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1. CMB 2013 (vol 57 pp. 439)
The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$ |
The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$ We prove that for every ordinary genus-$2$ curve $X$ over a finite
field $\kappa$ of characteristic $2$ with
$\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist
$\textrm{SL}(2,\kappa[\![s]\!])$-representations of $\pi_1(X)$ such
that the image of $\pi_1(\overline{X})$ is infinite. This result
produces a family of examples similar to Laszlo's counterexample
to de Jong's question regarding the finiteness of the geometric
monodromy of representations of the fundamental group.
Keywords:vector bundle, Frobenius pullback, representation, etale fundamental group Categories:14H60, 14D05, 14G15 |
2. CMB 2011 (vol 56 pp. 500)
The Lang--Weil Estimate for Cubic Hypersurfaces An improved estimate is provided for the number of $\mathbb{F}_q$-rational points
on a geometrically irreducible, projective, cubic hypersurface that is
not equal to a cone.
Keywords:cubic hypersurface, rational points, finite fields Categories:11G25, 14G15 |