Diss. ETH No. 25610

Compactification of the finite Drinfeld period domain as a moduli space of ferns

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH

(Dr. sc. ETH Zurich)

presented by

ALEXANDRE PUTTICK\
Master, Mathematics, Université de Paris-Sud 11\
born August 22, 1989\
citizen of the United States of America

accepted on the recommendation of

Prof. Dr. Richard Pink, examiner\
Prof. Dr. Ching-Li Chai, co-examiner

2018

Summary {#summary .unnumbered}
=======

Let ${{\mathbb{F}}}_q$ be a finite field with $q$ elements and let $V$ be a vector space over ${{\mathbb{F}}}_q$ of dimension $n>0$. Let $\OV$ be the Drinfeld period domain over ${{\mathbb{F}}}_q$. This is an affine scheme of finite type over ${{\mathbb{F}}}_q$, and its base change to ${{\mathbb{F}}}_q(t)$ is the moduli space of Drinfeld ${{\mathbb{F}}}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder’s smooth compactification $B_V$ of $\OV$.

Let $\hat V$ be the set $V\cup\{\infty\}$ for a new symbol $\infty$. We define the notion of a $V$-fern over an ${{\mathbb{F}}}_q$-scheme $S$, which consists of a stable $\hat V$-marked curve of genus $0$ over $S$ endowed with a certain action of the finite group $V\rtimes 