This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: RAY CLASS GROUPS AND RAY CLASS FIELDS: FIRST CLASSICALLY, THEN ADELICALLY PETE L. CLARK Contents 8. Intro 1 8.1. Moduli and ray class fields 1 8.2. The Hilbert Class Field 5 8.3. Remarks on real places, narrow class groups, et cetera 6 8.4. A word about class field towers 7 8.5. Class field theory over Q 8 8.6. Idelic interpretation 10 References 12 8. Intro We wish to explain our earlier claim that the idele class group C ( K ) is, as a topolog ical group, a “target group” for global class field theory: i.e., its profinite completion is canonically isomorphic to the Galois group Gal( K ab /K ). Note first that the statement that C ( K ) is such a target group is equivalent to the statement that C 1 ( K ) is such a target group: namely, we have in the number field case a short exact sequence 1 → C 1 ( K ) → C ( K ) → R > → 1 which shows that the inclusion C 1 ( K ) , → C ( K ) induces an isomorphism on the profinite completions. Exercise 8.1: What happens in the function field case? It may therefore be worth mentioning that the object C ( K ) is for many purposes “more fundamental”, whereas the subgroup C 1 ( K ) was – by virtue of being com pact – technically more convenient to establish the two finiteness theorems of the previous section. 8.1. Moduli and ray class fields. N.B.: On a first reading, I suggest that the reader make her task a little easier by: (i) restricting to the number field case, and (ii) ignoring – or mostly ignoring – the infinite part of the modulus. 1 2 PETE L. CLARK What we can do is define a certain family { K ( m ) } of abelian extensions which are parameterized solely by some arithmetic data m from K (you are not yet sup posed to know what this means; don’t worry). These field K ( m ) are called ray class fields of K . It is too much to hope for that every finite abelian extension of K is a ray class field, but what turns out to be true is that every abelian extension L is contained in some ray class field – in fact, in infinitely many ray class fields, but there will be a unique smallest ray class field containing L . The Galois the ory of subextensions of abelian extensions behaves beautifully – in particular every subextension is Galois – so that if we know all the ray class fields, we have a good chance at understanding all the finite abelian extensions. Let me now describe the objects m by which ray class fields are parameterized. Recall that if K/ Q is a number field of degree d , say given as Q [ t ] / ( P ( t )), then the embeddings of K into R correspond precisely to the real roots of P ( t ): in par ticular there is somewhere between 0 and [ K : Q ] such embeddings. Let us label these embeddings ∞ 1 , ∞ 2 ,..., ∞ r . We call such embeddings “real places.” The function field case is simpler: there are no real embeddings to worry about....
View
Full
Document
This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.
 Fall '11
 Staff
 Number Theory

Click to edit the document details