Iwasawa theory and modular forms
The main theme of this article is to study the \(\text{GL}_2\)-Iwasawa theory of \(p\)-adic Galois representations arising from modular forms. The author studies the (strict) Selmer groups associated to such Galois representations. If the image \(G\) of the Galois representation is an open subgroup of \(\text{GL}_2(\mathbb{Z}_p)\) (and hence non-abelian), the Selmer group is then a finitely generate discrete (left) module over the non-commutative Iwasawa algebra \(\Lambda(G)\). The author proves (Theorem 2.8), that the dual Selmer group tensored with \(\mathbb{Q}_p\) is infinite-dimensional over \(\mathbb{Q}_p\). Let \(L\) be a finite extension such that \(\text{Gal}(K_\infty/L)\) is pro-\(p\), where \(K_\infty\) denotes the trivializing extension associated to the modular form. The second main result of this article is Theorem 4.1. The dual Selmer group over \(L^{cyd}\) is infinite for the primitive modular forms of level one and even weight. The results (and methods) of this article are parallel to those proved for the Galois representations arising from elliptic curves (Coates, Howson, Ochi, Sujatha, Venjakob,\(\dots\)).
scientific article; zbMATH DE number 5074492