Overconvergent Eichler-Shimura isomorphisms for unitary Shimura curves over totally real fields
The \(p\)-adic Eichler-Shimura isomorphism, introduced by Faltings, describes the space of \(p\)-modular forms, defined as global section of certain automorphic bundle, using the cohomology of certain local systems. This construction works for modular forms of integral weight and can be used, for example, to build Galois representation from modular forms. Modular forms of integral weight can be interpolated by \(p\)-adic families of modular forms, and the theory of modular symbols introduced by Stevens provides a similar interpolation for the cohomology of the aforementioned local systems. One can then hope to interpolate the Eichler-Shimira isomorphisms. This has been done, at least over some subset of the weight space, by Andreatta, Iovita, and Stevens [\textit{F. Andreatta} et al., J. Inst. Math. Jussieu 14, No. 2, 221--274 (2015; Zbl 1379.11062)]. Since both the theory of \(p\)-adic families of modular forms and of modular symbols exist for a lot of Shimura varieties, a natural problem is to construct families of Eichler-Shimura isomorphisms for Shimura varieties other than the modular curve. In their previous work ``Overconvergent Eichler-Shimura isomorphisms for quaternionic modular forms over \(\mathbb{Q}\)'' [Int. J. Number Theory 13, No. 10, 2687--2715 (2017; Zbl 1428.11101)], the authors consider the case of Shimura curves associated to a quaternion algebra over \(\mathbb{Q}\). In the paper at hand they generalize this construction to Shimura curves associated to a quaternion algebra over a totally real field.
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