On a congruence prime criterion for cusp forms on \(\mathrm{GL}_2\) over number fields
Let \(f\) be a normalized Hecke eigenform on GL\(_2\) over a number field \(F\), and \(\mathfrak{P}\) be a prime ideal of a number field \(K\) which contains the Galois closure of the number field generated by all Fourier coefficients of \(f\) over \(F\). The aim of the paper under review is to give a sufficient condition for \(\mathfrak{P}\) to be a congruence prime for \(f\). The main result of the paper, whose precise statements are rather technical and will not be given here, extends the congruence prime criteria known previously for cusp forms on GL\(_2\) over \(\mathbb{Q}\) by \textit{H. Hida} [Invent. Math. 63, 225--261 (1981; Zbl 0459.10018)], the case of cusp forms on GL\(_2\) over imaginary quadratic fields by \textit{E. Urban} [Compos. Math. 99, No. 3, 283--324 (1995; Zbl 0846.11029)], and also the case of cusp forms on GL\(_2\) over totally real number fields by \textit{E. Ghate} [Compos. Math. 132, No. 3, 243--281 (2002; Zbl 1004.11019)] and \textit{M. Dimitrov} [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 505--551 (2005; Zbl 1160.11325)]. \ The strategy of the proof of the main theorem is similar to that of Hida's theorem for cusp forms on GL\(_2\) over \(\mathbb{Q}\). The author first computes the special value at \(s=1\) of the adjoint \(L\)-function associated with a cusp form \(f\), by using the cup product of the cuspidal cohomologies of the local system \(\mathscr{L}(n,\chi;\mathbb{C})\) on \(Y_U\), where \(\chi\) is the central character of \(f\) and \(Y_U\) is a manifold associated to certain subgroup \(U\) of GL\(_2(\widehat{\mathcal{O}}_F)\) (see section 4.1). The author extends Shimura's computation for the Hilbert modular case and that by Urban for the imaginary quadratic fields case to arbitrary number fields. Secondly, the author proves that the special values of the \(L\)-functions are contained in \(\mathcal{O}_{K,\mathfrak{P}}\) by using a nondegenerate bilinear form on an \(\mathcal{O}_{K,\mathfrak{P}}\)-lattice in the space of cusp forms on GL\(_2\). This is achieved by using the Eichler-Shimura isomorphism. The author gives a sufficient condition for a prime ideal \(\mathfrak{P}\) to be a congruence prime for \(f\) under the assumption that the bilinear form is perfect. Finally, the author gives a sufficient condition for the bilinear form to be perfect, and thus finishes the proof of the main theorem. In this last step the perfectness of the bilinear form is first reduced to that of the bilinear form on parabolic cohomology groups with integral coefficients, and then further reduced to the torsion-freeness of the boundary cohomology with integral coefficients. The author deduces the torsion-freeness by calculating the group cohomologies of local torus groups at every cusp under the assumptions of the main theorem, and the method is indebted to the above mentioned work of Hida, Urban and Ghate.
scientific article; zbMATH DE number 6507151