Efficient Computation of Rankin p-Adic L-Functions

Lauder, Alan G. B.

doi:10.1007/978-3-319-03847-6_7

Cited by 15 publications

(17 citation statements)

References 19 publications

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“…Whenγ and/orh arise from weight-two Eisenstein series, these expressions are also related to p-adic regulators of Beilinson elements in the K -groups K 2 (X 1 (N )) or in K 1 (X 1 (N ) 2 ) (see [BD2] and [BDR1], [BDR2], respectively). On the computational side, using a method for computing with overconvergent modular forms via Katz expansions developed in [La1], the article [La2] describes an algorithm for the efficient numerical evaluation of these p-adic iterated integrals, and uses it to calculate certain Chow-Heegner points on the elliptic curve E which were first defined and studied by Shouwu Zhang. These global points, which arise when k 2 andγ andh are eigenvectors for T N with the same eigenvalues, have a well-understood geometric provenance, and can also be calculated by complex analytic means following the strategy described in [DRS, DDLR].…”

Section: Hypothesis B (Global Vanishing Hypothesis) the L-function Lmentioning

confidence: 99%

“…It turns out to be expedient (albeit, not essential, as in the calculations relying on the algorithms of [DP]) to work at p = 17. This is because, with this choice of p, the classical modular form f of level 17 is p-adically of level 1, and the p-adic iterated integral can therefore be calculated by applying the ordinary projection algorithms of [La2] to a space of ordinary overconvergent 17-adic modular forms of weight one and relatively modest level 5 · 29.…”

mentioning

confidence: 99%

“…IP address: 52.183.12.225, on 11 Apr 2019 at 09:43:07, subject to the Cambridge Core our request using his descent code. We might also stress here that although the overconvergent projection algorithms used to compute p-adic iterated integrals are efficient, such computations still remain something of a technological tour de force: many of our examples pushed the available hardware and software (the algorithms of [La2] in particular) to their current limits. This section focusses on the case where V gh is reducible and at least one of g or h is an exotic cusp form whose associated projective Galois representation cuts out a nontotally real extension of Q with Galois group A 4 , S 4 , or A 5 .…”

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confidence: 99%

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Stark Points and -Adic Iterated Integrals Attached to Modular Forms of weight one

2015

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Let E be an elliptic curve over Q, and let and be odd two-dimensional Artin representations for which ⊗ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms f , g, and h of respective weights two, one, and one, giving rise to E, , and via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain p-adic iterated integrals attached to the triple ( f, g, h), which are p-adic avatars of the leading term of the Hasse-Weil-Artin L-series L(E, ⊗ , s) when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on E-referred to as Stark points-which are defined over the number field cut out by ⊗ . This formula can be viewed as an elliptic curve analogue of Stark's conjecture on units attached to weight-one forms. It is proved when g and h are binary theta series attached to a common imaginary quadratic field in which p splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing p-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell-Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark-Heegner points attached to Shintani-type cycles on H p × H), and extensions of Q with Galois group a central extension of the dihedral group D 2n or of one of the exceptional subgroups A 4 , S 4 , and A 5 of PGL 2 (C).

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Section: Hypothesis B (Global Vanishing Hypothesis) the L-function Lmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Stark Points and -Adic Iterated Integrals Attached to Modular Forms of weight one

2015

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“…are linearly independent in the Selmer group H 1 fin (Q, V fgh ) attached to E and V gh , for a suitable choice of π in (16). Theorem 3.4 and its corollary motivated the experimental study undertaken in [15] of the special values of p-adic L-functions appearing in (26). This led to a precise conjecture for these values up to a factor of L × rather than Q × p .…”

Section: Theorem 31 the Natural Image Ofmentioning

confidence: 97%

Elliptic curves of rank two and generalised Kato classes

Darmon

Rotger

2016

Res Math Sci

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Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell-Weil generators whose heights encode first derivatives of the associated Hasse-Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank > 1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569-604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse-Weil-Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted κ(f, g, h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations V g and V h of Gal (H/Q) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that κ(f, g, h) lies in the prop Selmer group of E over H precisely when L(E, V gh , 1) = 0, where L(E, V gh , s) is the L-function of E twisted by V gh := V g ⊗ V h. In the setting of interest, parity considerations imply that L(E, V gh , s) vanishes to even order at s = 1, and the Selmer class κ(f, g, h) is expected to be trivial when ord s=1 L(E, V gh , s) > 2. The main new contribution of this article is a conjecture expressing κ(f, g, h) as a canonical point in (E(H) ⊗ V gh) G Q when ord s=1 L(E, V gh , s) = 2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).

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“…to (1, 0) and (0, 1) respectively. A computer calculation using the algorithms based on [19] shows that…”

Section: 3amentioning

confidence: 99%

Gross–Stark units and p-adic iterated integrals attached to modular forms of weight one

2016

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This article can be read as a companion and sequel to the authors' earlier article on Stark points and p-adic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f , defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross's p-adic analogue of Stark's conjecture on Artin L-series at s = 0. Keywords Eisenstein series • p-adic modular forms • p-adic iterated integrals • Gross-Stark units Résumé Cet article peut se lire comme un supplément à l'article des mêmes auteurs sur les "points de Stark" et les intégrales itérées p-adiques. Dans cet article antérieur, il est conjecturé que les intégrales itérées p-adiques associées à un triplet (f, g, h) de formes modulaires de poids (2, 1, 1) s'expriment au moyen de points de Stark sur la variété abélienne associée à f , définis sur le corps de nombres découpé par le produit tensoriel des représentations d'Artin B Henri Darmon

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Efficient Computation of Rankin p-Adic L-Functions

Abstract: We present an efficient algorithm for computing certain special values of Rankin triple product p-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

Cited by 15 publications

References 19 publications

Stark Points and -Adic Iterated Integrals Attached to Modular Forms of weight one

Stark Points and -Adic Iterated Integrals Attached to Modular Forms of weight one

Elliptic curves of rank two and generalised Kato classes

Gross–Stark units and p-adic iterated integrals attached to modular forms of weight one

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