Question

What are the (cont.) endomorphisms resp. automorphisms of the adeles (for a given global field)

1) as a topological abelian group and

2) as a topological ring?

3) What are the endomorphisms and the automorphisms group of the ideles?

4) What is known for the adelic points of an algebraic group?

Answer 1

1. There are no nontrivial continuous homomorphisms between factors of different residue characteristic, so any endomorphism/automorphism decomposes into a collection of endomorphisms of factors, with a global condition that $(1,1,\cdots)$ has bounded denominators. Each factor over a rational prime $p$ (including infinity) is a product of copies of $\mathbb{Q}_p$, so continuous endomorphisms are elements of $M_{n_p}(\mathbb{Q}_p)$, where $n_p$ is the sum of inertia degrees over primes over $p$. Therefore, the endomorphism ring is the restricted product of $M_{n_p}(\mathbb{Q}_p)$, i.e., all but finitely many factors lie in $M_{n_p}(\mathbb{Z}_p)$ (when $p$ is finite). The automorphism group is given by replacing $M_{n_p}$ with $GL_{n_p}$.

2. An automorphism is a product of local automorphisms for a given residue charactristic, which are made out of permutations of $p$-adic fields in the local factors and Galois automorphisms of the fields. Endomorphisms also include projections that kill individual fields in the factors.

3. You can decompose the ideles as $\prod_{r_1} \mathbb{R}^\times \oplus \prod_{r_2} \mathbb{C}^\times \oplus \bigoplus_\mathfrak{p} \mathbb{Z} \oplus \prod_\mathfrak{p} \prod_{n,k=1}^\infty \mathbb{Z}/p^k\mathbb{Z} \oplus \prod_\mathfrak{p} \mathbb{Z}_p^{f_\mathfrak{p}}$, using the decomposition of units of a $p$-adic field into valuations and units of integers. It shouldn't be too hard to work it out on your own from here.