Drinfeld modules, Explicit class field theory and Lambda-structures
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2022
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Cheng, Derek
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This thesis translates some $\Lambda$-geometric results from number fields, to function fields. There are two major results in this thesis.
For each set $P$ of almost all primes of $\fp[t]$, we use $R_P$ to denote the ring corresponding to an affine open subsets of $\mathbb{P}^1_{\fp}$, i.e., the closed points of $R_P$ are the elements in $P$. We give an equivalent condition for a finite \'{e}tale $\Lambda$-scheme $S$ over $\mbox{Spec}(\fp(t))$ to have a $Q$-$\Lambda(P)$-model, by which we mean a finite flat and reduced scheme over $\mbox{Spec}(R_Q)$ together with a family of commuting Frobenius lifts for each place in $P$. We show that such $S$ has a $Q$-$\Lambda(P)$-model if and only if the action of $\bigoplus_P \mathbb{N}$ and the absolute Galois group of $\fp(t)$ on $\mbox{Hom}(\mbox{Spec}(\overline{\fp(t)}), S)$ factors through the monoid $\mathbb{N}^{P \setminus Q} \times \left( \left. \prod_{v \in P \cap Q} \mathcal{O}_v^{\circ} \times \prod_{v \notin P \cap Q} \mathcal{O}_v^{*} \times \hat{\mathbb{Z}} \right) \right/ \mathbb{F}_q^*$.
Secondly, we show that given a global function field $K$, the maximal abelian extension of $K$ is $\Lambda$-geometric. In other words, we present a scheme $X$ of finite type over a Dedekind domain $A$, with $K$ being the fraction field of $A$, such that the maximal abelian extension of $K$ is contained in the $\mathfrak{f}$-periodic locus of $X$ as $\mathfrak{f}$ runs over all ideals of $A$.
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