Congruence module for \(\text{GL}(2)\) of an imaginary quadratic field and Iwasawa theory of biquadratic CM field
Let \(M\) be a biquadratic CM-field and \(K\) one of its imaginary quadratic subfields. Let \(\tau\) be the nontrivial \(K\)-automorphism of \(M\) and \(p\geq 3\) be a prime number which is totally split in \(M/K\). Let \(M_\infty\) be the union of all ray class fields of conductor a \(p\)-power over \(M\) and \(M_\infty '\) the subfield of \(M_\infty\) which is fixed by the ``centralizer'' of \(\tau\) in the Galois group of \(M_\infty\) over \(M\). One wants to study a certain Galois group \textbf{X} (attached to \(M_\infty '\)) as a module under the Iwasawa-algebra \({\mathbb Z}_pW'\), where \(W'\) is a complement to the torsion subgroup of \(\text{ Gal}(M_\infty '/M)\). For every character \(v\) of this torsion subgroup there exists a \(\Lambda\)-adic modular form \(\Theta_v\), leading to a congruence module \(C_0\) for the ordinary Hecke-algebra. One of he main issues of the paper is to show that the characteristic series of \(C_0\) divides the characteristic series of \textbf{X}. The methods build on those developed by \textit{H. Hida} and \textit{J. Tilouine} in [Invent. Math. 117, No. 1, 89-147 (1994; Zbl 0819.11047)]. As a preparation, a good deal of results from the theory of automorphic forms for \(\text{GL}(2,K)\) is explained. Then \(\Lambda\)-adic forms are introduced, following Taylor and Hida. To \(\Theta_v\) there corresponds a Galois-representation \(\rho_v\), and a Selmer group for \(\text{Ad}(\rho_v)\) is linked to \textbf{X} by Mazur's deformation theory and this gives the final clue to the main theorem. The comparison of his results with loc.\ cit.\ lead the author to conjecture a relation between the congruence modules for \(K\) and for the real quadratic subfield of \(M\).
scientific article; zbMATH DE number 1425092
scientific article