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The “main conjectures” of Iwasawa theory for imaginary quadratic fields. (English) Zbl 0737.11030

This paper proves one- and two-variable “main conjectures” over imaginary quadratic fields for both split and non-split primes, and obtains very precise information on the conjecture of Birch and Swinnerton-Dyer.

Let K be an imaginary quadratic field, let p be a prime number not dividing the number of roots of unity in the Hilbert class field H of K, and let 𝔭 be a prime of K above p and K 𝔭 the corresponding completion. Fix an abelian extension K 0 of K containing H and let Δ=Gal(K 0 /K). Let K be an abelian extension of K containing K 0 such that Gal(K /K 0 ) p or p 2 . For each finite extension F of K inside K , let A(F) denote the p-part of the class group, (F) the global units, 𝒞(F) the elliptic units, U(F) the local units of F K K 𝔭 congruent to 1 modulo the primes above 𝔭, ¯(F) the closure of (F)U(F) in U(F), and similarly for 𝒞 ¯(F). When F is an infinite extension of K, define these groups to be the inverse limits of the corresponding groups for finite subextensions. Let X be the Galois group of the maximal abelian p-extension of K unramified outside the primes above 𝔭.

All the above modules for F=K are modules over the Iwasawa algebra Λ= p [[Gal(K /K]], which is a direct sum of power series rings in 1 or 2 variables, corresponding to Gal(K /K 0 ) p or p 2 . It is possible to define characteristic power series (denoted by “char”) for such modules.

The main theorem of the paper is the following. (i) Suppose p splits into two distinct primes in K. Then

char(A(K ))=char( ¯(K )/𝒞 ¯(K ))andchar(X )=char(U(K )/𝒞 ¯(K ))·

(ii) Suppose p remains prime or ramifies in K. Then

char(A(K ))divideschar( ¯(K )/𝒞 ¯(K ))·

If χ is an irreducible p -representation of Δ that is non-trivial on the decomposition group of 𝔭 in Δ, then

char(A(K ) χ )=char( ¯(K ) χ /𝒞 ¯(K ) χ )·

The first part of the theorem in the one-variable case was a question raised by J. Coates and A. Wiles [J. Aust. Math. Soc., Ser. A 26, 1-25 (1978; Zbl 0442.12007)]. Case (ii) has always been more problematic. The present result seems to be a good analogue for the non-split primes, and suffices for many applications to elliptic curves.

A very important consequence of the above theorem is the following application to elliptic curves: Suppose E is an elliptic curve defined over an imaginary quadratic field K, with complex multiplication by the ring of integers 𝒪 of K, and with minimal period lattice generated by Ω × . Write w=#(𝒪 × ). (i) If L(E/K,1)0 then E(K) is finite, the Tate- Shafarevich group Ш E/K of E is finite and there is a u𝒪[w -1 ] × such that

#(Ш E/K )=u#(E(K)) 2 L(E/K,1) ΩΩ ¯·

(ii) If L(E/K,1)=0 then either E(K) is infinite or the 𝔭-part of Ш E/K is infinite for all primes 𝔭 of K not dividing w.

The finiteness of E(K) was proved by J. Coates and A. Wiles [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)] and the finiteness of Ш E/K was proved by the author [Invent. Math. 89, 527-560 (1987; Zbl 0628.14018)]. The significance of part (i) of the present theorem is that it shows that the conjecture of Birch and Swinnerton-Dyer is true for such curves up to an element of K divisible only by primes dividing w. One application is that the full conjecture is true for the curves Y 2 =X 3 -p 2 X where p is a prime congruent to 3(mod8), since work of M. Razar [Am. J. Math. 96, 104-126 (1974; Zbl 0296.14015)] shows that L(E/,1)0 and that the 2-part of the conjecture holds in this case.

Part (ii) of the theorem was previously known under the additional assumptions that E is defined over and ord s=1 L(E/,s) is odd, by work of R. Greenberg [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)] and the author [Invent. Math. 88, 405-422 (1987; Zbl 0623.14006)].


MSC:
11R23Iwasawa theory
11G05Elliptic curves over global fields
11G40L-functions of varieties over global fields
14G10Zeta-functions and related questions
11R37Class field theory for global fields
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