Restriction of Eisenstein series and Stark–Heegner points

Hsieh, Ming-Lun; Yamana, Shunsuke

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

Dans un travail récent de Darmon, Pozzi et Vonk, les auteurs considèrent une famille $p$-adique de séries d’Eisenstein–Hilbert $E_k(1,\varphi )$ associées à un caractère impair $\varphi$ 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 $E_k(1,\varphi )$ et à une forme elliptique nouvelle $f$ de poids $2$ sur ${\Gamma }_0\left(p\right)$. 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 $\mathbb{Q}$, 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 $E_k(1,\varphi )$ associated with an odd character $\varphi$ 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 $E_k(1,\varphi )$ and an elliptic newform $f$ of weight $2$ on ${\Gamma }_0\left(p\right)$. 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 $\mathbb{Q}$, 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 $E$.

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

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