Hodge Cycles on Abelian Varieties

@inproceedings{Delign1982HodgeCO,
  title={Hodge Cycles on Abelian Varieties},
  author={Pierre Delign{\'e}},
  year={1982}
}
The main result proved in these notes is that any Hodge cycle on an abelian variety (in characteristic zero) is an absolute Hodge cycle — see §2 for definitions and (2.11) for a precise statement of the result. 
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255 Citations
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255 Citations

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References

SHOWING 1-10 OF 101 REFERENCES
Conjugates of Shimura Varieties
In the first three sections we review the definition of a Shimura variety of abelian type, describe how certain Shimura varieties are moduli varieties for abelian varieties with Hodge cycles and…
On the periods of abelian integrals and a formula of Chowla and Selberg
Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a…
NUMERICAL AND HOMOLOGICAL EQUIVALENCE OF ALGEBRAIC CYCLES ON HODGE MANIFOLDS.
with rational coefficients). We denote by 3h (M), (resp. (3(M)) the cycles which are homologous, (resp. numerically equivalent) to zero. In general g n D3-hand 3m, = m. Matsusaka's work [5] implies…
On the Periods of Certain Rational Integrals: II
In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how…
Resolution Of Singularities Of Embedded Algebraic Surfaces
0 Introduction.- 1. Local Theory.- 1 Terminology and preliminaries.- 2 Resolvers and principalizers.- 3 Dominant character of a normal sequence.- 4 Unramified local extensions.- 5 Main results.- 2.…
On Fermat varieties
(0.1) x? + x? + + x?+ί = 0 . Throughout this paper, we denote it by Xrm, or by X r m(p), when we need to specify the characteristic p of the base field k; we always assume that m ^ 0 (mod p). The…
Applications de la formule des traces aux sommes trigonométrigues
Dans cet expose, j’explique comment la formule des traces permet de calculer ou d’etudier diverses sommes trigonometriques et comment, jointe a la conjecture de Weil, elle peut permettre de les…
Introduction to Affine Group Schemes
I The Basic Subject Matter.- 1 Affine Group Schemes.- 1.1 What We Are Talking About.- 1.2 Representable Functors.- 1.3 Natural Maps and Yoneda's Lemma.- 1.4 Hopf Algebras.- 1.5 Translating from…
Nilpotent connections and the monodromy theorem: Applications of a result of turrittin
© Publications mathématiques de l’I.H.É.S., 1970, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www.…
La conjecture de Weil pour les surfacesK3
1. Enonc6 du th~or~me Soient Fq un corps ~t q ~l~ments, Fq une cl6ture alg6brique de Fq, r la substitution de Frobenius xv-~x q et F= tp -1 le << Frobenius g6om6trique >>. Soit X un sch6ma (s6par6 de…
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