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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 ${\germ p}$ 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 ${\germ p}$. 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. 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. 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 $w$. 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. 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)].

11R23Iwasawa theory
11G05Elliptic curves over global fields
11G40$L$-functions of varieties over global fields
14G10Zeta-functions and related questions
11R37Class field theory for global fields
Full Text: DOI EuDML
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