Jacobi-Eisenstein series and p-adic interpolation of symmetric squares of cusp forms
Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 605-624.

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.

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.

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     title = {Jacobi-Eisenstein series and $p$-adic interpolation of symmetric squares of cusp forms},
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Guerzhoy, Pavel I. Jacobi-Eisenstein series and $p$-adic interpolation of symmetric squares of cusp forms. Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 605-624. doi : 10.5802/aif.1467. http://archive.numdam.org/articles/10.5802/aif.1467/

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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