Selmer varieties for curves with CM Jacobians

TL;DR

This paper investigates Selmer varieties linked to curves with CM Jacobians, using Iwasawa theory to establish dimension bounds and provide a new proof of Diophantine finiteness over b1b1 for these curves.

Contribution

It introduces a novel approach combining Selmer varieties and Iwasawa theory to prove finiteness results for curves with CM Jacobians.

Findings

Dimension bounds for Selmer varieties established

New proof of Diophantine finiteness over b1b1 for curves with CM Jacobians

Application of elementary multi-variable Iwasawa theory

Abstract

We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi-variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over $\Q$ for such curves.