Non-Archimedean Rankin $L$-functions and their functional equations
Izvestiya. Mathematics , Tome 32 (1989) no. 2, pp. 339-358.

Voir la notice de l'article provenant de la source Math-Net.Ru

A functional equation is established for the $S$-adic $L$-functions obtained by non-Archimedean interpolation of the special values of the convolution of two cusp forms on the upper half-plane. Bibliography: 33 titles.
@article{IM2_1989_32_2_a4,
     author = {A. A. Panchishkin},
     title = {Non-Archimedean {Rankin} $L$-functions and their functional equations},
     journal = {Izvestiya. Mathematics },
     pages = {339--358},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1989_32_2_a4/}
}
TY  - JOUR
AU  - A. A. Panchishkin
TI  - Non-Archimedean Rankin $L$-functions and their functional equations
JO  - Izvestiya. Mathematics 
PY  - 1989
SP  - 339
EP  - 358
VL  - 32
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1989_32_2_a4/
LA  - en
ID  - IM2_1989_32_2_a4
ER  - 
%0 Journal Article
%A A. A. Panchishkin
%T Non-Archimedean Rankin $L$-functions and their functional equations
%J Izvestiya. Mathematics 
%D 1989
%P 339-358
%V 32
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1989_32_2_a4/
%G en
%F IM2_1989_32_2_a4
A. A. Panchishkin. Non-Archimedean Rankin $L$-functions and their functional equations. Izvestiya. Mathematics , Tome 32 (1989) no. 2, pp. 339-358. http://geodesic.mathdoc.fr/item/IM2_1989_32_2_a4/

[1] Vishik M. M., “Nearkhimedovy mery, svyazannye s ryadami Dirikhle”, Matem. sb., 99:2 (1976), 248–260 | MR | Zbl

[2] Leng S., Vvedenie v teoriyu modulyarnykh form, Mir, M., 1979, 254 pp. | MR

[3] Manin Yu. I., “Periody parabolicheskikh form i $p$-adicheskie ryady Gekke”, Matem. sb., 92:3 (1973), 378–401 | MR | Zbl

[4] Manin Yu. I., “Nearkhimedovo integrirovanie i $p$-adicheskie $L$-funktsii Zhake–Lenglendsa”, UMN, XXXI:1 (1976), 5–54 | MR

[5] Martin Yu. I., “Modular forms and number theory”, Proc. Int. Congr. Main. Helsinki, Helsinki, 1980, 177–186 | MR

[6] Manin Yu. I., Panchishkin A. A., “Svertki ryadov Gekke i ikh znacheniya v tselykh tochkakh”, Matem. sb., 104:4 (1977), 617–651 | MR | Zbl

[7] Panchishkin A. A., “Simmetricheskie kvadraty ryadov Gekke i ikh znacheniya v tselykh tochkakh”, Matem. sb., 108:3 (1979), 393–417 | MR | Zbl

[8] Panchishkin A. A., “O ryadakh Dirikhle, svyazannykh s modulyarnymi formami tselogo i polutselogo vesa”, Izv. AN SSSR. Ser. matem., 43:5 (1979), 1145–1158 | MR | Zbl

[9] Panchishkin A. A., “Kompleksnoznachnye mery, svyazannye s eilerovskimi proizvedeniyami”, Tr. seminara im. I. G. Petrovskogo, 7, 1981, 239–244 | MR | Zbl

[10] Pančiškin A. A., “Le prolongement $p$-adique analytique des fonctions $L$ de Rankin, I II”, Compte Rendus Acad. Sci. Paris, Ser. 1, 29 (1982), 51–53, 227–230 | MR

[11] Panchishkin A. A., “A functional equation of the non-archimedian Rankin convolution”, Duke Math. J., 54:1 (1987), 77–89 | DOI | MR | Zbl

[12] Arnaud B., “Interpolation $p$-adique d'un produit de Rankin”, Compte Rendus Acad. Sci. Paris, Ser. 1, 299 (1984), 527–530 | MR | Zbl

[13] Coates J., “Elliptic curves and Iwasawa theory”, Modul. Forms (Symp., Durham 30 June–10 July, 1983), Chichester, 1984, 51–73 | MR | Zbl

[14] Deligne J., “Valeurs de fonctions $L$ et périodes d'integrales”, Proc. Symp. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979, 247–290, part 2 | MR

[15] Deligne P., Ribet K. A., “Values of Abelian $L$-functions at negative integers over totally real fields”, Invent. Math., 59 (1980), 227–286 | DOI | MR | Zbl

[16] Gross B. H., Zagier Don B., “Heegner points and derivatives of $L$-series”, Invent. Math., 84 (1986), 225–320 | DOI | MR | Zbl

[17] Hecke E., “Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik”, Abh. Math. Sem. Hamburg, 5, 1927, 199–224 | Zbl

[18] Hida H., “A $p$-adic measure attached to the zefa-functions associated with two elliptic modular forms, I”, Invent. Math., 79 (1985), 159–195 | DOI | MR | Zbl

[19] Iwasawa K., Ann. of Math. Studies, 74, no. 6, Princeton, 1972 | MR | Zbl

[20] Katz N. M., “$p$-adic $L$-functions for CM-fields”, Invent. Math., 49 (1978), 199–297 | DOI | MR | Zbl

[21] Kubota T., Leopoldt H.-W., “Eine $p$-adische Theorie der Zeta-Werte”, Journal für die Reine Angew. Math., 214/215 (1964), 328–339 | MR | Zbl

[22] Li W.-C. W., “$L$-series of Rankin type and their functional equations”, Math. Ann., 244 (1979), 135–166 | DOI | MR | Zbl

[23] Mazur b., Swinnerton-Dyer H. P. F., “Arithmetic of Weil curves”, Invent. Math., 25 (1974), 1–61 | DOI | MR | Zbl

[24] Mazur b., Wiles A., “Class fields of Abelian extensions of $Q$”, Invent. Math., 76 (1984), 179–330 | DOI | MR | Zbl

[25] Rankin R. A., “Contribution to the theory of Ramanujans's function $\tau(n)$ and similar arithmetical functions, I, II”, Proc. Camb. Phil. Soc., 35 (1939), 351–372 | DOI | MR | Zbl

[26] Rankin R. A., “The scalar product of modular forms”, Proc. Lond. Math. Soc. Ser. 3, 2 (1952), 198–217 | DOI | MR | Zbl

[27] Schmidt C.-G., “The $p$-adic $L$-functions attached to Rankin convolutions of modular forms”, Journal für die Reine Angew. Math., 368 (1986), 201–220 | MR | Zbl

[28] Shimura G., “The special values of the zeta-functions associated with cusp forms”, Comm. Pure Appl. Math., 29 (1976), 783–804 | DOI | MR | Zbl

[29] Shimura G., “On the periods of modular forms”, Math. Ann., 229 (1977), 211–221 | DOI | MR | Zbl

[30] Selberg A., “Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist”, Arch. Math. Naturwid, 43 (1940), 47–50 | MR

[31] Siegel C.-L., “Die Funktionalgleichungen einiger Dirichletscher Reihen”, Math. Zeitschrift, 63 (1956), 363–373 | DOI | MR | Zbl

[32] Sturm J., “Special values of zeta-functions and Eisenstein series of half-integral weight”, Amer. J. Math., 102 (1980), 219–240 | DOI | MR | Zbl

[33] Zagier Don B., “Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields”, Lecture Notes Math., 627, 1977, 106–168 | MR