On congruence modules related to Hilbert Eisenstein series

Abstract

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary Λ-adic Hilbert modular forms following Hida's work on the space of multivariable ordinary Λ-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta's methods.for all cusps c, and hence, the assertion follows.Corollary 5.5. Let the notation and the assumption be as in Lemma 5.4. Then we haveMoreover, we have H 0 (S(c, Γ 1 (np r )) /W , ω k ) ⊗ W W m ≃ H 0 (S(c, Γ 1 (np r )) /W , ω k ⊗ W W m ) for all r ∈ Z >0 .