ℓ-adic algebraic monodromy groups, cocharacters, and the MumfordTate conjecture

Pink, Richard

doi:10.1515/crll.1998.018

Cited by 83 publications

(149 citation statements)

References 20 publications

Supporting

Mentioning

127

Contrasting

Unclassified

Order By: Relevance

“…Corollary (Pink). Every simple factor F 1 of G [Pi,Cor. 5.11]; see also [Ta2] for a slightly less general result).…”

Section: And 510])mentioning

confidence: 99%

“…But up to G Q l (Q l )-conjugation, we can choose the cocharacter µé fact that T v,Q l is generated by images of certain cocharacters in the set C l (H A , X A ) is only a weaker form of [Pi,Thm. 3.16].…”

Section: Corollary We Assume Thatmentioning

confidence: 99%

“…By passing to a finite field extension of E, we can choose A such that it has a principal polarization λ A (cf. (c) One can not get a general form of [Pi,Thm. 5.14] in the pattern of Theorem 1.3.4 (c) by using only the ideas of Sections 1 to 7.…”

Section: Remarks (A)mentioning

confidence: 99%

See 2 more Smart Citations

Some cases of the Mumford-Tate conjecture and Shimura varieties

Vasiu¹

2008

Indiana Univ. Math. J.

View full text Add to dashboard Cite

ABSTRACT. 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.

show abstract

“…Corollary (Pink). Every simple factor F 1 of G [Pi,Cor. 5.11]; see also [Ta2] for a slightly less general result).…”

Section: And 510])mentioning

confidence: 99%

Section: Corollary We Assume Thatmentioning

confidence: 99%

See 1 more Smart Citation

Some cases of the Mumford-Tate conjecture and Shimura varieties

Vasiu¹

2008

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Par un argument de projection (voir la démonstration du corollaire 5 dans [Rat04], qui s'adapte à l'identique ici), on déduit du théorème 1.1 : La seule restriction qui empêche d'obtenir une minoration inconditionnelle est donc l'hypothèse H. Elle est en fait l'objet de la conjecture suivante (voir [Pin98], § 7) :…”

Section: Introductionunclassified

Une minoration du minimum essentiel sur les variétés abéliennes

Galateau¹

2010

Comment. Math. Helv.

View full text Add to dashboard Cite

Résumé. On étend à la codimension générale la minoration du minimum essentiel sur les variétés abéliennes obtenue dans [Gal08], sous une conjecture concernant leurs idéaux premiers ordinaires. Cette minoration est optimale "à " près" en le degré de la sous-variété et rend inconditionnelle la démonstration, par Viada, de la conjecture de Zilber-Pink pour une courbe plongée dans une puissance de courbe elliptique.Abstract. We extend to the general codimension the lower bound for the essential minimum on abelian varieties found in [Gal08], under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, "up to an "", in the degree of the subvariety and completes Viada's proof of the Zilber-Pink conjecture for a curve embedded in a power of an elliptic curve.Classification mathématique par sujets (2010). 11G10, 11J81, 14G40.

show abstract

“…It follows from [Pin98,5.13] that there exist a connected reductive group G over Q and a representation V of G such that (G, V ) ⊗ Q ∼ = (G , V ) for all primes belonging to a set of primes of Dirichlet density 1. Of course, Lie(G) has one dimensional centre c and…”

Section: Introductionmentioning

confidence: 99%

Abelian varieties with ℓ-adic Galois representation of Mumford's type

Noot¹

2000

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

This paper is devoted to the study of 4-dimensional abelian varieties over number fields with the property that the Lie algebra of the image of some associated -adic Galois representation is Q -isomorphic to c ⊕ (sl 2 ) 3 . Such varieties will be referred to as 'abelian varieties with -adic Galois representation of Mumford's type'. We show that such abelian varieties have potentially good reduction at all prime ideals and we determine the possible Newton polygons and the possible isogeny types of these reductions.

show abstract

ℓ-adic algebraic monodromy groups, cocharacters, and the MumfordTate conjecture

Cited by 83 publications

References 20 publications

Some cases of the Mumford-Tate conjecture and Shimura varieties

Some cases of the Mumford-Tate conjecture and Shimura varieties

Une minoration du minimum essentiel sur les variétés abéliennes

Abelian varieties with ℓ-adic Galois representation of Mumford's type

Contact Info

Product

Resources

About