[Show abstract][Hide abstract] ABSTRACT: This is a continuation of our work [44] on the torsion and integral cohomology groups of PEL-type 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 — PEL-type 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 Zp-cohomology groups are concentrated in the middle degree, that the Zp-cohomology groups are free of p-torsion, and that every Fp-cohomology class lifts to a Zp-cohomology class.
Duke Mathematical Journal 04/2012; 161(2012). DOI:10.1215/00127094-1548452 · 1.58 Impact Factor
[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 PEL-type Shimura varieties (including all noncompact ones). As a consequence, we deduce that for a good prime p no smaller than the dimension of a PEL-type Shimura variety, any mod p cusp form of positive cohomological parallel weight is liftable to characteristic zero.
International Mathematics Research Notices 12/2010; DOI:10.1093/imrn/rnq145 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This is a continuation of our work [44] on the torsion and integral cohomology groups of PEL-type Shimura varieties. We consider the interior cohomology (and the Hodge graded pieces in the case of the de Rham real-ization) of general — not necessarily compact — PEL-type Shimura varieties with coefficients in the local systems corresponding to sufficiently regular al-gebraic representations of the associated reductive group. For primes p bigger than an effective bound, we prove that the Fp-and Zp-cohomology groups are concentrated in the middle degree, that the Zp-cohomology groups are free of p-torsion, and that every Fp-cohomology class lifts to a Zp-cohomology class.
Advances in Mathematics 01/2010; 242. DOI:10.1016/j.aim.2013.04.004 · 1.29 Impact Factor
[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
American Journal of Mathematics 05/2008; 132(3). DOI:10.1353/ajm.0.0112 · 1.18 Impact Factor