On Artin representations and nearly ordinary Hecke algebras over totally real fields
The main theme of this paper is to prove several modularity theorems for Hilbert cusp forms over totally real fields.NEWLINENEWLINELet \(F\) be a totally real number field and \(\rho : \text{Gal}(\overline F/F)\longrightarrow \mathrm{GL}_2(\mathbb C)\) a totally odd, irreducible continuous Galois representation. The author considers the case of icosahedral representations, namely, those having projective image in \(\mathrm{GL}_2(\mathbb C)/\mathbb C^\times\) is isomorphic to \(A_5\). The case when \(F=\mathbb Q\) is treated in [\textit{K. Buzzard} et al., Duke Math. J. 109, No. 2, 283--318 (2001; Zbl 1015.11021)] and the author follows basically the approach of [loc. cit.] Under some technical assumptions, the author shows that such a representation is modular, namely, comes form a Hilbert cusp eigenform of weight 1, and therefore the \(L\)-function \(L(\rho,s)\) of \(\rho\) is entire, proving a conjecture of Artin.NEWLINENEWLINEThe above results are obtained as consequences of the main result of the paper, which can be stated as follows. Let \(F\) be a totally real field, \(K\) be a finite extension of \(\mathbb Q_p\) with ring of integers \(\mathcal O\) and maximal ideal \(\mathfrak m\), \(p\) a rational prime which splits completely in \(F\) and \(\rho:\text{Gal}(\overline F/F)\longrightarrow \mathrm{GL}_2(\mathcal O)\) a continuous representation which is ramified only at finitely many primes. One supposes that \(\bar\rho=\rho\mod\mathfrak m\) admits a modular lifting, which is potentially ordinary and potentially Barsotti-Tate at every prime of \(F\) over \(p\) and that the restriction of \(\bar\rho\) to \(\text{Gal}(\overline F/F(\zeta_p))\) is absolutely irreducible. One finally assumes that the restriction of \(\bar\rho\) at the decomposition group at every prime of \(F\) above \(p\) is a direct sum of characters, distinct modulo \(\mathfrak m\), such that the images of the inertia subgroups via each of these characters are finite. (If \(p=2\) one also needs stronger assumptions). Then there exists a Hilbert cusp eigenform \(f\) of weight 1 whose associated representation is isomorphic to \(\rho\).
scientific article; zbMATH DE number 6220362