself in certain cases; consequently, we refer to this conjecture as the Morita conjecture from now on. Recently, Noot [13], Paugam [15], Vasiu [24] and Lan [10] proved the conjecture in more general situations. Paugam, among other things, generalized the well-known criterion on good reduction of abelian varieties over local fields of residue characteristic p in terms of the Galois representation on the l( = p)-adic Tate module to the p-adic case.…”

“…). Now, a theorem of Paugam[15, Proposition 4.1.2] says that, for two abelian varieties A and B related to each other by this condition, A has potentially good reduction at a prime p of F if and only if B has the same property; to be precise, Paguam assumes that the Mumford-Tate groups of A and B are simple and non-commutative (his original assumption was that A, B are simple as abelian varieties, which is however not a correct one; cf. [15, Erratum (2005)]), but one easily sees that this hypothesis is dispensable in his argument.…”

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

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This is an erratum to the article [Pau04].The conceptual mistake underlying this errata is the following: the fact that a simple abelian variety has non-commutative Mumford-Tate group does not imply (on contrary to what we wrote at some point in the article) that its adjoint Mumford-Tate group is simple. This was explained to the author by Laurent Clozel and Rutger Noot.

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Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

“…2.7] we argue that the normalization of in is a projective ‐scheme provided the Morita conjecture holds for all abelian varieties over number fields. We recall from [47, 57], and [33], that the Morita conjecture predicts that each abelian variety over a number field with the property that a pullback of it over has a Mumford–Tate group whose adjoint has ‐rank 0, has potentially good reduction everywhere. As the Morita conjecture holds (see [33]), we get that is a projective ‐scheme.…”