$p$-adic measures attached to Siegel modular forms

Böcherer, Siegfried; Schmidt, Claus-Günther

Annales de l'Institut Fourier, Tome 50 (2000) no. 5, pp. 1375-1443.

On étudie les valeurs critiques de la fonction $L$ complexe standard, associée à une forme modulaire de Siegel holomorphe et des fonctions $L$ tordues des caractères de Dirichlet. Notre objet principal est, pour un nombre premier rationnel $p$ donné, l’interpolation $p$-adique des valeurs critiques essentiellement algébriques en laissant varier les caractères de Dirichlet afin d’obtenir un contrôle systématique des dénominateurs des valeurs critiques par des congruences de Kummer généralisées. Pour organiser cette information on montre l’existence de mesures $p$-adiques telles que l’intégration d’un caractère de Dirichlet de conducteur une $p$-puissance sur la mesure, donne la valeur critique normalisée de la fonction $L$ complexe tordue du caractère de Dirichlet. D’une manière standard les mesures $p$-adiques définissent des fonctions $Lp$-adiques qui par conséquent interpolent $p$-adiquement les valeurs critiques normalisées.

We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of $p$-adic measures such that integration of any Dirichlet character of $p$-power conductor over the measure yields the suitably normalized critical value of the complex $L$-function twisted by the Dirichlet character. In a standard manner the $p$-adic measures naturally define $p$-adic $L$-functions which hence $p$-adically interpolate the normalized critical values.

[1] A.N. Andrianov, V.L. Kalinin, On the analytic properties of standard zeta functions of Siegel modular forms, Math. USSR Sbornik, 35 (1979), 1-17. | Zbl

[2] A.N. Andrianov, Quadratic Forms and Hecke Operators. Grundlehren der mathematischen Wissenschaften 286, Berlin-Heidelberg-New York, Springer, 1987. | Zbl

[3] S. Böcherer, Über die Fourier-Jacobi-Entwicklung der Siegelschen Eisensteinreihen, Math.Z., 183 (1983), 21-43. | Zbl

[4] S. Böcherer, Über die Fourier-Jacobi-Entwicklung der Siegelschen Eisensteinreihen II, Math.Z., 189 (1985), 81-100. | Zbl

[5] S. Böcherer, Über die Fourierkoeffizienten Siegelscher Eisensteinreihen, Manuscripta Math., 45 (1984), 273-288. | Zbl

[6] S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. reine angew. Math., 362 (1985), 146-168. | Zbl

[7] S. Böcherer, Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe, Abh. Math. Sem. Univ. Hamburg, 56 (1986), 35-47. | Zbl

[8] U. Christian, Selberg's Zeta-, L-, and Eisenstein Series. Lecture Notes in Math. 1030. Berlin-Heidelberg-New York, Springer, 1983. | MR | Zbl

[9] U. Christian, Maaßsche L-Reihen und eine Identität für Gaußsche Summen, Abh. Math. Sem. Univ. Hamburg, 54 (1984), 163-175. | MR | Zbl

[10] P. Deligne, Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math., 33, Part 2 (1979), 313-346. | MR | Zbl

[11] P. Feit, Poles and residues of Eisenstein series for symplectic and unitary groups, Memoirs AMS, 61 (1986), no 346. | MR | Zbl

[12] E. Freitag, Siegelsche Modulfunktionen. Grundlehren der mathematischen Wissenschaften 254, Berlin-Heidelberg-New York, 1983. | MR | Zbl

[13] P.B. Garrett, M. Harris, Special values of triple product L-functions, Amer. J. Math., 115 (1993), 159-238. | MR | Zbl

[14] S. Gelbart, I. Piatetski-Shapiro, S. Rallis, Explicit Constructions of Automorphic L-Functions. Springer Lecture Notes in Math. 1254, Berlin-Heidelberg-New York, Springer, 1987. | MR | Zbl

[15] M. Harris, Special values of zeta functions attached to Siegel modular forms, Ann Sci. Ecole Norm. Sup., 14 (1981), 77-120. | Numdam | MR | Zbl

[16] L.K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Transl. Math. Monographs 6, AMS 1963. | Zbl

[17] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J., 95 (1984), 73-84. | MR | Zbl

[18] H. Maaß, Siegel's modular forms and Dirichlet series, Lecture Notes in Math. 216, Berlin-Heidelberg-New York, Springer, 1971. | MR | Zbl

[19] Sh.-I. Mizumoto, Poles and residues of standard L-functions attached to Siegel modular forms, Math. Ann., 289 (1991), 589-612. | MR | Zbl

[20] A.A. Panchishkin, Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Springer Lecture Notes in Math. 1471, Berlin-Heidelberg-New York, Springer, 1991. | MR | Zbl

[21] A.A. Panchishkin, Admissible Non-Archimedean Standard Zeta Functions associated with Siegel Modular Forms, Proc. Symp. Pure Math., 55, Part 2, 251-292. | MR | Zbl

[22] I. Piatetski-Shapiro, S. Rallis, A new way to get Euler products, J. reine angew. Math., 392 (1988), 110-124. | MR | Zbl

[23] C.G. Schmidt, P-adic measures attached to automorphic representations of G1(3), Invent. Math., 92 (1988), 597-631. | MR | Zbl

[24] G. Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen, 1975, 261-268. | MR | Zbl

[25] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., XXIX (1976), 783-804. | MR | Zbl

[26] G. Shimura, Arithmetic of differential operators on symmetric domains, Duke Math. J., 48 (1981), 813-843. | MR | Zbl

[27] G. Shimura—, Confluent hypergeometric functions on tube domains, Math. Ann., 269 (1982), 269-302. | MR | Zbl

[28] G. Shimura, On Eisenstein Series, Duke Math. J., 50 (1983), 417-476. | MR | Zbl

[29] G. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math., 77 (1984), 463-488. | MR | Zbl

[30] G. Shimura, Differential operators and the singular values of Eisenstein series, Duke Math. J., 51 (1984), 261-329. | MR | Zbl

[31] G. Shimura, On Eisenstein series of half-integral weight, Duke Math. J., 52 (1985), 281-314. | MR | Zbl

[32] G. Shimura, On a class of nearly holomorphic automorphic forms, Annals of Math., 123 (1986), 347-406. | MR | Zbl

[33] G. Shimura, Nearly holomorphic functions on hermitian symmetric spaces, Math. Ann., 278 (1987), 1-28. | MR | Zbl

[34] G. Shimura, Invariant differential operators on hermitian symmetric spaces, Annals of Math., 132 (1990), 237-272. | MR | Zbl

[35] G. Shimura, Differential Operators, Holomorphic Projection, and Singular Forms, Duke Math. J., 76 (1994), 141-173. | MR | Zbl

[36] J. Sturm, The critical values of zeta functions associated to the symplectic group, Duke Math. J., 48 (1981), 327-350. | MR | Zbl

[37] T. Tamagawa, On the zeta functions of a division algebra, Annals of Math., 77 (1963), 387-405. | MR | Zbl

[38] L. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math. 83, Berlin-Heidelberg-New York, Springer, 1982. | MR | Zbl