# Number Theory Seminar

This talk is based on a joint work with Adel Betina studying the geometry of the p-adic eigencurve at a weight one theta series f 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 L-invariant L(\varphi) does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst L(\varphi) and L(\varphi^{−1}) is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz p-adic L-function of \varphi has a simple (trivial) zero at s=0, if L(\varphi) is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz p-adic L-function of \varphi at s=0 thus extending a conjecture of Gross, and check its compatibility with a conjecture made by Benois.