The “main conjectures” of Iwasawa theory for imaginary quadratic fields.

*(English)*Zbl 0737.11030This 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)].

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)].

Reviewer: L.Washington (College Park)

##### 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 |

##### Keywords:

Birch-Swinnerton-Dyer conjecture; Galois group; maximal abelian \(p\)-extension; non-split primes; elliptic curves; Tate-Shafarevich group##### References:

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