Arithmetic & Homotopic Galois Theory IRN | On representations of automorphism groups of free profinite groups and of the absolute Galois group over Q

February 6, 2023	On representations of automorphism groups of free profinite groups and of the absolute Galois group over Q Frauke Bleher, University of Iowa, USA Zoom only · JP:21:00 · FR:13:00 This is joint work with Ted Chinburg and Alex Lubotzky. Let $d \geq 2$ , let $F_{d}$ be a free discrete group of rank $d$ , and let ${\hat{F}}_{d}$ be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical conditions, representations of finite index subgroups of $Aut (F_{d})$ that have as images certain large arithmetic groups. In this talk, I will first show how their method leads to a stronger result for $Aut ({\hat{F}}_{d})$ . I will then discuss an application of this result to Galois theory. This uses a result by Belyi who showed that there is a natural embedding of the absolute Galois group $G_{Q}$ of $\overset{―}{Q}$ over $Q$ into $Aut ({\hat{F}}_{2})$ . In particular, I will show how the natural action of certain subgroups of $G_{Q}$ on the Tate modules of generalized Jacobians of covers of $P^{1}$ over $\overset{―}{Q}$ that are unramified outside ${0, 1, \infty}$ can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $Aut ({\hat{F}}_{2})$ . I will also give a criterion for this action to define, up to a finite index subgroup, a compatible action on the Tate modules of the usual Jacobians of the covers. · Twitter · GCalendar · Outlook

February 6, 2023

On representations of automorphism groups of free profinite groups and of the absolute Galois group over Q

Frauke Bleher, University of Iowa, USA Zoom only · JP:21:00 · FR:13:00

This is joint work with Ted Chinburg and Alex Lubotzky. Let $d \geq 2$ , let $F_{d}$ be a free discrete group of rank $d$ , and let ${\hat{F}}_{d}$ be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical conditions, representations of finite index subgroups of $Aut (F_{d})$ that have as images certain large arithmetic groups. In this talk, I will first show how their method leads to a stronger result for $Aut ({\hat{F}}_{d})$ . I will then discuss an application of this result to Galois theory. This uses a result by Belyi who showed that there is a natural embedding of the absolute Galois group $G_{Q}$ of $\overset{―}{Q}$ over $Q$ into $Aut ({\hat{F}}_{2})$ . In particular, I will show how the natural action of certain subgroups of $G_{Q}$ on the Tate modules of generalized Jacobians of covers of $P^{1}$ over $\overset{―}{Q}$ that are unramified outside ${0, 1, \infty}$ can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $Aut ({\hat{F}}_{2})$ . I will also give a criterion for this action to define, up to a finite index subgroup, a compatible action on the Tate modules of the usual Jacobians of the covers.