## 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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