“God exists since mathematics is consistent, and the Devil exists since we cannot prove it.”
André Weil
I’ve recently finished my final project for my algebraic number theory course on adeles and ideles, which can be found here. Adeles is an object in algebraic number theory that solves the technical problem of doing analysis over \(\mathbb Q\) so to speak. It lets us work over all completions of a global field simultaneously. One of the applications of it is that certain compactness theorems on it proves the Dirichlet unit theorem and class number theorem. If I had more time, I would have elaborated further at the last part in my project but i unfortunately did not have enough time.
In this post, I’ll give a quick summery of my project. Fix a global field \(K\).
Definition. The adele ring \(\mathbb A_K\) associated to \(K\) is the restricted direct product
\[\mathbb A_K=\prod_{\nu}(K_\nu,\mathcal O_\nu)\] where \(\mathcal O_\nu\) is the valuation ring of \(K_\nu\), as a topological ring. More generally, for \(S\) a set of places of \(K\), the \(S\)-adele ring is the product \[\mathbb A_{K,S}=\prod_{\nu\in S}K_\nu\times \prod_{\nu\not\in S}\mathcal O_\nu\] and we have further that \(\mathbb{A}_K \simeq \displaystyle\lim_{\longrightarrow} \mathbb{A}_{K, S}\).
This is a locally compact and Hausdorff topological ring, by giving it a basis consisting of \(\prod_{\nu}U_\nu\) where \(U_\nu\subseteq K_{\nu}\) open with almost all \(U_\nu=\mathcal O_\nu\). We have a product formula \[\prod_{\nu}|x|_\nu=1\] for \(x\in K^\times\) for normalized absolute values. The ideles \(\mathbb I_K\) are the unit group in \(\mathbb A_K^\times\), given the subspace topology by diagonal embedding in \(\mathbb A_K\times \mathbb A_K\). The \(1\)-ideles is the subgroup of the ideles with norm \(1\), which inherits the same topology from \(\mathbb I_K\) and \(\mathbb A_K\). The important result is
Proposition. \(\mathbb A_K/K\) and \(\mathbb I^1_K/K^\times\) are compact.
This is what leads to Dirichlet unit theorem and class number theorem.
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