Jacobi-Eisenstein series and p-adic interpolation of symmetric squares of cusp forms
Annales de l'Institut Fourier, Volume 45 (1995) no. 3, p. 605-624
The aim of this paper is to construct and calculate generating functions connected with special values of symmetric squares of modular forms. The Main Theorem establishes these generating functions to be Jacobi-Eisenstein series i.e. Eisenstein series among Jacobi forms. A theorem on p-adic interpolation of the special values of the symmetric square of a p-ordinary modular form is proved as a corollary of our Main Theorem.
On construit et calcule une fonction génératrice liée aux valeurs spéciales du carré symétrique des formes modulaires. Le théorème principal établit que cette fonction est égale à la série de Jacobi-Eisenstein. Le théorème d’interpolation p-adique pour les valeurs spéciales du carré symétrique d’une forme modulaire p-ordinaire est prouvé comme corollaire du théorème principal.
@article{AIF_1995__45_3_605_0,
     author = {Guerzhoy, Pavel I.},
     title = {Jacobi-Eisenstein series and $p$-adic interpolation of symmetric squares of cusp forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     pages = {605-624},
     doi = {10.5802/aif.1467},
     zbl = {0820.11035},
     mrnumber = {96d:11053},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1995__45_3_605_0}
}
Guerzhoy, Pavel I. Jacobi-Eisenstein series and $p$-adic interpolation of symmetric squares of cusp forms. Annales de l'Institut Fourier, Volume 45 (1995) no. 3, pp. 605-624. doi : 10.5802/aif.1467. http://www.numdam.org/item/AIF_1995__45_3_605_0/

[1] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., 217 (1975), 271-285. | MR 52 #3080 | Zbl 0311.10030

[2] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics, vol. 55, Birkhauser, Boston-Basel-Stuttgart, 1985. | MR 86j:11043 | Zbl 0554.10018

[3] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms 1, Invent. Math., 79 (1985), 159-195. | MR 86m:11097 | Zbl 0573.10020

[4] Yu. I. Manin, A.A. Panchishkin, Convolutions of Hecke series and their values at integer points, Mat. Sbornik, 104 (1977), 617-651. | Zbl 0392.10028

[5] A.A. Panchishkin, Uber nichtarchimedische symmetrische Quadrate von Spitzenformen, Max-Plank-Institut fur Mathematik, Bonn, preprint.

[6] A.A. Panchishkin, Non-Archimedian ζ-functions, Publishing house of Moscow University, 1988 (in Russian).

[7] A.A. Panchishkin, Non-Archimedian L-functions of Siegel and Hilbert Modular Forms, Springer Lecture Notes, 1471, Springer Verlag, 1991. | MR 93a:11044 | Zbl 0732.11026

[8] G. Shimura, On modular forms of half-integral weight, Ann. of Math., 97 (1973), 440-481. | MR 48 #10989 | Zbl 0266.10022

[9] D. Zagier, Periods of modular forms and Jacobi theta-functions, Invent. Math., 104 (1991), 449-465. | MR 92e:11052 | Zbl 0742.11029

[10] D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in Modular Functions of One Variable 4, Springer Lecture Notes 627, 105-169, Springer Verlag, 1977. | MR 58 #5525 | Zbl 0372.10017