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Notes on Class Field Theory and Complex Multiplication

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Notes on Class Field Theory and Complex Multiplication Notes on class ¯eld theory and complex multiplication Kartik Prasanna University of California, Los Angeles, CA July 5, 2005 Disclaimer: These notes are in draft form and have not been carefully proofread. Please use at your own risk ! Contents 1 Introduction 2 2 Class ¯eld theory 2 2.1 Number ¯elds and their completions . 2 2.1.1 Number ¯elds, prime ideals . 2 2.1.2 Fractional Ideals . 3 2.1.3 Completions . 4 2.1.4 Adeles and ideles . 4 2.1.5 Cycles . 5 2.1.6 The trace and the norm . 6 2.2 Main theorems of class ¯eld theory . 7 2.3 Examples . 10 2.3.1 Cyclotomic ¯elds and the Kronecker-Weber theorem . 10 2.3.2 Quadratic ¯elds and quadratic reciprocity . 11 3 Complex Multiplication 11 3.1 Introduction . 11 3.2 Imaginary quadratic ¯elds . 12 3.2.1 Elliptic curves . 12 3.2.2 Elliptic curves with complex multiplication . 13 3.2.3 Idelic actions on lattices . 15 3.2.4 Main theorem for CM elliptic curves . 17 3.2.5 The j-invariant, automorphisms and Weber functions . 18 3.2.6 Class ¯elds of imaginary quadratic ¯elds as an applica- tion of the main theorem . 19 1 3.3 CM ¯elds . 21 3.3.1 Abelian Varieties . 21 3.3.2 CM ¯elds and CM Abelian varieties . 22 3.3.3 Properties of CM ¯elds : the reflex ¯eld . 24 3.3.4 Examples . 25 3.3.5 The main theorem for CM Abelian varieties . 25 1 Introduction These notes are meant to be an introduction to class ¯eld theory, the theory of complex multiplication and some aspects of the theory of Shimura curves for participants of a summer school in Hangzhou. There is absolutely nothing original in these notes, either in terms of content or presentation. Indeed, we have often simply copied from standard texts, notably Lang's book for class ¯eld theory, and Shimura's books and articles for the rest of the notes. The only virtue of these notes then is (hopefully) that they provide to a beginning student clear statements of the main theorems and motivation to read the more comprehensive books and articles cited. 2 Class ¯eld theory 2.1 Number ¯elds and their completions 2.1.1 Number ¯elds, prime ideals A number ¯eld is a ¯nite extension of Q. The ring of integers of L, oL is de¯ned to be the subring of L consisting of x 2 L that satisfy a monic polynomial with coe±cients in Z. oL is an integrally closed integral domain of dimension 1 i.e. all the nonzero prime ideals of oL are maximal. Let p be a nonzero prime ideal of oL. Then p \ Z = (p) for some prime number p. The prime ideal p is said to lie over p. Conversely, for any integer prime p, the set of prime ideals p of oL that lie over p is ¯nite. If fp1; p2;:::; prg denotes the set of these primes, one has Y epi poL = pi (1) i for some integers epi . If fpi := [o=pi : Z=pZ] is the degree of the ¯nite ¯eld o=pi over Fp, one has also X [L : Q] = epi fpi i 2 More generally, let K=L be an extension of number ¯elds, and p a prime ideal in oL. Then the set of primes fp1; p2;:::; prg of oK lying over p (i.e. such that pi \ oL = p) is ¯nite. We have Y e pi=p poK = pi (2) i for some integers epi=p, generalising (1) above. If fpi=p := [oK =pi : oL=p] is the degree of the ¯nite ¯eld oK =pi over oL=p, one has also X [K : L] = epi=pfpi=p (3) i generalising (2) above. De¯nition 2.1 A prime p in L is said to be rami¯ed in K if epi=p > 1 for some pi lying over p. It is known that only ¯nitely many primes of L ramify in K. In fact one may de¯ne an ideal in L called the discriminant ideal dK=L such that p rami¯es in K if and only if pjdK=L. p is said to be unrami¯ed in K if it is not rami¯ed. 2.1.2 Fractional Ideals A fractional ideal a in L is an oL-submodule of L, such that xa ⊆ oL for some x 2 L£. If a and b are fractional ideals, X ab := f aibi; ai 2 a; bi 2 bg (4) i is also a fractional ideal. Likewise, if a 6= 0, ¡1 a = f® 2 L; ®a ⊆ oLg (5) is also a fractional ideal. The set of fractional ideals in L then forms a group under the multiplication law (4), with inverses being de¯ned as in (5). We denote this group by the symbol IL or just by I if the ¯eld L is ¯xed. Associated to every ® 2 L is the principal fractional ideal (®) := ®o. Since (®)(¯) = (®¯) and (®)¡1 = (®¡1), the set of principal fractional ideals forms a subgroup P of I. Theorem 2.2 (Unique factorization) Every fractional ideal a factors uniquely as Y ni a = pi i for some set of distinct prime ideals pi and non-zero integers ni. Theorem 2.3 The quotient group I=P is ¯nite. I=P is called the class group of L. 3 2.1.3 Completions There are two kinds of completions that one wishes to study corresponding to two di®erent kinds of metrics (places) on L. Let us denote oL just by the symbol o. Non-archimedean places. Let p be a nonzero prime ideal in o. One may de¯ne on L a norm j ¢ jp in the following way: 1 vp(a) jajp = ( ) (6) p1=ep m where vp(a) is de¯ned to be the largest integer m such that a 2 p . Let Lp denote the completion of L for the metric de¯ned by j¢jp. Then L ⊆ Lp and the norm j¢jp extends naturally to a norm on Lp, also denoted by the same symbol j ¢ jp. Lp is a topological ¯eld with respect to the associated metric. Let op and i i p op denote the closures of o and p in Lp respectively. Then op is a topological i i ring in which pop is the unique nonzero prime ideal and p op = (pop) . The ideal pop is principal and every non-zero ideal in op is a power of pop. In other words, op is a discrete valuation ring with maximal ideal pop. The formula (6) continues to hold for a 2 Lp, except that vp(a) must now be de¯ned to be the m largest integer m such that a 2 p op. The additive group (Lp; +) is locally compact, and the ¯ltration of additive i+1 i subgroups ::: ⊆ p op ⊆ p op ⊆ ::: gives a fundamental system of compact open neighborhoods of 0. Let Up denote the units of op i.e. Up = opnpop. De¯ne i £ Up;i = 1+p op. Each Up;i is a multiplicative subgroup of Lp . The multiplicative £ group Lp is also locally compact, with the ¯ltration ::: ⊆ Up;i+1 ⊆ Up;i ⊆ ::: being a fundamental system of open neighborhoods of 1. Archimedean places. These correspond to metrics obtained by embedding L in C and restricting the usual absolute value on C. The number of distinct embeddings of L in C is well known to equal the degree [L : Q]. An embedding σ is said to be real if σ(L) ⊆ R and imaginary otherwise. The imaginary embeddings clearly occur in pairs of complex conjugate embeddings that induce the same absolute value. Thus if r is the number of real embeddings and 2s the number of imaginary embeddings, there are r + s distinct metrics thus obtained. Correspondingly the completions Lσ = R for the real embeddings and = C for the imaginary ones. We denote by §L;f (resp. §L;1, resp. §L) the set of non-archimedean places (resp. the set of archimedean places of L, resp. the set of all places) of L. 2.1.4 Adeles and ideles For many purposes it turns out to be useful to bundle together all the comple- tions of a number ¯eld L into a single object, called the adele ring of L. 4 De¯nitionQ 2.4 The Adele ring of L, denoted AL, is de¯ned to be the subring of L consisting of those (a ) such that a 2 o for almost all v2§L v v v2§L p p p 2 §L;f (i.e. for all but ¯nitely many p.) We also denote by AL;f the subring of AL consisting of those (av)v2§L such that aσ = 1 for all σ 2 §L;1 and AL;1 the subring of AL consisting of those (av)v2§L such that ap = 1 for all p 2 §L;f . Addition and multiplication on AL is de¯ned component-wise. It will also be important to put a topology on AL in the following way. For each ¯nite subset S ⊆ §L containing §L;1, de¯ne AL;S to be the subringQ of AL consistingQ of those (av)v2§L such that ap 2 op for p 62 S. Since AL;S = p62S op £ v2S Lv, and each op is compact, we can make AL;S into a locally compact group by giving it the product topology. Now we make AL into a topological group by declaring each AL;S to be an open subgroup.
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