On Extra Zeros of p-Adic L-Functions: The Crystalline Case

Benois, Denis

doi:10.1007/978-3-642-55245-8_3

Cited by 11 publications

(33 citation statements)

References 47 publications

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“…Let V be a p-adic representation with coefficients in A. Then there are functorial isomorphisms [7], [48]. Let A be an affinoid algebra over Q p .…”

Section: (R)mentioning

confidence: 99%

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Selmer complexes and p-adic Hodge theory

Benois

2015

Arithmetic and Geometry

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“…Let V be a p-adic representation with coefficients in A. Then there are functorial isomorphisms [7], [48]. Let A be an affinoid algebra over Q p .…”

Section: (R)mentioning

confidence: 99%

“…Proof. The reader can compare this proof to the proof of Theorem 5.1.3 of [7]. We will use the following elementary lemma.…”

Section: Remarkmentioning

confidence: 99%

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Selmer complexes and p-adic Hodge theory

Benois

2015

Arithmetic and Geometry

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“…Our setting is rather closer to Benois' fundamental work on

scriptL

‐invariants at non‐central critical points: cf. [2, 3] and Horte's recent Ph.D. thesis [11].…”

Section: Introductionmentioning

confidence: 99%

On the L‐invariant of the adjoint of a weight one modular form

Roset

Rotger

Vatsal

2021

J. London Math. Soc.

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The purpose of this article is proving the equality of two natural L-invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L-invariant as a universal norm of a canonical Zp-extension of Qp associated to the representation. We relate it to a certain 2 × 2 regulator of p-adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic L-invariant computed in Rivero and Rotger [J.

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“…cit. We refer also to the seminal works of Benois [Ben1], [Ben2] where similar questions are addressed.…”

Section: Introductionmentioning

confidence: 99%

Cyclotomic derivatives of Beilinson--Flach classes and a new proof of a Gross--Stark formula

Rivero

2021

Preprint

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We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the L-invariant of the adjoint of a weight one modular form in terms of units and p-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson-Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be seen as a counterpart to the earlier work of Loeffler, Venjakob and Zerbes. We hope that this method could be applied to other scenarios regarding exceptional zeros, and illustrate how this could lead to a better understanding of this setting by conjecturally introducing a new p-adic L-function whose special values involve information just about the unit of the adjoint (and not also the p-unit), in the spirit of the conjectures of Harris and Venkatesh.

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On Extra Zeros of p-Adic L-Functions: The Crystalline Case

Cited by 11 publications

References 47 publications

Selmer complexes and p-adic Hodge theory

Selmer complexes and p-adic Hodge theory

On the L‐invariant of the adjoint of a weight one modular form

Cyclotomic derivatives of Beilinson--Flach classes and a new proof of a Gross--Stark formula

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