On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

Betina, Adel; Dimitrov, Mladen; Pozzi, Alice

doi:10.48550/arxiv.1804.00648

Cited by 4 publications

(19 citation statements)

References 21 publications

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“…Proof. The first claim follows from similar arguments that have already been used in [6,Lemma 1.8]. In fact, since dim (and hence any of their quotients) are generated by the trace of their universal deformations.…”

Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 62%

“…, and in which ρ univ (γ 0 ) is a diagonal matrix. Now the assertion follows from exactly the same arguments as those in the proof of [6,Lemma 1.4].…”

Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 67%

“…The following definition (similar to [6,Def.1.7]) is an attempt to describe Galois theoretically the local ring at f of the closed analytic subspace E ⊥ of E ord defined by the union of irreducible components without CM by K. As we will later show in Theorem 3.3 one indeed has f ∈ E ⊥ . Definition 2.6.…”

Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 99%

“…If ψis trivial then the corresponding theta series θ ψ coincides with the Eisenstein series attached to ψ and ψǫ K , and the local structure of the eigencurve at such points has been extensively studied in [6].…”

Section: Components Of the Eigencurve Containing Fmentioning

confidence: 99%

“…It is enough to show that for all n ⩾ 1 the length of the Λ-module Hom Λ (J J 2 , Λ (X n )) is at most 1. We use an homological machinery as in [6,Prop.4.11]. Let us fix a realization ρ R ord ρ (g) = a(g) b(g) c(g) d(g) of the universal representation in an ordinary basis.…”

Section: 4mentioning

confidence: 99%

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

Betina¹,

Dimitrov²

2019

Preprint

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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.

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Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 62%

“…, and in which ρ univ (γ 0 ) is a diagonal matrix. Now the assertion follows from exactly the same arguments as those in the proof of [6,Lemma 1.4].…”

Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 67%

Section: Let D Fil Be the Functor Assigning Tomentioning

confidence: 99%

Section: Components Of the Eigencurve Containing Fmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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

Betina¹,

Dimitrov²

2019

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On Sharifi's conjecture: exceptional case

Shih¹,

Wang²

2020

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In the present article, we study the conjecture of Sharifi on the surjectivity of the map ̟ θ . Here θ is a primitive even Dirichlet character of conductor N p, which is exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawa algebra related to the trivial zero of the Kupota-Leopoldt p-adic L-function Lp(s, θ −1 ω 2 ), we compute the image of ̟ θ,p in local Galois cohomology groups and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization. Sharifi's ConjectureThroughout this paper, we will denote by N a positive integer, denote by p ≥ 5 a prime number not dividing N φ(N ). The goal of this section is to review Sharifi's conjecture on the map ̟ following [13] and to state Theorem 1.1. We refer the reader to loc. cit. for more details on Sharifi's conjecture.

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

Betina,

Williams

2020

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Let fnew be a classical eigenform of weight ≥ 2 and level N prime to p. We study the arithmetic of fnew and its unique p-stabilisation f when fnew is p-irregular, that is, when its Hecke polynomial at p admits a single repeated root. In particular, we study p-adic weight families through f and its base-change to an imaginary quadratic field F where p splits, and prove that the respective eigencurves are both Gorenstein at f . We use this to construct a two-variable p-adic L-function over a Coleman family through f , and a three-variable p-adic L-function over the base-change of this family to F . We relate the two-and three-variable p-adic L-functions via p-adic Artin formalism. These results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in this case.In an appendix, we prove results towards Hida duality for modular symbols, constructing a pairing between Hecke algebras and families of overconvergent modular symbols and proving that it is non-degenerate locally around any cusp form. This allows us to control the sizes of (classical and Bianchi) Hecke algebras in families.

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On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

Cited by 4 publications

References 21 publications

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

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

On Sharifi's conjecture: exceptional case

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

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