##
**Hopf orders and Galois module structure. With contributions by N.P.Byott.**
*(English)*
Zbl 0811.11068

Group rings and class groups, Notes Talks DMV-Semin., Günzburg/Ger. 1990, DMV Semin. 18, 153-210 (1992).

[For the entire collection see Zbl 0742.00085.]

This article gives a survey of some of the author’s important work on global Galois module structure and \(L\)-functions.

Let \(K\) be a number field with ring of integers \({\mathcal O}\), \(G\) be a finite group, and \({\mathfrak A}\) an order over \({\mathcal O}\) in \(KG\). The author begins by reviewing the tame theory: Noether’s theorem (if \(N/K\) is tamely ramified then \({\mathcal O}_ N\) is locally free over \({\mathfrak A}={\mathcal O} G\)), the idelic description of the class group \(\text{Cl} ({\mathfrak A})\), and Taylor’s theorem, which in \(\text{Cl} (\mathbb{Z} G)\) identifies the class of \({\mathcal O}_ N\) with a class derived from the Artin root numbers associated with the functional equation of the Artin \(L\)-functions of characters of \(G\). No proofs are given.

The next two sections review Hopf orders over \({\mathcal O}\) in \(KG\) and principal homogeneous spaces for these Hopf orders. The attractive exposition includes a description of the Tate-Oort classification for Hopf orders in \(KG\), \(G\) of prime order \(p\), and a description of the map \(\psi\) from the group \(\text{PHS} ({\mathfrak A}^*)\) of principal homogeneous spaces for \({\mathfrak A}^*\) to \(\text{Cl} ({\mathfrak A})\), in terms of resolvants, following L. R. McCulloh [J. Algebra 82, 102- 134 (1983; Zbl 0508.12008)]. If \(G\) has prime order, \(K\) contains \(p\)-th roots of unity and \({\mathfrak A}\) is a Hopf order in \(KG\) with \({\mathfrak B}={\mathfrak A}^*\), then \(\text{PHS} ({\mathfrak B})\) and \(\ker \psi\) are identified as subgroups of \(K^*/ (K^*)^ p\), following the first author’s paper in Semin. Theor. Nombres Bordx., II. Ser 2, 255-271 (1990; Zbl 0731.14032).

The fourth section fixes \(K= \mathbb{Q}[ \zeta]\), \(\zeta\) a primitive \(p\)-th root of unity, \(G\) of order \(p\) and \({\mathfrak A}={\mathcal O} G\). Decomposing \(\ker\psi\) into eigenspaces under the action of \(\Delta= \text{Gal} (K/\mathbb{Q})\), the author shows that whether or not the eigenspace corresponding to an even power of the Teichmüller character is zero depends on a congruence condition on the corresponding \(p\)-adic \(L\)- function. The proof uses a result of Mazur and Wiles and the validity of Leopoldt’s conjecture for \(K\).

In the final section the author considers \(E\), an elliptic curve defined over an imaginary quadratic extension \(F\) of \(\mathbb{Q}\), admitting complex multiplication by \({\mathcal O}_ F\). If \(p\) is a prime which splits, \(p= \pi\pi^*\) in \({\mathcal O}_ F\), let \(E_ \pi= \ker \pi: E(\mathbb{Q}^ c)\to E(\mathbb{Q}^ c)\), a group of order \(p\), and let \(K\) be a field extension of \(F\) over which the points of \(E_ \pi\) are defined. Let \({\mathcal E}\) be the Neron minimal model of \(E\) over \(K\), let \({\mathcal E}_ \pi= \ker \pi:{\mathcal E}\to {\mathcal E}\), then \({\mathcal E}_ \pi= \text{Spec}({\mathfrak B})\) for some Hopf order \({\mathfrak B}\) in \(\operatorname{Hom}(E_ \pi, K)\). Then we have the homomorphism \(\psi: \text{PHS}({\mathfrak B})\to \text{Cl}({\mathfrak A})\) where \({\mathfrak A}={\mathfrak B}^*\). Under certain additional conditions on \(p\), there is an \(\mathbb{F}_{p^ \Delta}\)-action on \(\ker \psi\); the vanishing of the \(i\)-th eigenspace is then equivalent to a congruence condition on a certain \(p\)-adic \(L\)-function, quite analogous to the cyclotomic result above. The result is stated without proof. In addition, the authors define a map \(\alpha: E(K)\to \text{PH} ({\mathfrak B})\) and observe that given the Birch-Swinnerton-Dyer conjecture for \(E/K\), the non-vanishing of the Hasse-Weil \(L\)-function \(L(E/K,s)\) at \(s=1\) implies that for any finite extension \(N\) of \(K\) and any \(P\) in \(E(N)\), \(\alpha_ N(P)\cong{\mathfrak A}_ N\) as \({\mathfrak A}\)-modules: in particular, for any \(P\) in \(E(K)\), \(\alpha(P)\) is in \(\ker\psi\).

This article gives a survey of some of the author’s important work on global Galois module structure and \(L\)-functions.

Let \(K\) be a number field with ring of integers \({\mathcal O}\), \(G\) be a finite group, and \({\mathfrak A}\) an order over \({\mathcal O}\) in \(KG\). The author begins by reviewing the tame theory: Noether’s theorem (if \(N/K\) is tamely ramified then \({\mathcal O}_ N\) is locally free over \({\mathfrak A}={\mathcal O} G\)), the idelic description of the class group \(\text{Cl} ({\mathfrak A})\), and Taylor’s theorem, which in \(\text{Cl} (\mathbb{Z} G)\) identifies the class of \({\mathcal O}_ N\) with a class derived from the Artin root numbers associated with the functional equation of the Artin \(L\)-functions of characters of \(G\). No proofs are given.

The next two sections review Hopf orders over \({\mathcal O}\) in \(KG\) and principal homogeneous spaces for these Hopf orders. The attractive exposition includes a description of the Tate-Oort classification for Hopf orders in \(KG\), \(G\) of prime order \(p\), and a description of the map \(\psi\) from the group \(\text{PHS} ({\mathfrak A}^*)\) of principal homogeneous spaces for \({\mathfrak A}^*\) to \(\text{Cl} ({\mathfrak A})\), in terms of resolvants, following L. R. McCulloh [J. Algebra 82, 102- 134 (1983; Zbl 0508.12008)]. If \(G\) has prime order, \(K\) contains \(p\)-th roots of unity and \({\mathfrak A}\) is a Hopf order in \(KG\) with \({\mathfrak B}={\mathfrak A}^*\), then \(\text{PHS} ({\mathfrak B})\) and \(\ker \psi\) are identified as subgroups of \(K^*/ (K^*)^ p\), following the first author’s paper in Semin. Theor. Nombres Bordx., II. Ser 2, 255-271 (1990; Zbl 0731.14032).

The fourth section fixes \(K= \mathbb{Q}[ \zeta]\), \(\zeta\) a primitive \(p\)-th root of unity, \(G\) of order \(p\) and \({\mathfrak A}={\mathcal O} G\). Decomposing \(\ker\psi\) into eigenspaces under the action of \(\Delta= \text{Gal} (K/\mathbb{Q})\), the author shows that whether or not the eigenspace corresponding to an even power of the Teichmüller character is zero depends on a congruence condition on the corresponding \(p\)-adic \(L\)- function. The proof uses a result of Mazur and Wiles and the validity of Leopoldt’s conjecture for \(K\).

In the final section the author considers \(E\), an elliptic curve defined over an imaginary quadratic extension \(F\) of \(\mathbb{Q}\), admitting complex multiplication by \({\mathcal O}_ F\). If \(p\) is a prime which splits, \(p= \pi\pi^*\) in \({\mathcal O}_ F\), let \(E_ \pi= \ker \pi: E(\mathbb{Q}^ c)\to E(\mathbb{Q}^ c)\), a group of order \(p\), and let \(K\) be a field extension of \(F\) over which the points of \(E_ \pi\) are defined. Let \({\mathcal E}\) be the Neron minimal model of \(E\) over \(K\), let \({\mathcal E}_ \pi= \ker \pi:{\mathcal E}\to {\mathcal E}\), then \({\mathcal E}_ \pi= \text{Spec}({\mathfrak B})\) for some Hopf order \({\mathfrak B}\) in \(\operatorname{Hom}(E_ \pi, K)\). Then we have the homomorphism \(\psi: \text{PHS}({\mathfrak B})\to \text{Cl}({\mathfrak A})\) where \({\mathfrak A}={\mathfrak B}^*\). Under certain additional conditions on \(p\), there is an \(\mathbb{F}_{p^ \Delta}\)-action on \(\ker \psi\); the vanishing of the \(i\)-th eigenspace is then equivalent to a congruence condition on a certain \(p\)-adic \(L\)-function, quite analogous to the cyclotomic result above. The result is stated without proof. In addition, the authors define a map \(\alpha: E(K)\to \text{PH} ({\mathfrak B})\) and observe that given the Birch-Swinnerton-Dyer conjecture for \(E/K\), the non-vanishing of the Hasse-Weil \(L\)-function \(L(E/K,s)\) at \(s=1\) implies that for any finite extension \(N\) of \(K\) and any \(P\) in \(E(N)\), \(\alpha_ N(P)\cong{\mathfrak A}_ N\) as \({\mathfrak A}\)-modules: in particular, for any \(P\) in \(E(K)\), \(\alpha(P)\) is in \(\ker\psi\).

Reviewer: L.N.Childs (Albany)

### MSC:

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11G05 | Elliptic curves over global fields |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R54 | Other algebras and orders, and their zeta and \(L\)-functions |