Restriction of Eisenstein series and Stark–Heegner points
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 887-944.

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 , 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.

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 , 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.

Published online:
DOI: 10.5802/jtnb.1182
Classification: 11F67, 11F33
Keywords: $p$-adic $L$-functions, Stark-Heegner points, Hida families
Hsieh, Ming-Lun 1; Yamana, Shunsuke 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
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     title = {Restriction of {Eisenstein} series and {Stark{\textendash}Heegner} points},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Hsieh, Ming-Lun; Yamana, Shunsuke. Restriction of Eisenstein series and Stark–Heegner points. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 887-944. doi : 10.5802/jtnb.1182.

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