Non-Archimedean Sato-Tate Distributions for Hilbert Modular Forms of Weight One

We investigate the spectral distribution of Hecke eigenvalues associated with Hilbert modular forms of weight one over totally real fields. While the classical Sato-Tate conjecture predicts a continuous distribution for forms of weight k 2 governed by the Haar measure on SU(2), the weight one case is known to exhibit a discrete distribution reflecting the finite image of the associated Artin representations. In this monograph, we construct a non-Archimedean analogue of the Sato-Tate measure, taking values in the space of p-adic distributions. We analyze the variation of these distributions within Hida families, specifically at the weight one boundary. By utilizing the theory of overconvergent modular forms and the geometry of the eigencurve, we define a p-adic Sato-Tate group and prove an equidistribution theorem for the Frobenius classes in the pro-p setting. Our results provide a measure-theoretic framework for understanding the arithmetic statistics of weight one forms in the p-adic topology, linking the classical Chebotarev density with the deformation theory of Galois representations.