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  1. Abstract

    We show that for every integer$$m > 0$$m>0, there is an ordinary abelian variety over $${{\mathbb {F}}}_2$$F2that has exactlymrational points.

     
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  2. Let $X$ be a smooth scheme over a finite field of characteristic $p$.Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adicWeil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell\neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$.Using the Langlands correspondence for global function fields in both the\'etale and crystalline settings (work of Lafforgue and Abe, respectively), onesees that on a curve, any coefficient object in one category has "companions"in the other categories with matching characteristic polynomials of Frobeniusat closed points. A similar statement is expected for general $X$; building onwork of Deligne, Drinfeld showed that any \'etale coefficient object has\'etale companions. We adapt Drinfeld's method to show that any crystallinecoefficient object has \'etale companions; this has been shown independently byAbe--Esnault. We also prove some auxiliary results relevant for theconstruction of crystalline companions of \'etale coefficient objects; thissubject will be pursued in a subsequent paper. 
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  3. Anni, Samuele ; Karemaker, Valentijn ; Lorenzo García, Elisa (Ed.)