Geometry of the eigencurve at CM points and trivial zeros of Katz $p$-adic $L$-functions

Abstract

The primary goal of this paper is to investigate the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight one cuspidal theta series irregular at the prime number $p$. We show that $f$ belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we show that the congruence ideal of a CM component has a simple zero at $f$ if and only if a certain anti-cyclotomic $\mathcal{L}$-invariant ${\mathcal{L}}_{-}(\phi )$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst ${\mathcal{L}}_{-}(\phi )$ and ${\mathcal{L}}_{-}({\phi }^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $\phi$ has a simple (trivial) zero at $s=0$ if ${\mathcal{L}}_{-}(\phi )$ 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 $\phi$ at $s=0$ extending a conjecture of Gross.

revised version to appear in Advances in Mathematics