Math 254B, UC Berkeley, Spring 2002 (Kedlaya) Adeles and Ideles in Field Extensions

Reference. Neukirch, Section VI.1 and VI.2.

Adeles in Field Extensions.

If L/K is an extension of number fields, we get an embedding 𝔸_{K} ↪ 𝔸_{L} as follows: given α ∈ 𝔸_{K}, each place w of L restricts to a place v of K, so set the w-component of the image of α to α_{v}. This embedding induces an inclusion I_{K} ↪ I_{L} of idele groups as well.

If L/K is Galois with Galois group G, then G acts naturally on 𝔸_{L} and I_{L}; more generally, if g ∈ Gal(\overline{K}/K), then g maps L to some other extension L^{g} of K, and G induces isomorphisms of 𝔸_{L} with 𝔸_{L^{g}} and of I_{L} with I_{L^{g}}. Namely, if (α_{v})_{v} is an idele over L and g ∈ G, then g maps the completion L_{v} of L to a completion L_{v^{g}} of L^{g}. (Remember, a place v corresponds to an absolute value | · |_{v} on L; the absolute value | · |_{v^{g}} on L^{g} is given by |a^{g}|_{v^{g}}| = |a|_{v}.) As you might expect, this action is compatible with the embeddings of L in I_{L} and L^{g} in I_{L^{g}}, so it induces an isomorphism C_{L} → C_{L^{g}} of idele class groups.

Aside. Neukirch points out that you can regard 𝔸_{L} as the tensor product 𝔸_{K} ⊗_{K} L; in particular, this is a good way to see the Galois action on 𝔸_{L}. Details are left to the reader.

We can define trace and norm maps as well:

Trace_{𝔸_{L}/𝔸_{K}}(x) = ∑_{g} x^{g}, Norm_{I_{L}/I_{K}}(x) = ∏_{g} x^{g}

where g runs over coset representatives of Gal(\overline{K}/L) in Gal(\overline{K}/K), the sum and product taking places in the adele and idele rings of the Galois closure of L over K. In particular, if L/K is Galois, g simply runs over Gal(L/K).

In terms of components, these definitions translate as

(Trace_{𝔸_{L}/𝔸_{K}}(α))_{v} = ∑_{w|v} Trace_{L_{w}/K_{v}}(α_{w})
(Norm_{I_{L}/I_{K}}(α))_{v} = ∏_{w|v} Norm_{L_{w}/K_{v}}(α_{w}).

The trace and norm do what you expect on principal adeles/ideles. In particular, the norm descends to a map Norm_{L/K} : C_{L} → C_{K}.

Aside. You can also define the trace of an adele α ∈ 𝔸_{L} as the trace of addition by α as an endomorphism of the 𝔸_{K}-module 𝔸_{L}, and the norm of an idele α ∈ I_{L} as the determinant of multiplication by α as an automorphism of the 𝔸_{K}-module 𝔸_{L}. (Yes, the action is on the adeles in both cases. Remember, ideles should be thought of as automorphisms of the adeles, not as elements of the adele ring.)

If L/K is a Galois extension, then Gal(L/K) acts on 𝔸_{L} and I_{L} fixing 𝔸_{K} and I_{K}, respectively, and we have the following.

Proposition 1. If L/K is a Galois extension with Galois group G, then 𝔸_{L}^{G} = 𝔸_{K} and I_{L}^{G} = I_{K}.

Proof. If v is a place of K, then for each place w of K above v, the decomposition group G_{w} of w is isomorphic to Gal(L_{w}/K_{v}). So if (α) is an adele or idele which is G-invariant, then α_{w} is Gal(L_{w}/K_{v})-invariant for each w, so belongs to K_{v}. Moreover, G acts transitively on the places w above v, so α_{w} = α_{w′} for any two places w, w′ above v. Thus (α) is an adele or idele over K.

This has the following nice consequence for the idele class group, a fact which is quite definitely not true for the ideal class group: the map Cl_{K} → Cl_{L}^{G} is neither injective nor surjective in general. This is our first hint of why the idele class group will be a more convenient target for a reciprocity map than the ideal class group.

Proposition 2 (Galois descent). If L/K is a Galois extension with Galois group G, then G acts on C_{L}, and the G-invariant elements are precisely C_{K}.

Proof. We start with an exact sequence

1toL^{∗} → I_{L} → C_{L} → 1

of G-modules. Taking G-invariants, we get a long exact sequence

1 → (L^{∗})^{G} = K^{∗} → (I_{L})^{G} = I_{K} → C_{L}^{G} → H^{1}(G, L^{∗}),

and the last term is 1 by (Hilbert-Noether) Theorem 90. So we again have a short exact sequence, and C_{L}^{G} ≅ I_{K}/K^{∗} = C_{K}.