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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 $${\mathfrak p}$$ be a prime of $$K$$ above $$p$$ and $$K_{\mathfrak p}$$ the corresponding completion. Fix an abelian extension $$K_ 0$$ of $$K$$ containing $$H$$ and let $$\Delta=\text{Gal}(K_ 0/K)$$. Let $$K_ \infty$$ be an abelian extension of $$K$$ containing $$K_ 0$$ such that $$\text{Gal}(K_ \infty/K_ 0)\simeq\mathbb{Z}_ p$$ or $$\mathbb{Z}^ 2_ p$$. For each finite extension $$F$$ of $$K$$ inside $$K_ \infty$$, let $$A(F)$$ denote the $$p$$-part of the class group, $${\mathcal E}(F)$$ the global units, $${\mathcal C}(F)$$ the elliptic units, $$U(F)$$ the local units of $$F\otimes_ KK_{\mathfrak p}$$ congruent to 1 modulo the primes above $${\mathfrak p}$$, $$\overline {\mathcal E}(F)$$ the closure of $${\mathcal E}(F)\cap U(F)$$ in $$U(F)$$, and similarly for $$\overline {\mathcal C}(F)$$. When $$F$$ is an infinite extension of $$K$$, define these groups to be the inverse limits of the corresponding groups for finite subextensions. Let $$X_ \infty$$ be the Galois group of the maximal abelian $$p$$-extension of $$K_ \infty$$ unramified outside the primes above $${\mathfrak p}$$.
All the above modules for $$F=K_ \infty$$ are modules over the Iwasawa algebra $$\Lambda=\mathbb{Z}_ p[[\text{Gal}(K_ \infty/K]]$$, which is a direct sum of power series rings in 1 or 2 variables, corresponding to $$\text{Gal}(K_ \infty/K_ 0)\simeq\mathbb{Z}_ p$$ or $$\mathbb{Z}^ 2_ p$$. 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 $\text{char}(A(K_ \infty))=\text{char}(\overline {\mathcal E}(K_ \infty)/\overline {\mathcal C}(K_ \infty))\text{ and }\text{char}(X_ \infty)=\text{char}(U(K_ \infty)/\overline {\mathcal C}(K_ \infty)).$ (ii) Suppose $$p$$ remains prime or ramifies in $$K$$. Then $\text{char}(A(K_ \infty)) \text{ divides } \text{char}(\overline {\mathcal E}(K_ \infty)/\overline {\mathcal C}(K_ \infty)).$ If $$\chi$$ is an irreducible $$\mathbb{Z}_ p$$-representation of $$\Delta$$ that is non-trivial on the decomposition group of $${\mathfrak p}$$ in $$\Delta$$, then $\text{char}(A(K_ \infty)^ \chi)=\text{char} (\overline {\mathcal E}(K_ \infty)^ \chi/\overline{\mathcal C}(K_ \infty)^ \chi).$ 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 $${\mathcal O}$$ of $$K$$, and with minimal period lattice generated by $$\Omega\in\mathbb{C}^ \times$$. Write $$w=\#({\mathcal O}^ \times)$$. (i) If $$L(E/K,1)\neq0$$ then $$E(K)$$ is finite, the Tate- Shafarevich group $$\text Ш_{E/K}$$ of $$E$$ is finite and there is a $$u\in{\mathcal O}[w^{-1}]^ \times$$ such that $\#(\text Ш_{E/K})=u\#(E(K))^ 2{L(E/K,1)\over \Omega\overline\Omega}.$ (ii) If $$L(E/K,1)=0$$ then either $$E(K)$$ is infinite or the $${\mathfrak p}$$-part of $$\text Ш_{E/K}$$ is infinite for all primes $${\mathfrak p}$$ 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 $$\text{ Ш}_{E/K}$$ 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 $$Y^ 2=X^ 3-p^ 2X$$ where $$p$$ is a prime congruent to $$3 \pmod 8$$, since work of M. Razar [Am. J. Math. 96, 104-126 (1974; Zbl 0296.14015)] shows that $$L(E/\mathbb{Q},1)\neq 0$$ 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 $$\mathbb{Q}$$ and $$\text{ord}_{s=1}L(E/\mathbb{Q},s)$$ 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:
 11R23 Iwasawa theory 11G05 Elliptic curves over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11R37 Class field theory
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