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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 ${\Delta }=\text{Gal}\left({K}_{0}/K\right)$. Let ${K}_{\infty }$ be an abelian extension of $K$ containing ${K}_{0}$ such that $\text{Gal}\left({K}_{\infty }/{K}_{0}\right)\simeq {ℤ}_{p}$ or ${ℤ}_{p}^{2}$. For each finite extension $F$ of $K$ inside ${K}_{\infty }$, let $A\left(F\right)$ denote the $p$-part of the class group, $ℰ\left(F\right)$ the global units, $𝒞\left(F\right)$ the elliptic units, $U\left(F\right)$ the local units of $F{\otimes }_{K}{K}_{𝔭}$ congruent to 1 modulo the primes above $𝔭$, $\overline{ℰ}\left(F\right)$ the closure of $ℰ\left(F\right)\cap U\left(F\right)$ in $U\left(F\right)$, and similarly for $\overline{𝒞}\left(F\right)$. 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 $𝔭$.

All the above modules for $F={K}_{\infty }$ are modules over the Iwasawa algebra ${\Lambda }={ℤ}_{p}\left[\left[\text{Gal}\left({K}_{\infty }/K\right]\right]$, which is a direct sum of power series rings in 1 or 2 variables, corresponding to $\text{Gal}\left({K}_{\infty }/{K}_{0}\right)\simeq {ℤ}_{p}$ or ${ℤ}_{p}^{2}$. 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

$\text{char}\left(A\left({K}_{\infty }\right)\right)=\text{char}\left(\overline{ℰ}\left({K}_{\infty }\right)/\overline{𝒞}\left({K}_{\infty }\right)\right)\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{char}\left({X}_{\infty }\right)=\text{char}\left(U\left({K}_{\infty }\right)/\overline{𝒞}\left({K}_{\infty }\right)\right)·$

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

$\text{char}\left(A\left({K}_{\infty }\right)\right)\phantom{\rule{4.pt}{0ex}}\text{divides}\phantom{\rule{4.pt}{0ex}}\text{char}\left(\overline{ℰ}\left({K}_{\infty }\right)/\overline{𝒞}\left({K}_{\infty }\right)\right)·$

If $\chi$ is an irreducible ${ℤ}_{p}$-representation of ${\Delta }$ that is non-trivial on the decomposition group of $𝔭$ in ${\Delta }$, then

$\text{char}\left(A{\left({K}_{\infty }\right)}^{\chi }\right)=\text{char}\left(\overline{ℰ}{\left({K}_{\infty }\right)}^{\chi }/\overline{𝒞}{\left({K}_{\infty }\right)}^{\chi }\right)·$

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 ${\Omega }\in {ℂ}^{×}$. Write $w=#\left({𝒪}^{×}\right)$. (i) If $L\left(E/K,1\right)\ne 0$ then $E\left(K\right)$ is finite, the Tate- Shafarevich group ${\text{Ш}}_{E/K}$ of $E$ is finite and there is a $u\in 𝒪{\left[{w}^{-1}\right]}^{×}$ such that

$#\left({\text{Ш}}_{E/K}\right)=u#{\left(E\left(K\right)\right)}^{2}\frac{L\left(E/K,1\right)}{{\Omega }\overline{{\Omega }}}·$

(ii) If $L\left(E/K,1\right)=0$ then either $E\left(K\right)$ is infinite or the $𝔭$-part of ${\text{Ш}}_{E/K}$ is infinite for all primes $𝔭$ of $K$ not dividing $w$.

The finiteness of $E\left(K\right)$ 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 $3\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}8\right)$, since work of M. Razar [Am. J. Math. 96, 104-126 (1974; Zbl 0296.14015)] shows that $L\left(E/ℚ,1\right)\ne 0$ 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 $ℚ$ and ${\text{ord}}_{s=1}L\left(E/ℚ,s\right)$ 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 14G10 Zeta-functions and related questions 11R37 Class field theory for global fields