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Eisenstein points on the Hilbert cuspidal eigenvariety

We present a comprehensive study of the geometry of Hilbert $p$-adic eigenvarieties at classical parallel weight one intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at $p$ and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.