On the image of Galois lll-adic representations for abelian varieties of type III (original) (raw)
A class of p-adic Galois representations arising from abelian varieties over
Mathematische Annalen, 2005
Let V be a p-adic representation of the absolute Galois group G of Qp that becomes crystalline over a finite tame extension, and assume p = 2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module Vp(A) of an abelian variety A defined over Qp. These conditions are stated on the filtered (ϕ, G)module attached to V .
A support problem for the intermediate Jacobians of l-adic representations
Eprint Arxiv Math 0212420, 2002
This is a revised version of ANT-0332: "A support problem for the intermediate Jacobians of l-adic representations", by G. Banaszak, W. Gajda & P. Krason, which was placed on these archives on the 29th of January 2002. Following a suggestion of the referee we have subdivided the paper into two separate parts: "Support problem for the intermediate Jacobians of l-adic representations", and "On Galois representations for abelian varieties with complex and real multiplications". Our results on the image of Galois and the Mumford-Tate conjecture for some RM abelian varieties are contained in the second paper. Both papers were accepted for publication.
On the modularity level of modular abelian varieties over number fields
Journal of Number Theory, 2010
Let f be a weight two newform for Γ 1 (N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res L/Q (B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N L (B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N L (B) belongs to Z and N L (B) f dim B L
Abelian Varieties and the Mordell{Lang Conjecture
Model Theory Algebra and Geometry 2000 Isbn 0 521 78068 3 Pags 199 227, 2000
This is an introductory exposition to background material useful to appreciate various formulations of the Mordell-Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (for example, via the method of Chabauty) relevant to Mordell-Lang. The article turns technical at one point (the step in the proof of the Mordell-Lang Conjecture in characteristic zero which passes from number fields to general fields). Two different procedures are sketched for doing this, with more details given than are readily found in the literature. There is also some discussion of issues of effectivity.
Abelian varieties,l-adic representations and Lie algebras
Inventiones Mathematicae, 1979
Lie(Im p~) = ~-1 Lie(Im p2) r 0.1. If K is a global field, then the representations above constitute a strictly compatible system of integral /-adic representations of K. The set of places at which X has bad reduction is the exceptional set of this system (I-11]). One conjectures ([1, 10]) that Lie(Imp3 "does not depend" on l, i.e., that there exists a Lie algebra go over Q, such that Lie(Impz)~go| for all /+charK. 0.2. Let K be a global field of characteristic p, with p>2. The following statements are proved in [12]: a) Lie(Imp3 is reductive. b) Let's decompose Lie(Imp3 in the direct sum Lie(Im Pl) = g~S | c, where 9~ ' is semi-simple, and c is the center of Lie(Imp3. Then c is contained in C | QI and dimQ, c = 1. 0.2. The main result of this paper is the following statement: 0.2.1. Theorem. Under the assumptions of (0.2) above, the rank of g7 "~ does not depend on I. 0.2.2. Remark. It is enough to prove the theorem in case where X is a product of simple Abelian varieties and End(X |163 = End X, see (0.0.1). 0.2.3. In the next section, a formula for the rank of g~s will be given. Theorem 0.2.1 is an immediate corollary of this formula. When writing this paper, conversations with Yu.I. Manin and correspondence with P. Deligne were very useful. I am glad to express them my deep acknowledgement.
Connectedness results for lll-adic representations associated to abelian varieties
Compositio Mathematica, 1995
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Support problem for the intermediate Jacobians of l-adic representations
Journal of Number Theory, 2003
We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homology of the general linear group of rings of integers. We answer the question of Corrales-Rodrigáñez and Schoof concerning the support problem for higher dimensional abelian varieties.
Some cases of the Mumford-Tate conjecture and Shimura varieties
Indiana University Mathematics Journal, 2008
We prove the Mumford-Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the B n and D R n Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.