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Zbl 0737.11030
Rubin, Karl
The ``main conjectures" of Iwasawa theory for imaginary quadratic fields. (English)
[J] Invent. Math. 103, No. 1, 25-68 (1991). ISSN 0020-9910; ISSN 1432-1297/e

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.\par 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 ${\germ p}$ be a prime of $K$ above $p$ and $K\sb{\germ p}$ the corresponding completion. Fix an abelian extension $K\sb 0$ of $K$ containing $H$ and let $\Delta=\text{Gal}(K\sb 0/K)$. Let $K\sb \infty$ be an abelian extension of $K$ containing $K\sb 0$ such that $\text{Gal}(K\sb \infty/K\sb 0)\simeq\bbfZ\sb p$ or $\bbfZ\sp 2\sb p$. For each finite extension $F$ of $K$ inside $K\sb \infty$, let $A(F)$ denote the $p$-part of the class group, ${\cal E}(F)$ the global units, ${\cal C}(F)$ the elliptic units, $U(F)$ the local units of $F\otimes\sb KK\sb{\germ p}$ congruent to 1 modulo the primes above ${\germ p}$, $\overline {\cal E}(F)$ the closure of ${\cal E}(F)\cap U(F)$ in $U(F)$, and similarly for $\overline {\cal C}(F)$. When $F$ is an infinite extension of $K$, define these groups to be the inverse limits of the corresponding groups for finite subextensions. Let $X\sb \infty$ be the Galois group of the maximal abelian $p$-extension of $K\sb \infty$ unramified outside the primes above ${\germ p}$.\par All the above modules for $F=K\sb \infty$ are modules over the Iwasawa algebra $\Lambda=\bbfZ\sb p[[\text{Gal}(K\sb \infty/K]]$, which is a direct sum of power series rings in 1 or 2 variables, corresponding to $\text{Gal}(K\sb \infty/K\sb 0)\simeq\bbfZ\sb p$ or $\bbfZ\sp 2\sb p$. It is possible to define characteristic power series (denoted by ``char'') for such modules.\par The main theorem of the paper is the following. (i) Suppose $p$ splits into two distinct primes in $K$. Then $$\text{char}(A(K\sb \infty))=\text{char}(\overline {\cal E}(K\sb \infty)/\overline {\cal C}(K\sb \infty))\text{ and }\text{char}(X\sb \infty)=\text{char}(U(K\sb \infty)/\overline {\cal C}(K\sb \infty)).$$ (ii) Suppose $p$ remains prime or ramifies in $K$. Then $$\text{char}(A(K\sb \infty)) \text{ divides } \text{char}(\overline {\cal E}(K\sb \infty)/\overline {\cal C}(K\sb \infty)).$$ If $\chi$ is an irreducible $\bbfZ\sb p$-representation of $\Delta$ that is non-trivial on the decomposition group of ${\germ p}$ in $\Delta$, then $$\text{char}(A(K\sb \infty)\sp \chi)=\text{char} (\overline {\cal E}(K\sb \infty)\sp \chi/\overline{\cal C}(K\sb \infty)\sp \chi).$$ The first part of the theorem in the one-variable case was a question raised by {\it J. Coates} and {\it 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.\par 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 ${\cal O}$ of $K$, and with minimal period lattice generated by $\Omega\in\bbfC\sp \times$. Write $w=\#({\cal O}\sp \times)$. (i) If $L(E/K,1)\ne0$ then $E(K)$ is finite, the Tate- Shafarevich group $\text{\cyr Sh}\sb{E/K}$ of $E$ is finite and there is a $u\in{\cal O}[w\sp{-1}]\sp \times$ such that $$\#(\text{\cyr Sh}\sb{E/K})=u\#(E(K))\sp 2{L(E/K,1)\over \Omega\overline\Omega}.$$ (ii) If $L(E/K,1)=0$ then either $E(K)$ is infinite or the ${\germ p}$-part of $\text{\cyr Sh}\sb{E/K}$ is infinite for all primes ${\germ p}$ of $K$ not dividing $w$.\par The finiteness of $E(K)$ was proved by {\it J. Coates} and {\it A. Wiles} [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)] and the finiteness of ${\cyr Sh}\sb{E/K}$ 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 $Y\sp 2=X\sp 3-p\sp 2X$ where $p$ is a prime congruent to $3 \pmod 8$, since work of {\it M. Razar} [Am. J. Math. 96, 104-126 (1974; Zbl 0296.14015)] shows that $L(E/\bbfQ,1)\ne 0$ and that the 2-part of the conjecture holds in this case.\par Part (ii) of the theorem was previously known under the additional assumptions that $E$ is defined over $\bbfQ$ and $\text{ord}\sb{s=1}L(E/\bbfQ,s)$ is odd, by work of {\it R. Greenberg} [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)] and the author [Invent. Math. 88, 405-422 (1987; Zbl 0623.14006)].
[L.Washington (College Park)]

MSC 2000:: *11R23 Iwasawa theory
11G05 Elliptic curves over global fields
11G40 L-functions of varieties over global fields
14G10 Zeta-functions and related questions
11R37 Class field theory for global fields

Keywords: Birch-Swinnerton-Dyer conjecture; Galois group; maximal abelian $p$-extension; non-split primes; elliptic curves; Tate-Shafarevich group

Citations: Zbl 0623.14006; Zbl 0359.14009; Zbl 0628.14018; Zbl 0296.14015; Zbl 0546.14015; Zbl 0442.12007

Cited in: Zbl 1235.11048 Zbl 1230.11135 Zbl 1214.11122 Zbl 1193.11103 Zbl 1183.11032 Zbl 1144.11075 Zbl 1200.11082 Zbl 1183.11067 Zbl 1129.11048 Zbl 1201.11051 Zbl 1055.14047 Zbl 1159.11311 Zbl 0993.11033 Zbl 0991.11028 Zbl 0918.11037 Zbl 0913.11026 Zbl 0849.11089 Zbl 0809.11066 Zbl 0744.11053

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