Selmer varieties for curves with CM Jacobians

Abstract

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.