LAUSR

We show that the p-adic eigenvariety constructed by AndreattaIovita-Pilloni, parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field F , is smooth at certain classical parallel weight one points which are regular at every place of F above p and also determine whether the map to the weight space at those points is étale or not. We prove these results assuming the Leopoldt conjecture for certain quadratic extensions of F in some cases, assuming the p-adic Schanuel conjecture in some cases and unconditionally in some cases, using the deformation theory of Galois representations. As a consequence, we also determine whether the cuspidal part of the 1-dimensional parallel weight eigenvariety, constructed by Kisin-Lai, is smooth or not at those points.