Restriction of Eisenstein series and Stark–Heegner points

Hsieh, Ming-Lun; Yamana, Shunsuke

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 $E_k(1,\varphi )$ 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 $E$.

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 $E$.

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

[1] Bertolini, Massimo; Darmon, Henri The rationality of Stark-Heegner points over genus fields of real quadratic fields, Ann. Math., Volume 170 (2009) no. 1, pp. 343-370 | DOI | MR | Zbl

[2] Blanco-Chacón, Iván; Fornea, Michele Twisted triple product $p$-adic $L$-functions and Hirzebruch-Zagier cycles, J. Inst. Math. Jussieu, Volume 19 (2020) no. 6, pp. 1947-1992 | DOI | MR | Zbl

[3] Bump, D. Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, 1998

[4] Casselman, W. On some results of Atkin and Lehner, Math. Ann., Volume 201 (1973), pp. 301-314 | DOI | MR | Zbl

[5] Chen, Shih-Yu; Hsieh, Ming-Lun On primitive $p$-adic Rankin-Selberg $L$-functions, Development of Iwasawa theory - The Centennial of K. Iwasawa’s Birth (Advanced Studies in Pure Mathematics), Volume 86, 2020 no. 11, pp. 195-242 | DOI | Zbl

[6] Darmon, Henri Integration on ${\mathscr{H}}_p\times \mathscr{H}$ and arithmetic applications, Ann. Math., Volume 154 (2001) no. 3, pp. 589-639 | DOI | MR | Zbl

[7] Darmon, Henri; Dasgupta, Samit Elliptic units for real quadratic fields, Ann. Math., Volume 163 (2006) no. 1, pp. 301-346 | DOI | MR | Zbl

[8] Darmon, Henri; Pozzi, Alice; Vonk, Jan Diagonal restrictions of $p$-adic Eisenstein families, Math. Ann., Volume 379 (2021) no. 1-2, pp. 503-548 | DOI | MR | Zbl

[9] Greenberg, R.; Stevens, G. $p$-adic $L$-functions and $p$-adic periods of modular forms, Inventiones mathematiace, Volume 111 (1993), pp. 407-447 | DOI | MR | Zbl

[10] Hida, Haruzo Congruence of cusp forms and special values of their zeta functions, Invent. Math., Volume 63 (1981) no. 2, pp. 225-261 | DOI | MR

[11] Hida, Haruzo Modules of congruence of Hecke algebras and $L$-functions associated with cusp forms, Am. J. Math., Volume 110 (1988) no. 2, pp. 323-382 | DOI | MR | Zbl

[12] Hida, Haruzo On $p$-adic Hecke algebras for GL(2) over totally real fields, Ann. Math., Volume 128 (1988), pp. 295-384 | DOI | MR | Zbl

[13] Hida, Haruzo A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier, Volume 38 (1988) no. 3, pp. 1-83 | DOI | MR | Zbl

[14] Hida, Haruzo Elementary Theory of $L$-functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge University Press, 1993 | DOI

[15] Hsieh, M.-L.; Yamana, S. Four variable p-adic triple product L-functions and the trivial zero conjecture (2020) (submitted. arXiv:1906.10474)

[16] Hsieh, Ming-Lun Hida families and $p$-adic triple product $L$-functions, Am. J. Math., Volume 143 (2021) no. 2, pp. 411-532 | DOI | MR | Zbl

[17] Ishikawa, Isao On the construction of twisted triple product $p$-adic $L$-functions, 2017 Thesis (Ph.D.)–Kyoto University

[18] Jacquet, H. Automorphic Forms on $\mathrm{GL}\left(2\right)$ II, Lecture Notes in Mathematics, 278, Springer-Verlag, Berlin and New York, 1972 | DOI

[19] Jacquet, H.; Langlands, R. Automorphic Forms on $\mathrm{GL}\left(2\right)$, Lecture Notes in Mathematics, 114, Springer-Verlag, Berlin and New York, 1970 | DOI

[20] Keaton, Rodney; Pitale, Ameya Restrictions of Eisenstein series and Rankin-Selberg convolution, Doc. Math., Volume 24 (2019), pp. 1-45 | MR | Zbl

[21] Kitagawa, Koji On standard $p$-adic $L$-functions of families of elliptic cusp forms, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) (Contemporary Mathematics), Volume 165, American Mathematical Society, 1994, pp. 81-110 | DOI | MR | Zbl

[22] Rohrlich, David E. On $L$-functions of elliptic curves and cyclotomic towers, Invent. Math., Volume 75 (1984) no. 3, pp. 409-423 | DOI | MR | Zbl

[23] Saha, Abhishek Large values of newforms on $\mathrm{GL}\left(2\right)$ with highly ramified central character, Int. Math. Res. Not. (2016) no. 13, pp. 4103-4131 | DOI | MR | Zbl

[24] Saito, H. On Tunnell’s formula for characters of $\mathrm{GL}\left(2\right)$, Compos. Math., Volume 85 (1993) no. 1, pp. 99-108 | Numdam | MR | Zbl

[25] Schmidt, R. Some remarks on local newforms for $\mathrm{GL}\left(2\right)$, Journal of Ramanujan Mathematical Society, Volume 17 (2002) no. 2, pp. 115-147 | MR | Zbl

[26] Shimura, G. On the periods of modular forms, Math. Ann., Volume 229 (1977), pp. 211-221 | DOI | MR | Zbl

[27] Tunnell, J. Local $ϵ$-factors and characters of $\mathrm{GL}\left(2\right)$, Am. J. Math., Volume 105 (1983) no. 6, pp. 1277-1307 | DOI | MR

[28] Wiles, A. On ordinary $\lambda$-adic representations associated to modular forms, Inventiones mathematiace, Volume 94 (1988) no. 3, pp. 529-573 | DOI | MR | Zbl

[29] Wiles, A. Modular elliptic curves and Fermat last theorem, Ann. Math., Volume 141 (1995) no. 3, pp. 443-551 | DOI | MR | Zbl