A weight independence result for quaternionic Hecke algebras (Q2870222)

A famous result of \textit{Hida} [Invent. Math. 85, 545--613 (1986; Zbl 0612.10021)] states that his \(p\)-adic Hecke algebra of tame level~\(N\) is independent of the weight. In the article under review, an analogue of this theorem is proved in the setting of quaternionic Hecke algebras for indefinite quaternion algebras over the rationals which are ramified at~\(p\).NEWLINENEWLINEThe proof is cohomological, based on the Matshushima--Shimura isomorphism relating quaternionic automorphic forms to the cohomology of the associated Shimura curve, and similar to \textit{H. Hida}'s proof in [Am. J. Math. 110, No. 2, 323--382 (1988; Zbl 0645.10029)].NEWLINENEWLINEThe author discusses the relationship with Hida's Hecke algebra via the Jacquet--Langlands correspondence and places the present article into a research program of hers outlined in [\textit{L. Terracini}, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 5, No. 3, 677--700 (2002; Zbl 1177.11046)]. Moreover, the author points out that the absence of the standard Hecke operator at~\(p\) could possibly be remedied by exploiting the action of \(\mathbb{Z}_{p^2}^\times\) coming from the ramification of the quaternion algebra at~\(p\).