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Assuming the generalised Riemann hypothesis (GRH) for CM-fields, we prove a conjecture of Yves André's asserting that a curve contained in a Shimura variety and containing an infinite set of special points is of Hodge type
We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in ). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action
We describe the cohomology groups of a homogeneous vector bundle on any Hermitian symmetric variety of ADE-type as the cohomology of a complex explicitly described. The main tool is the equivalence (introduced by Bondal, Kapranov, and Hille) between the category of homogeneous bundles and the category of representations of a certain quiver with relations. We prove that the relations are the commutative ones on projective spaces, but they involve additional scalars on general Grassmannians. In addition, we introduce moduli spaces of homogeneous bundles
We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma in conjunction with a result of Buzzard [Bu, Theorem 5.2] to give a proof of (a generalization of) Coleman's theorem, which states that overconvergent modular forms of small slope are classical. The proof is geometric in nature and is suitable for generalization to other Shimura varieties
In this article we show that if a stable-stationary harmonic map from a domain in into has an isolated singular point , then the mapping degree of at is or . Furthermore, the complete characterization of stable-stationary tangent maps from to is given
For any , let be a geometrically integral algebraic variety of degree . This article is concerned with the number of -rational points on which have height at most . For any , we establish the estimate , provided that . As indicated, the implied constant depends at most on , and
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