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

Betina, Adel; Dimitrov, Mladen

doi:10.1016/j.aim.2021.107724

Cited by 11 publications

(15 citation statements)

References 28 publications

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“…In this case, the extrazero conjecture for M = Q(ρ χ) generalizes the Gross-Stark conjecture for abelian characters of totally real fields proved by Dasgupta, Kakde and Ventullo [27]. However, our methods are also applicable beyond the totally real and CM cases, and should provide some insight on a computation of Betina and Dimitrov [18]. On the other hand, it seems interesting to compare the formalism of [8] with the approach of Büyükboduk and Sakamoto [21].…”

Section: Introductionsupporting

confidence: 54%

On extra-zeros of p-adic Rankin-Selberg L-functions

Benois,

Horte

2020

Preprint

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We prove a version of the Extra-zero conjecture, formulated by the first named author in [8], for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied. CONTENTS DENIS BENOIS AND ST ÉPHANE HORTE 6.3. The second improved p-adic L-function 47 6.4. The functional equation 48 6.5. Functional equation for zeta elements 49 7. Extra zeros of Rankin-Selberg L-functions 51 7.1. The p-adic regulator 51 7.2. The L -invariant 53 7.3. The main theorem 55 References 57 3 Here η α f denotes the canonical eigenvector associated to α( f ). See Section 3.2 below. 4 In the weight 2 case, the motivic version of this element was first constructed by Beilinson[2]. In [29], FlachProof. This theorem was first proved by Bertolini-Darmon-Rotger [15] for modular forms of weight 2. Kings, Loeffler and Zerbes extended the proof to the higher weight case [43, Theorem 7.2.6], [47, Theorem 7.1.5]. Note that the results proved in [43] and [47] are in fact more general and include also the case of modular forms of different weights.We remark that, combining this formula with the computaiton of the special value of the complex L-function in terms of the Beilinson regulator, one can write this theorem in the form (2) (see [43, Theorem 7.2.6]). Also, this result suggests that L p,α ( f , g, s + k 0 ) satisfies the conjectural interpolation properties of L p (M f ,g (k 0 ), D, s) up to "bad" Euler factors at primes dividing N f N g . 0.5. The main result. We keep previous notation and conventions. In this paper, we prove a result toward the extra-zero conjecture for the p-adic L-finction L p,α ( f , g, s) as s = k 0 . In addition to M1-2) we assume that the following conditions hold: M3) ε f , ε g and ε f ε g are primitives modulo N f , N g and lcm(N f , N g ) respectively. M4) ε f (p)ε g (p) = 1.5 More explicitly, e 1 = ε ⊗ t −1 , where ε is a compatible system of p n th roots of unity, and t = log[ε] the associated

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Section: Introductionsupporting

confidence: 54%

On extra-zeros of p-adic Rankin-Selberg L-functions

Benois,

Horte

2020

Preprint

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“…the Q p -points) of C . Theorem 1.5 was inspired by Betina-Dimitrov's work [BD21] where the authors show the non-vanishing of a certain anticyclotomic L-invariant for Katz's p-adic L-function. In fact, their result generalizes to any Z p -extension with non-transcendental slope.…”

Section: Introductionmentioning

confidence: 99%

On Leopoldt's and Gross's defects for Artin representations

Maksoud¹

2022

Preprint

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We generalize Waldschmidt's bound for Leopoldt's defect and prove a similar bound for Gross's defect for an arbitrary extension of number fields. As an application, we prove new cases of the generalized Gross conjecture (also known as the Gross-Kuz'min conjecture) beyond the classical abelian case, and we show that Gross's p-adic regulator has at least half of the conjectured rank. We also describe and compute non-cyclotomic analogues of Gross's defect.

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“…• det log p u χ p log p u χ log p u χ p log p u χ (cf. [BD21,(45), page 36]). Note that L (χ) is a Gross-style regulator for imaginary quadratic fields.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, we adapt the ideas in [DDP11], replacing the Eisenstein congruence with the CM congruence for elliptic modular forms. This approach is inspired by a series of works of Hida and Tilouine [HT91], [HT93] and [HT94] on CM congruences and the anticyclotomic main conjecture for CM fields and a recent work [BD21]. This method is units-free and more amenable to general CM fields as in [DDP11] at least under some suitable Leopoldt conjecture (see [BH22] for the case of general CM fields).…”

Section: Introductionmentioning

confidence: 99%

The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros

Chida,

Hsieh

2022

Preprint

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The rank one Gross conjecture for Deligne-Ribet p-adic L-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz p-adic L-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the p-adic Rankin-Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the 1 + ε specialization.Résumé. Le conjecture de Gross en rang 1 pour les fonctions L p-adiques de Deligne-Ribet a été résolue par Darmon-Dasgupta-Pollack et Ventullo au moyen de congruences d'Eisenstein parmi les formes modulaires de Hilbert. Le but de cet article est de prouver un analogue de la conjecture de Gross pour les fonctions L p-adiques de Katz des corps quadratiques imaginaires, via les congruences entre formes CM et formes non-CM. Le nouvel ingrédient est l'application de la méthode de Rankin-Selberg p-adique pour construire une famille de Hida non-CM qui est congruente à une famille de Hida de formes CM pour la spécialisation 1 + ε.

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

Cited by 11 publications

References 28 publications

On extra-zeros of p-adic Rankin-Selberg L-functions

On extra-zeros of p-adic Rankin-Selberg L-functions

On Leopoldt's and Gross's defects for Artin representations

The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros

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