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 be an imaginary quadratic field, let be a prime number not dividing the number of roots of unity in the Hilbert class field of , and let be a prime of above and the corresponding completion. Fix an abelian extension of containing and let . Let be an abelian extension of containing such that or . For each finite extension of inside , let denote the -part of the class group, the global units, the elliptic units, the local units of congruent to 1 modulo the primes above , the closure of in , and similarly for . When is an infinite extension of , define these groups to be the inverse limits of the corresponding groups for finite subextensions. Let be the Galois group of the maximal abelian -extension of unramified outside the primes above .
All the above modules for are modules over the Iwasawa algebra , which is a direct sum of power series rings in 1 or 2 variables, corresponding to or . 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 splits into two distinct primes in . Then
(ii) Suppose remains prime or ramifies in . Then
If is an irreducible -representation of that is non-trivial on the decomposition group of in , then
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 is an elliptic curve defined over an imaginary quadratic field , with complex multiplication by the ring of integers of , and with minimal period lattice generated by . Write . (i) If then is finite, the Tate- Shafarevich group of is finite and there is a such that
(ii) If then either is infinite or the -part of is infinite for all primes of not dividing .
The finiteness of was proved by J. Coates and A. Wiles [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)] and the finiteness of 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 divisible only by primes dividing . One application is that the full conjecture is true for the curves where is a prime congruent to , since work of M. Razar [Am. J. Math. 96, 104-126 (1974; Zbl 0296.14015)] shows that 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 is defined over and 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)].