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Register a new userFiniteness of cohomology of local systems on rigid analytic spaces
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We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite extension of Q_p or an algebraically closed nonarchimedean field containing Q_p. This result manifests as a special case of a more general finiteness result for the higher direct images of a relative (phi, Gamma)-module along a smooth proper morphism of rigid analytic spaces over a mixed-characterstic nonarchimedean field.
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Let fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{fa}(M), the finiteness dimension of M with respect to fa, and, its dual notion q_{fa}(M), the Artinianess dimension of M with respect to fa. When (R,fm) is local and r:=f_{fa}(M) is less than f_{fa}^{fm}(M), the fm-finiteness dimension of M relative to fa, we prove that H^r_{fa}(M) is not Artinian, and so the filter depth of fa on M doesnt exceeds f_{fa}(M). Also, we show that if M has finite dimension and H^i_{fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{fa}(M)/fa H^t_{fa}(M) is Artinian. It immediately implies that if q:=q_{fa}(M)>0, then H^q_{fa}(M) is not finitely generated, and so f_{fa}(M)leq q_{fa}(M).
We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.
We describe how a systematic use the deep methods from $ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.
A $textit{portrait}$ $mathcal{P}$ on $mathbb{P}^N$ is a pair of finite point sets $Ysubseteq{X}subsetmathbb{P}^N$, a map $Yto X$, and an assignment of weights to the points in $Y$. We construct a parameter space $operatorname{End}_d^N[mathcal{P}]$ whose points correspond to degree $d$ endomorphisms $f:mathbb{P}^Ntomathbb{P}^N$ such that $f:Yto{X}$ is as specified by a portrait $mathcal{P}$, and prove the existence of the GIT quotient moduli space $mathcal{M}_d^N[mathcal{P}]:=operatorname{End}_d^N//operatorname{SL}_{N+1}$ under the $operatorname{SL}_{N+1}$-action $(f,Y,X)^phi=bigl(phi^{-1}circ{f}circphi,phi^{-1}(Y),phi^{-1}(X)bigr)$ relative to an appropriately chosen line bundle. We also investigate the geometry of $mathcal{M}_d^N[mathcal{P}]$ and give two arithmetic applications.
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $ggeq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
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