# zbMATH — the first resource for mathematics

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 $$\ker\psi$$.

##### 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