On the ideal class groups of real abelian number fields.

*(English)*Zbl 0665.12003This paper presents a new relationship between the ideal class group \(A\) and the unit group \(E\) of a real abelian field \(K\). In fact, the author finds annihilators of the \(p\)-class group \((A)_ p\) related to the structure of \(W=E/C\) (\(C=\) circular units), and in this way obtains smooth statements about relations between annihilators of \((A)_ p\) and \((W)_ p\).

Let \(m\) denote the conductor of \(K\), let \(\zeta_ n\) be a primitive \(n\)th root of unity, and put \(\Delta =\text{Gal}(K/\mathbb Q)\). Define the group \(C\) of circular units of \(K\) to consist of all units of the form \(f(1)\), where \[ f(X)=\pm \prod^{j}_{i=1}\prod^{m-1}_{k=1}(X^ i-\zeta^ k_ m)^{a_{ik}} \in K(X) \] with \(a_{ik}\in\mathbb Z\) and \(j\geq 1\) (the author points out that this definition differs from the known and may produce a larger group). Consider first a field \(L=K(\zeta_ q)\), where \(q\) is a sufficiently large prime that splits completely in \(K\). Starting with a unit \(\delta =f(1)\in C\) the author observes that a generating automorphism of \(L/K\) maps certain elements \(\alpha\in L\) to \(f(\zeta_ q)\alpha\). On studying the prime factorization of \(\alpha\) he infers, back to \(K\), a decomposition \[ (N_{L/K} (\alpha)) = R^ b \prod_{\sigma\in\Delta} \sigma^{-1} (Q)^{r_\sigma}, \tag{\(*\)} \] valid under certain conditions, where \(R\) and \(Q\) are ideals of \(K\), \(Q\) a prime above \(q\), and \(b\) is a divisor of \(q-1\). The exponents \(r_{\sigma}\) have a deep connection to the units of \(K\). Here a crucial role is played by a local-global theorem depending on Chebotarev’s density theorem.

From (*) it follows that if \(R^ b\) is principal then \(\sum_{\sigma \in \Delta}r_{\sigma}\sigma^{-1} \) annihilates the ideal class \({\mathcal C}\) containing \(Q\). By use of the properties of \(r_{\sigma}\) this implies the following results: If \(b=p^ n\) is the exponent of \((A)_ p\) and if \(\delta\) satisfies, for all \(\sigma\in \Delta\), a congruence \(\sigma (\delta)\equiv \delta^{c_{\sigma}}\bmod E^{p^ n}\) with \(p\nmid c_{\sigma}\), then \(\sum_{\sigma \in \Delta}c_{\sigma}\sigma^{-1} \), multiplied by a certain integer \(2d\), annihilates all classes \({\mathcal C}\in (A)_ p\) containing an ideal \(Q\). If fact, \(d\) depends on \(| \delta |\) in a simple way.

Suppose that \(p\nmid [K:\mathbb Q]\) and \(\chi\) is a non-trivial \(p\)-adic valued character of \(\Delta\), with the corresponding idempotent \(e_{\chi}\) of \(\mathbb Z[\Delta]\). Then, by taking \(c_{\sigma}=\chi (\sigma)\) one proves that \(e_{\chi}(A)_ p\) is annihilated by the exact exponent \(p^ a\) of \(e_{\chi}(W)_ p\). As corollary it follows that if \(K\) is a subfield of \(\mathbb Q(\zeta_ p+\zeta_ p^{-1})\), then every annihilator of \((W)_ p\) annihilates \((A)_ p\). Finally, a generalization of the above ideas leads to the beautiful theorem:

If \(p\nmid [K:\mathbb Q]\) and if \(\theta\in\mathbb Z[\Delta]\) annhilates \((W)_ p\), then \(2\theta\) annihilates \((A)_ p\).

Let \(m\) denote the conductor of \(K\), let \(\zeta_ n\) be a primitive \(n\)th root of unity, and put \(\Delta =\text{Gal}(K/\mathbb Q)\). Define the group \(C\) of circular units of \(K\) to consist of all units of the form \(f(1)\), where \[ f(X)=\pm \prod^{j}_{i=1}\prod^{m-1}_{k=1}(X^ i-\zeta^ k_ m)^{a_{ik}} \in K(X) \] with \(a_{ik}\in\mathbb Z\) and \(j\geq 1\) (the author points out that this definition differs from the known and may produce a larger group). Consider first a field \(L=K(\zeta_ q)\), where \(q\) is a sufficiently large prime that splits completely in \(K\). Starting with a unit \(\delta =f(1)\in C\) the author observes that a generating automorphism of \(L/K\) maps certain elements \(\alpha\in L\) to \(f(\zeta_ q)\alpha\). On studying the prime factorization of \(\alpha\) he infers, back to \(K\), a decomposition \[ (N_{L/K} (\alpha)) = R^ b \prod_{\sigma\in\Delta} \sigma^{-1} (Q)^{r_\sigma}, \tag{\(*\)} \] valid under certain conditions, where \(R\) and \(Q\) are ideals of \(K\), \(Q\) a prime above \(q\), and \(b\) is a divisor of \(q-1\). The exponents \(r_{\sigma}\) have a deep connection to the units of \(K\). Here a crucial role is played by a local-global theorem depending on Chebotarev’s density theorem.

From (*) it follows that if \(R^ b\) is principal then \(\sum_{\sigma \in \Delta}r_{\sigma}\sigma^{-1} \) annihilates the ideal class \({\mathcal C}\) containing \(Q\). By use of the properties of \(r_{\sigma}\) this implies the following results: If \(b=p^ n\) is the exponent of \((A)_ p\) and if \(\delta\) satisfies, for all \(\sigma\in \Delta\), a congruence \(\sigma (\delta)\equiv \delta^{c_{\sigma}}\bmod E^{p^ n}\) with \(p\nmid c_{\sigma}\), then \(\sum_{\sigma \in \Delta}c_{\sigma}\sigma^{-1} \), multiplied by a certain integer \(2d\), annihilates all classes \({\mathcal C}\in (A)_ p\) containing an ideal \(Q\). If fact, \(d\) depends on \(| \delta |\) in a simple way.

Suppose that \(p\nmid [K:\mathbb Q]\) and \(\chi\) is a non-trivial \(p\)-adic valued character of \(\Delta\), with the corresponding idempotent \(e_{\chi}\) of \(\mathbb Z[\Delta]\). Then, by taking \(c_{\sigma}=\chi (\sigma)\) one proves that \(e_{\chi}(A)_ p\) is annihilated by the exact exponent \(p^ a\) of \(e_{\chi}(W)_ p\). As corollary it follows that if \(K\) is a subfield of \(\mathbb Q(\zeta_ p+\zeta_ p^{-1})\), then every annihilator of \((W)_ p\) annihilates \((A)_ p\). Finally, a generalization of the above ideas leads to the beautiful theorem:

If \(p\nmid [K:\mathbb Q]\) and if \(\theta\in\mathbb Z[\Delta]\) annhilates \((W)_ p\), then \(2\theta\) annihilates \((A)_ p\).

Reviewer: Tauno Metsänkylä (Turku)

##### MSC:

11R18 | Cyclotomic extensions |

11R29 | Class numbers, class groups, discriminants |

11G16 | Elliptic and modular units |

11R27 | Units and factorization |