Restriction of Eisenstein series and Stark–Heegner points

Ming-Lun Hsieh; Shunsuke Yamana

doi:10.5802/jtnb.1182

Ming-Lun Hsieh ¹ ; Shunsuke Yamana ²

¹ Institute of Mathematics Academia Sinica and National Center for Theoretic Sciences Taipei 10617, Taiwan
² Department of Mathematics Graduate School of Science Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585, Japan

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 887-944.

Résumé
Abstract

Dans un travail récent de Darmon, Pozzi et Vonk, les auteurs considèrent une famille $p$ -adique de séries d’Eisenstein–Hilbert $E_{k} (1, ϕ)$ associées à un caractère impair $ϕ$ du groupe de classes d’idéaux au sens restreint d’un corps quadratique réel $F$ . Ils calculent la dérivée première d’une certaine série $L$ $p$ -adique à une variable d’un produit triple tordu attachée à $E_{k} (1, ϕ)$ et à une forme elliptique nouvelle $f$ de poids $2$ sur $Γ_{0} (p)$ . Dans cet article, nous généralisons leur construction afin de prendre en compte la variable cyclotomique, et obtenons ainsi une série $L$ $p$ -adique à deux variables du produit triple tordu. De plus, quand $f$ est associée à une courbe elliptique $E$ sur $ℚ$ , nous prouvons que la dérivée première de cette série $L$ $p$ -adique par rapport au poids est le produit du logarithme $p$ -adique d’un point de Stark–Heegner de $E$ sur $F$ introduit par Darmon et de la fonction $L$ $p$ -adique cyclotomique de $E$ .

In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular $p$ -adic family of Hilbert–Eisenstein series $E_{k} (1, ϕ)$ associated with an odd character $ϕ$ of the narrow ideal class group of a real quadratic field $F$ and compute the first derivative of a certain one-variable twisted triple product $p$ -adic $L$ -series attached to $E_{k} (1, ϕ)$ and an elliptic newform $f$ of weight $2$ on $Γ_{0} (p)$ . In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product $p$ -adic $L$ -series. Moreover, when $f$ is associated with an elliptic curve $E$ over $ℚ$ , we prove that the first derivative of this $p$ -adic $L$ -series along the weight direction is a product of the $p$ -adic logarithm of a Stark–Heegner point of $E$ over $F$ introduced by Darmon and the cyclotomic $p$ -adic $L$ -function for $E$ .

Reçu le : 2020-02-26
Révisé le : 2021-04-08
Accepté le : 2021-07-09
Publié le : 2022-01-26

DOI : 10.5802/jtnb.1182

Classification : 11F67, 11F33
Mots clés : $p$-adic $L$-functions, Stark-Heegner points, Hida families

Affiliations des auteurs :

Ming-Lun Hsieh ¹ ; Shunsuke Yamana ²

¹ Institute of Mathematics Academia Sinica and National Center for Theoretic Sciences Taipei 10617, Taiwan
² Department of Mathematics Graduate School of Science Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Ming-Lun Hsieh and Shunsuke Yamana},
     title = {Restriction of {Eisenstein} series and {Stark{\textendash}Heegner} points},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {887--944},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     doi = {10.5802/jtnb.1182},
     language = {en},
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Ming-Lun Hsieh; Shunsuke Yamana. Restriction of Eisenstein series and Stark–Heegner points. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 887-944. doi : 10.5802/jtnb.1182. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1182/

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