Selmer complexes and <i>p</i>-adic Hodge theory

Benois, Denis

doi:10.1017/cbo9781316106877.008

Cited by 10 publications

(13 citation statements)

References 34 publications

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“…In fact in [Nek93] it is assumed that is semistable; we will recall the definitions under this assumption, and at the same time see that they can be made compatible with extending the ground field (in particular, to reduce the potentially semistable case to the semistable case). Compare also [Ben14] for a very general treatment.…”

Section: -Adic Heightsmentioning

confidence: 99%

The -adic Gross–Zagier formula on Shimura curves

Disegni¹

2017

Compositio Math.

117

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Abstract. -We prove a general formula for the p-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg p-adic L-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is +1 rather than −1, by an anticyclotomic version of the Waldspurger formula.When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a p-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

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Section: -Adic Heightsmentioning

confidence: 99%

The -adic Gross–Zagier formula on Shimura curves

Disegni¹

2017

Compositio Math.

117

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“…The third follows from the p-adic Gross-Zagier formula of Kobayashi in [Kob13]. We will deduce 4 the last portion from the discussion in [PR93, §3.3.3], as enhanced by Benois [Ben15,Ben14] building the work of Pottharst, combined with Kobayashi's p-adic Gross-Zagier formula.…”

Section: Logarithms Of Heegner Points and Beilinson-kato Classesmentioning

confidence: 89%

Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture

Büyükboduk

2015

Preprint

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Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae.Our second goal is to set up a conceptual framework in the context of Λ-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for higher motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of CM abelian varieties and the (conjectural) Coleman-Rubin-Stark elements we introduce here. This is a higher dimensional version of Rubin's results on rational points on CM elliptic curves.

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“…We refer the reader to [9] for a discussion of these conditions. Here we only remark that if V is the p-adic realization of a pure motive of weight −2 having good reduction at p, then C3-5) are deep conjectures which are known only in some special cases.…”

Section: Complements On the L -Invariantmentioning

confidence: 99%

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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Selmer complexes and p-adic Hodge theory

Cited by 10 publications

References 34 publications

The -adic Gross–Zagier formula on Shimura curves

The -adic Gross–Zagier formula on Shimura curves

Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture

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

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