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

2019/07/22 by Betina, Adel, Dimitrov, Mladen
#11F33 #11F80 #11R23 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.09422

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 \mathscrL-invariant \mathscrL_-(φ) does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst \mathscrL_-(φ) and \mathscrL_-(φ^-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 φ has a simple (trivial) zero at s=0 if \mathscrL_-(φ) 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 φ at s=0 extending a conjecture of Gross.

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