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Register a new userNonlinear descent on moduli of local systems
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Junho Peter Whang
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We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.
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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.
Recently Pelayo-V~{u} Ngoc classified semitoric integrable systems in terms of five symplectic invariants. Using this classification we define a family of metrics on the space of semitoric integrable systems. The resulting metric space is incomplete and we construct the completion.
The concept of a covering system was first introduced by ErdH{o}s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli. Arguably, the most famous open question regarding covering systems is the odd covering problem. In this paper, we explore a variation of the odd covering problem, allowing a single odd prime to appear as a modulus in the covering more than once, while all other moduli are distinct, odd, and greater than $1$. We also consider this variation while further requiring the moduli of the covering system to be square-free.
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.
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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