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

• Published in: arXiv.org 2020-04
• Main Author: Puttick, Alexandre R
• Format: Article
• Language: English
• Subjects: Domains; Fields (mathematics); Isomorphism; Modular construction; Modular structures; Modules
• ISSN: 2331-8422

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 \(\Omega_V\) 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 \(\Omega_V\). 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 \(V\rtimes \mathbb{F}_q^\times\). Our main result is that the scheme \(B_V\) represents the functor that associates an \(\mathbb{F}_q\)-scheme \(S\) to the set of isomorphism classes of \(V\)-ferns over \(S\). Thus \(V\)-ferns over \(\mathbb{F}_q(t)\)-schemes can be regarded as generalizations of Drinfeld \(\mathbb{F}_q[t]\)-modules with level \((t)\) structure and rank \(n\). To prove this theorem, we construct an explicit universal \(V\)-fern over \(B_V\). We then show that any \(V\)-fern over a scheme \(S\) determines a unique morphism \(S\to B_V\), depending only its isomorphism class, and that the \(V\)-fern is isomorphic to the pullback of the universal \(V\)-fern along this morphism. We also give several functorial constructions involving \(V\)-ferns, some of which are used to prove the main result. These constructions correspond to morphisms between various modular compactifications of Drinfeld period domains over \(\mathbb{F}_q\). We describe these morphisms explicitly.