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Selmer varieties for curves with CM Jacobians. (English) Zbl 1283.11092
Summary: We study the Selmer variety associated to a canonical quotient of the $$\mathbb{Q}_{p}$$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $$\mathbb{Q}$$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multivariable Iwasawa theory is used to prove bounds for the dimension of the Selmer variety, which, in turn, leads to a new proof of finiteness of rational points on such curves.

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11R23 Iwasawa theory
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