Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus Three

On Grothendieck's tame topology

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A …

直线与超平面 GT 变体

Line and Hyperplane GT-variants

本文利用超平面排列补集与 pro-ℓ 两步幂零基本群构造 GT 群的新变体，并证明其与有理数域的绝对 Galois 群同构。

A Motivic Grothendieck-Teichmüller Group

This paper proves the Beilinson-Soul{é} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches gro…

The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $Γ_{g,n}$ and $H_{g,n}$, the so called Teichm{ü}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrowΓ_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $Γ_{g,n}$ is the group of iso…