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A geometric description of the class invariant homomorphism. (English) Zbl 0833.11055

Let \(F\) be a number field with ring of integers \(O_F\), \(G\) a finite abelian group, \(B\) a Hopf order over \(O_F\) in \(FG\), \(C\) a principal homogeneous space for \(B\); then \(B^*\) acts on \(C\). The class invariant homomorphism \(\varphi_n\), also known as the Picard invariant homomorphism, maps \(C\) in \(PHS (B)\) to the class of \(C\) in \(\text{Pic} (B^*)\).
The author examines this map for \(B_n\) representing the \(O_F\)-group scheme \(\ker [p^n ]= A_{p^n}\) on the Neron model \(A\) of an abelian variety \(A/F\) with everywhere good reduction. The author shows that if \(Q\) is a point in the Cartier dual of \(A(F)\), \(L_Q\) is the line bundle on \(A\) associated to \(Q\), and \(L^n\) is \(L_q\) restricted to \(A_{p^n}\), then the class of \(L_n\) in \(\text{Pic} (A_{p^n})= \text{Pic} (B_n)\) is the same as that of the Kummer order of \(B_n^*\) in the Kummer extension of \(F\) defined by taking \(p^n\)th roots of \(Q\) on \(A\). As a consequence, the author verifies that the map \(\varphi_n\) maps into the primitive elements of \(\text{Pic} (A_{p^n})\). The article concludes with some questions about the image and kernel of \(\varphi_\infty= \varinjlim \varphi_n\).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
14L15 Group schemes
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References:

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