This talk is based on a joint work with Mladen Dimitrov studying the geometry of the eigencurve at a \(p\)-stabilisation \(f\) of a weight one theta series \(\theta_\psi\) irregular at \(p\).
We show that \(f\) belongs to exactly four Hida families and study their mutual congruences. In particular, we show that the congruence ideal of one of the CM families has a simple zero at \(f\) if, and only if, a certain \(\mathcal{L}\)-invariant \(\mathcal{L}(\psi/\bar\psi)\) does not vanish. Combined with a divisibility proved by Hida-Tilouine, we deduce that the anti-cyclotomic Katz \(p\)-adic \(L\)-function \(L_p^{-}(\psi/\bar\psi,s)\) has a simple (trivial) zero at \(s=0\) if \(\mathcal{L}(\psi/\bar\psi) \ne 0\), which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg.