Motives over totally real fields and $p$-adic $L$-functions

Panchishkin, Alexei A.

Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 989-1023.

On étudie des valeurs spéciales des fonctions $L$ de type $L(M,s)$ où $M$ est un motif sur un corps totalement réel $F$ à coefficients dans un corps de nombres $T$, et

$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$

est un produit eulérien étendu sur tous les idéaux maximaux $𝔭$ de l’ordre maximal ${𝒪}_F$ de $F$ et

$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$

est un polynôme à coefficients dans $T$. À l’aide des polygones de Newton et de Hodge de $M$ on formule des conditions conjecturales de l’existence d’un prolongement $p$-adique analytique de ces valeurs spéciales. On vérifie cette conjecture dans une série d’exemples liés aux formes modulaires de Hilbert.

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and

$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$

is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and

$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$

being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to Hilbert modular forms.

[AmV] Y. Amice, J. Vélu, Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque n°24/25, Soc. Math. France, Paris, (1975), 119-131. | MR | Zbl

[Ba] D. Barsky, Fonctions zeta p-adiques d'une classe de rayon des corps de nombres totalement réels, Groupe d'Étude d'Analyse Ultramétrique (Y. Amice, G. Christol, P.Robba), 5e année, 1977/1978, n°16, 23 p. | Numdam | Zbl

[Bl1] D. Blasius, On the critical values of Hecke L-series, Ann. Math., 124 (1986), 23-63. | MR | Zbl

[Bl2] D. Blasius, Appendix to Orloff critical values of certain tensor product L-function, Invent. Math., 90 (1987), 181-188. | MR | Zbl

[Bl3] D. Blasius, A p-adic property of Hodge classes on Abelian variety, in Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20-August 2 1991, Seattle, Providence, R.I., 1993. | Zbl

[BlRo] D. Blasius and J.D. Rogawski, Motives for Hilbert modular forms, Invent. Math., 114 (1993), 55-87. | MR | Zbl

[Ca] H. Carayol, Sur les représentations p-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4), 19 (1986), 409-468. | Numdam | MR | Zbl

[Cass-N] P. Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zeta et fonctions zeta p-adiques, Invent. Math., 51 (1979), 29-59. | MR | Zbl

[Co] J. Coates, On p-adic L-functions, Sém. Bourbaki, 40ème année, 1987-1988, Astérisque n°701 (1989), 177-178. | Numdam | Zbl

[CoPe-Ri] J. Coates, B. Perrin-Riou, On p-adic L-functions attached to motives over Q, Advanced Studies in Pure Math., 17 (1989), 23-54. | MR | Zbl

[CoSch] J. Coates, C.-G. Schmidt, Iwasawa theory for the symmetric square of an elliptic curve, J. Reine Angew. Math., 375/376 (1987), 104-156. | MR | Zbl

[Da] A. Dabrowski, p-adic L-functions of Hilbert modular forms, Ann. Inst. Fourier (Grenoble), 44-4 (1994). | Numdam | MR | Zbl

[De1] P. Deligne, Formes modulaires et représentations l-adiques, Sém. Bourb. 1968/1969, exp. n°335. Springer-Verlag, Lect. Notes in Math., 179 (1971), 139-172. | Numdam | Zbl

[De2] P. Deligne, La conjecture de Weil. I, Publ. Math. IHES, 43 (1974), 273-307. | Numdam | MR | Zbl

[De3] P. Deligne, Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math AMS, 33 (part 2) (1979), 313-342. | MR | Zbl

[DeR] P. Deligne, K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math., 59 (1980), 227-286. | MR | Zbl

[H] S. Haran, p-adic L-functions for modular forms, Compos. Math., 62 (1986), 31-46. | Numdam | MR | Zbl

[Ha1] M. Harris, Arithmetical vector bundles and automorphic forms on Shimura varieties. I, Invent. Math., 59 (1985), 151-189. | Zbl

[Ha2] M. Harris, Period invariants of Hilbert modular forms, I. Lecture Notes in Math., 1447 (1990), 155-200. | MR | Zbl

[Ha3] M. Harris, Hodge-de Rham structures and periods of automorphic forms, in Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20-August 2 1991, Seattle, Providence, R.I., 1993.

[Hi1] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic cusp forms. I, Invent. Math., 79 (1985), 159-195. | MR | Zbl

[Hi2] H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Invent. Math., 85 (1986), 545-613. | MR | Zbl

[Hi3] H. Hida, On p-adic L-functions of GL(2) x GL(2) over totally real fields, Ann. Inst. Fourier (Grenoble), 40-2 (1991), 311-391. | Numdam | MR | Zbl

[Iw] K. Iwasawa, Lectures on p-adic L-functions, Ann. of Math. Studies, 74, Princeton University Press, 1972. | MR | Zbl

[Ja] U. Jannsen, Mixed motives and algebraic K-theory, Springer-Verlag, Lecture Notes in Math., 1400 (1990). | Zbl

[Ka1] N.M. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math., 104 (1976), 459-571. | MR | Zbl

[Ka2] N.M. Katz, The Eisenstein measure and p-adic interpolation, Amer. J. Math., 99 (1977), 238-311. | MR | Zbl

[Ka3] N.M. Katz, p-adic L-functions for CM-fields, Invent. Math., 48 (1978), 199-297. | MR | Zbl

[Kl1] H. Klingen, Über die Werte Dedekindscher Zetafunktionen, Math. Ann., 145 (1962), 265-272. | MR | Zbl

[Kl2] H. Klingen, Über den arithmetischen Charakter der Fourier-koefficienten von Modulformen, Math. Ann., 147 (1962), 176-188. | MR | Zbl

[Ko1] N. Koblitz, p-adic numbers, p-adic analysis and zeta functions, 2nd ed. Springer-Verlag, 1984.

[Ko2] N. Koblitz, p-adic analysis: a short course on recent work, London Math. Soc. Lect. Notes Series, 46, Cambridge University Press, London, Cambridge, 1980. | MR | Zbl

[KuLe] T. Kubota, H.-W. Leopoldt, Eine p-adische Theorie der Zetawerte, J. Reine Angew. Math., 214/215 (1964), 328-339. | MR | Zbl

[Kurč] P.F. Kurčanov, Local measures connected with cusp forms of Jacquet-Langlands over CM-fields, Mat. Sbornik, 108 (1979), 483-503 (in Russian). | Zbl

[Man1] Y.I. Manin, Cyclotomic fields and modular curves, Uspekhi Mat. Nauk, 26 (1971), 7-78 (in Russian). | Zbl

[Man2] Y.I. Manin, Cusp forms and zeta functions of modular curves, Izvestija Akad. Nauk. Ser. Matem., 36 (1972), 19-66 (in Russian).

[Man3] Y.I. Manin, Explicit formulas for the eigenvalues of Hecke operators, Acta Arithm., 24 (1973), 239-249. | MR | Zbl

[Man4] Y.I. Manin, Periods of cusp forms and p-adic Hecke series, Mat. Sbornik, 92 (1973), 378-401 (in Russian). | MR | Zbl

[Man5] Y.I. Manin, The values of p-adic Hecke series at integer points of the critical strip, Mat. Sbornik, 93 (1974), 621-626 (in Russian).

[Man6] Y.I. Manin, Non-Archimedean integration and p-adic L- functions of Jacquet-Langlands, Uspekhi Mat. Nauk, 31 (1976), 5-54 (in Russian). | Zbl

[Man7] Y.I. Manin, Modular forms and number theory, Proc. Int. Congr. Math. Helsinki, (1978), 177-186. | Zbl

[ManPa] Y.I. Manin, A.A. Panchishkin, Convolutions of Hecke series and their values at integral points. Mat. Sbornik, 104 (1977), 617-651 (in Russian). | Zbl

[Maz1] B. Mazur, On the special values of L-functions, Invent. Math., 55 (1979), 207-240. | MR | Zbl

[Maz2] B. Mazur, Modular curves and arithmetic, Proc. Int. Congr. Math. Warszawa, 16-24 August 1982, North Holland, Amsterdam (1984), 185-211. | Zbl

[MazSD] B. Mazur, H.P.F. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math., 25 (1974), 1-61. | MR | Zbl

[MazW1] B. Mazur, A. Wiles, Analogies between function fields and number fields, Am. J. Math., 105 (1983), 507-521. | MR | Zbl

[MazW2] B. Mazur, A. Wiles, Class fields of Abelian extensions of Q, Invent. Math., 76 (1984), 179-330. | MR | Zbl

[MazW3] B. Mazur, A. Wiles, On p-adic analytic families of Galois representations, Compos. Math., 59 (1986), 231-264. | Numdam | MR | Zbl

[Miy] T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of Math., 94 (1971), 174-189. | MR | Zbl

[My] My Vinh Quang, Convolutions p-adiques non bornées de formes modulaires de Hilbert, C.R. Acad. Sci. Paris Sér. I Math., 315 n°11 (1992), 1121-1124. | MR | Zbl

[Oda] T. Oda, Periods of Hilbert modular surfaces, Boston, Birkhäuser, Progress in Math., 19 (1982). | MR | Zbl

[Oh] M. Ohta, On the zeta-functions of an Abelian scheme over the Shimura curve, Japan J. of Math., 9 (1983), 1-26. | MR | Zbl

[Pa1] A.A. Panchishkin, Symmetric squares of Hecke series and their values at integral points, Mat. Sbornik, 108 (1979), 393-417 (in Russian). | MR | Zbl

[Pa2] A.A. Panchishkin, On p-adic Hecke series, in “Algebra” (Ed. by A. I. Kostrikin), Moscow Univ. Press (1980), 68-71 (in Russian). | Zbl

[Pa3] A.A. Panchishkin, Complex valued measures attached to Euler products, Trudy Sem. Petrovskogo, 7 (1981), 239-244 (in Russian). | MR | Zbl

[Pa4] A.A. Panchishkin, Modular forms, in the series “Algebra. Topology. Geometry.” Vol. 19. VINITI Publ., Moscow (1981), 135-180 (in Russian). | MR | Zbl

[Pa5] A.A. Panchishkin, Local measures attached to Euler products in number fields, in “Algebra” (Ed. by A. I. Kostrikin), Moscow Univ. Press (1982), 119-138 (in Russian). | MR | Zbl

[Pa6] A.A. Panchishkin, Automorphic forms and the functoriality principle, in “Automorphic forms, representations and L-functions”, Mir Publ., Moscow (1984), 249-286 (in Russian).

[Pa7] A.A. Panchishkin, Le prolongement p-adique analytique de fonctions L de Rankin, C. R. Acad. Sci. Paris, Sér. I Math., 294 (1982), 51-53 ; 227-230. | Zbl

[Pa8] A.A. Panchishkin, A functional equation of the non-Archimedean Rankin convolution, Duke Math. J., 54 (1987), 77-89. | MR | Zbl

[Pa9] A.A. Panchishkin, Non-Archimedean convolutions of Hilbert modular forms, Abstracts of the 19th USSR Algebraic Conference, Septembre 1987, Lvov. vol. 1, 211.

[Pa10] A.A. Panchishkin, Non-Archimedean Rankin L-functions and their functional equations, Izvestija Akad. Nauk., Ser. Matem., 52 (1988), 336-354 (in Russian). | MR | Zbl

[Pa11] A.A. Panchishkin, Convolutions of Hilbert modular forms and their non-Archimedean analogues, Mat. Sbornik, 136 (1988), 574-587 (in Russian). | MR | Zbl

[Pa12] A.A. Panchishkin, Non-Archimedean automorphic zeta-functions, Moscow University Press, (1988), 166p. | Zbl

[Pan13] A.A. Panchishkin, Convolutions of Hilbert modular forms, motives and p-adic zeta functions, preprint MPI, Bonn, 43 (1990).

[Pan14] A.A. Panchishkin, Non-Archimedean L-functions associated with Siegel and Hilbert modular forms, Lecture Notes in Math., 1471, Springer-Verlag, (1991), 166 p. | MR | Zbl

[Pa15] A.A. Panchishkin, Admissible Non-Archimedean standard zeta functions of Siegel modular forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20-August 2 1991, Seattle, Providence, R.I., vol. 2 (1993), 251-292. | MR | Zbl

[Ran1] R.A. Rankin, Contribution to the theory of Ramanujan's function τ(n) and similar arithmetical functions. I.II, Proc. Camb. Phil. Soc., 35 (1939), 351-372. | JFM | MR | Zbl

[Ran2] R.A. Rankin, The scalar product of modular forms, Proc. London Math., Soc., 2 (1952), 198-217. | MR | Zbl

[RoTu] J.D. Rogawski, J.B. Tunnel, On Artin L-functions associated to Hilbert modular forms, Invent. Math., 74 (1983), 1-42. | MR | Zbl

[Schm1] C.-G. Schmidt, The p-adic L-functions attached to Rankin convolutions of modular forms, J. Reine Angew. Math., 368 (1986), 201-220. | MR | Zbl

[Schm2] C.-G. Schmidt, p-adic measures attached to automorphic representations of GL(3), Invent. Math., 92 (1988), 597-631. | MR | Zbl

[Scho] A.J. Scholl, Motives for modular forms, Invent. Math., 100 (1990), 419-430. | MR | Zbl

[Shi1] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971. | Zbl

[Shi2] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc., 31 (1975), 79-98. | MR | Zbl

[Shi3] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783-804. | MR | Zbl

[Shi4] G. Shimura, On the periods of modular forms, Math. Annalen, 229 (1977), 211-221. | MR | Zbl

[Shi5] G. Shimura, On certain reciprocity laws for theta functions and modular forms, Acta Math., 141 (1978), 35-71. | MR | Zbl

[Shi6] G. Shimura, The special values of zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978), 637-679. | MR | Zbl

[Shi7] G. Shimura, Algebraic relations between critical values of zeta functions and inner products, Amer. J. Math., 105 (1983), 253-285. | MR | Zbl

[Shi9] G. Shimura, On Eisenstein series, Duke Math. J., 50 (1983), 417-476. | MR | Zbl

[Shi10] G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Invent. Math., 94 (1988), 245-305. | MR | Zbl

[Sie] C.-L. Siegel, Über die Fourierschen Koeffizienten von Modulformen, Nachr. Acad. Wiss. Göttingen. II. Math.-Phys. Kl., 3 (1970), 15-56. | MR | Zbl

[Ta] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math., 98 (1989), 265-280. | MR | Zbl

[V1] M.M. Višik, Non-Archimedean measures associated with Dirichlet series, Mat. Sbornik, 99 (1976), 248-260.

[V2] M.M. Višik, Non-Archimedean spectral theory, in the series “Modern Problems of Mathematics”, Moscow, VINITI Publ., 25 (1984), 51-114.

[Wa] L. Washington, Introduction to cyclotomic fields, Springer-Verlag, N.Y. e.a., 1982. | MR | Zbl

[Wi] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. Math., 131 (1990), 493-540. | MR | Zbl

[Yo] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, Duke Math. J., 75, n°1 (1994), 121-191. | MR | Zbl