Publications (6)7.14 Total impact
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ABSTRACT: This is a continuation of our work [44] on the torsion and integral cohomology groups of PELtype Shimura varieties. We consider the interior cohomology (and the Hodge graded pieces in the case of the de Rham realization) of general — not necessarily compact — PELtype Shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group. For primes p bigger than an effective bound, we prove that the Fp and Zpcohomology groups are concentrated in the middle degree, that the Zpcohomology groups are free of ptorsion, and that every Fpcohomology class lifts to a Zpcohomology class.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.  [Show abstract] [Hide abstract]
ABSTRACT: We present a vanishing theorem for automorphic line bundles on good reduction fibers of PELtype Shimura varieties (including all noncompact ones). As a consequence, we deduce that for a good prime p no smaller than the dimension of a PELtype Shimura variety, any mod p cusp form of positive cohomological parallel weight is liftable to characteristic zero.  [Show abstract] [Hide abstract]
ABSTRACT: This is a continuation of our work [44] on the torsion and integral cohomology groups of PELtype Shimura varieties. We consider the interior cohomology (and the Hodge graded pieces in the case of the de Rham realization) of general — not necessarily compact — PELtype Shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group. For primes p bigger than an effective bound, we prove that the Fpand Zpcohomology groups are concentrated in the middle degree, that the Zpcohomology groups are free of ptorsion, and that every Fpcohomology class lifts to a Zpcohomology class.  [Show abstract] [Hide abstract]
ABSTRACT: We construct general type surfaces in mixed characteristic whose geometric genera can be made to jump by an arbitrarily prescribed positive amount under specialization. We then show that this phenomenon of jumping geometric genus presents itself in some compact Shimura surfaces. Finally, we find a set of conditions, met by the latter Shimura surfaces, that forces the higher plurigenera to remain constant in reduction modulo p.  [Show abstract] [Hide abstract]
ABSTRACT: We show that conjugation by an automorphism of C may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups. Let V be an algebraic variety over C, and let τ be an automorphism of C (as an abstract field). On applying τ to the coefficients of the polynomials defining V, we obtain a conjugate algebraic variety τV over C. The cohomology groups H i (V an, Q) and H i ( (τV) an, Q) have the same dimension, and hence are isomorphic, because, when tensored with Qℓ, they become isomorphic to the étale cohomology groups which are unchanged by conjugation. Similarly, the profinite completions of the fundamental groups of V an and (τV) an are isomorphic because they are isomorphic to the étale fundamental groups. However, Serre [Se] gave an example in which the fundamental groups themselves are not isomorphic (see [EV] for a discussion of this and other examples). It seems to have been known (or, at least, expected) for some time that the theory of Shimura varieties
Publication Stats
51  Citations  
7.14  Total Impact Points  
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Institutions

20102011

Harvard University
 Department of Mathematics
Cambridge, MA, United States


20082010

Massachusetts Institute of Technology
 Department of Mathematics
Cambridge, Massachusetts, United States
