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Register a new userCongruence for rational points over finite fields and coniveau over local fields
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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.
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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 $p>0$).
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pilas algorithm for abelian varieties.
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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