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On a Shioda-Mitani formula. (Sur une formule de Shioda et Mitani.) (French) Zbl 0699.14038

The author computes for certain abelian surfaces \(X\) (over the complex numbers \(\mathbb C\)) the numbers of curves \(C\) whose Jacobians are isomorphic to \(X\). The \(X\) under question have multiplication by an indefinite quaternion-algebra, and are simple.
The idea of proof is as follows: One has to classify the principal polarisations on \(X\). This becomes a problem in the arithmetic of quaternions, and is governed by certain class numbers.

MSC:

14H40 Jacobians, Prym varieties
14K22 Complex multiplication and abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G18 Arithmetic aspects of modular and Shimura varieties
14K05 Algebraic theory of abelian varieties
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References:

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