In this talk, we will report a joint work with Adel Betina and Mladen Dimitrov about the geometry of the Hilbert cuspidal eigenvarity at a point \(f\) coming from a weight one Eisenstein series irregular at a single prime \(\mathfrak{p}\) of the totally real field \(F\) above \(p\). 

Assuming the Leopoldt conjecture for \(F\) at \(p\), we show that the ordinary cuspidal eigencurve is etale at \(f\) over the weight space when \( [ F_{\mathfrak{p}}:\mathbb{Q}_p] \geq [F:\mathbb{Q}]-1 \), and hence, the nearly ordinary eigenvariety is etale over the weight space as well. When \(F_{\mathfrak{p}}=\mathbb{Q}_p\) we show that the eigenvariety is smooth, and in all remaining cases, we prove that the eigenvarieties are not smooth at \(f\).

As applications, we give a new proof of the Gross-Stark conjecture in the rank one case relating the leading term of the Kubota-Leopoldt \(p\)-adic \(L\)-function and a non-zero arithmetic \(L\)-invariant. This was first proved by Dasgupta-Darmon-Pollack under the assumption that a sum of two analytic \(L\)-invariances is non-zero, which is not assumed in our proof. Another application, when \(F\) is a real quadratic field, is the study of Gross-Stark units by Darmon-Pozzi-Vonk which relies on our result of the uniqueness of the \(p\)-adic deformation of \(f\) in the anti-parallel direction.