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 study the geometry of the p-adic eigencurve at a point f corresponding to a weight one theta series θ ψ irregular at p. We show that f belongs to exactly three or four irreducible components and study their mutual congruences.In particular, we show that the congruence ideal of one of the CM components has a simple zero at f if, and only if, a certain L -invariant L-(ψ-) does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst L-(ψ-) and L-(ψ −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 ψ-has a simple (trivial) zero at s = 0, if L-(ψ-) 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 ψ-at s = 0 thus extending a conjecture of Gross.

Arithmetic of p-irregular modular forms: families and p-adic L-functions

“…There are plenty of examples of weight one classical eigenforms which are irregular at p. Such eigenforms have critical slope. The recent works [BDP18,BD19] studied the geometry of the eigencurve at such points following a new approach, hence deducing some arithmetic properties on trivial zeros of their adoint p-adic L-functions (that is, the Kubota-Leopoldt and Katz p-adic L-functions). In contrast to the results of this paper, the local ring at these irregular weight one eigenforms is never Gorenstein and their associated overconvergent generalised eigenspace contains non-classical padic modular forms; hence the construction of the two-variable p-adic L-function around these points remains an open and challenging question in Iwasawa theory.…”

A geometric view on Iwasawa theory

“…The question, analogous to the Ferrero-Greenberg Theorem, of whether the trivial zeros of Katz' anti-cyclotomic p-adic Lfunctions are simple, has remained open while being reformulated in terms of certain Iwasawa modules, via the mysterious bridge envisioned by Iwasawa. In [7] we use Hida Theory together with Mazur's Galois Deformations Theory, to show that these trivial zeros are indeed simple, provided that a certain anti-cyclotomic L -invariant does not vanish, as predicted by the Four Exponentials Conjecture in Transcendence Theory.…”

“…This has been the leitmotiv in [8], resp. [7], where the geometry of C at certain p-irregular Eisenstein, resp. CM, weight 1 points is related to trivial zeros of the Kubota-Leopoldt, resp.…”

“…Here ε K denotes the quadratic Dirichlet character attached to K Q, and Ver denotes the transfer homomorphism. Let f = ∑ n⩾1 a n q n denote the unique p-stabilization of θ ψ (see (7)).…”

“…Using the resulting q-expansion Principle we prove a perfect Hida duality between the space of Coleman families and the corresponding Hecke algebra. Exploiting this duality and the results of [7] on the local geometry of C at f , allows us in §3 to compute infinitesimally the Fourier coefficients of all families passing through f . The resulting formulas involve p-adic logarithms of algebraic numbers in the field cut out by the projective Galois representation attached to f .…”