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

Content Provider	Springer Nature Link
Author	Darmon, Henri Lauder, Alan Rotger, Victor
Copyright Year	2015
Abstract	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 $$s=0$$ .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 associées à g et à h, lorsque f est cuspidale. Le but premier de cet article est d’énoncer une conjecture du même genre quand f provient plutôt d’une série d’Eisenstein. Cette conjecture fait intervenir des logarithmes p-adiques d’unités de Stark et de p-unités de Gross-Stark, et peut donc être envisagée comme une nouvelle variante des conjectures de Gross étendant au cadre p-adique les conjectures de Stark sur les valeurs des fonctions L d’Artin en $$s=0$$ .
Starting Page	325
Ending Page	354
Page Count	30
File Format	PDF
ISSN	21954755
Journal	Annales mathématiques du Québec
Volume Number	40
Issue Number	2
e-ISSN	21954763
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-01-05
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Algebra p-adic iterated integrals Analysis p-adic modular forms Gross-Stark units Elliptic curves over global fields Eisenstein series $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Mathematics Number Theory Elliptic and modular units
Content Type	Text
Resource Type	Article
Subject	Mathematics

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