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Problems arising from the Tate and Beilinson conjectures in the context of Shimura varieties. (English) Zbl 0714.14022
Automorphic forms, Shimura varieties, and L-functions, Vol. II, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 11, 227-252 (1990).
[For the entire collection see Zbl 0684.00004.]
The general conjectures of Tate, Beilinson, Deligne, Bloch, et al. relating the poles and zeros, and special values of L-functions of algebraic varieties to the arithmetic and geometry are discussed in the context of Shimura varieties. Problems and questions are formulated explicitly, and some examples are presented to illustrate the conjectures.
Let $$S_ K$$ denote a Shimura variety defined over a canonical number field E, $$S^*_ K$$ the Baily-Borel-Satake compactification /E of $$S_ K$$ and $$\tilde S^*_ K({\mathbb{C}})$$ a smooth toroidal compactification of $$S^*_ K({\mathbb{C}})$$. Questions like: the existence of motivic decomposition on $$\tilde S_ K$$, the construction of nontrivial primitive algebraic cohomology classes in the intersection cohomology $$IH^{2m}(\tilde S_ K)$$, the existence of simple non-CM abelian varieties which do not occur as factors of Jacobians of Shimura curves, the evaluation of height pairings on special cycles, analytic expression for the derivatives of the relevant L-functions, among others, are formulated.
Reviewer: N.Yui
##### MSC:
 14G35 Modular and Shimura varieties 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14A20 Generalizations (algebraic spaces, stacks)