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Given a curve $X$, the group action of the automorphism group $G=$Aut$(X)$ on $X$ may allow one to decompose its Jacobian $J$ as a product of abelian varieties of lower dimension. Idempotent relations in the group ring $\mathbb{Q}[G]$ correspond, via a natural map of $\mathbb{Q}$-algebras $\rho: \mathbb{Q}[G] \to \text{End}(J) \otimes_{\mathbb{Z}} \mathbb{Q}$, to factorizations of the Jacobian variety into abelian varieties that are the images of the Jacobian under the endomorphisms determined by those idempotents. The dimensions of the factors in this group algebra decomposition are computed as inner products of certain characters of $G$ (and are thus canonical).

Note: factors in the group algebra decomposition need not be simple (it may be possible to decompose them further), and factors of the same dimension may be isogenous.

For further details see the source of the automorphisms of curves data.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2018-07-31 23:15:28
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