57 research outputs found
Comparison Isomorphisms for Smooth Formal Schemes
For a smooth proper scheme or formal scheme over an unramified, complete DVR
of mixed characteristics we prove a comparison isomorphism relating etale
cohomology of the generic fiber with values in a crystalline etale sheaf to the
crystalline cohomology of its special fiber with values in the associated
F-isocrystal
Faltings heights of abelian varieties with complex multiplication
Let M be the Shimura variety associated with the group of spinor similitudes
of a rational quadratic space over of signature (n,2). We prove a conjecture of
Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of
special divisors and big CM points on M to the central derivatives of certain
-functions. As an application of this result, we prove an averaged version
of Colmez's conjecture on the Faltings heights of CM abelian varieties.Comment: Final version. To appear in Annals of Mat
Crystalline realizations of 1-motives
We consider the crystalline realization of Deligne's 1-motives in positive
characteristics and prove a comparison theorem with the De Rham realization of
liftings to zero characteristic. We then show that one dimensional crystalline
cohomology of an algebraic variety, defined by universal cohomological descent
via de Jong's alterations, coincide with the crystalline realization of the
(cohomological) Picard 1-motive, over perfect fields.Comment: 54 pages, exposition improved, references & appendix adde
Motivic periods and Grothendieck arithmetic invariants
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti\u2013de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Th\ue9l\ue8ne\u2013Raskind)
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