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We introduce a natural way to define Selmer groups and -adic -functions for modular forms of weight 1. The corresponding Galois representation of is a 2-dimensional Artin representation with odd determinant. Thus, the dimension of the (+1)-eigenspace for complex conjugation is 1. Choose a prime such that the restriction of to the local Galois group has a 1-dimensional constituent with multiplicity 1. If we fix the choice of such an , we can define a Selmer group and a -adic -function. On the algebraic side, we prove that the Selmer group over the cyclotomic -extension of is a cotorsion module over the Iwasawa algebra . That result is valid for an Artin representation of arbitrary dimension under the assumption that and that such an can be chosen. On the analytic side, the corresponding complex -function has no critical values and the definition of the -adic -function depends on deforming the Galois representation by Hida theory.
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Published: 1 January 2020
First available in Project Euclid: 12 January 2021