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Register a new userCM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields
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Authors
Adrian Vasiu
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We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.
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Let $k$ be an algebraically closed field of positive characteristic $p$. We first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and then we study stratifications defined by inner isomorphism classes of these $D$-truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat $F$-decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over $k$.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
Elkies proposed a procedure for constructing explicit towers of curves, and gave two towers of Shimura curves as relevant examples. In this paper, we present a new explicit tower of Shimura curves constructed by using this procedure.
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