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Two variable $$p$$-adic $$L$$ functions attached to eigenfamilies of positive slope. (English) Zbl 1065.11025
Let $$p\geq 5$$ be a prime number, and $$N$$ a positive integer prime to $$p$$. Any classical cusp eigenform $$f$$ of weight $$k\geq 2$$ for $$\Gamma_0(N)$$, with a Dirichlet character $$\psi\bmod N$$ can be induced (under suitable assumptions on $$p$$ and $$f$$) into a family of cusp eigenforms $$f_{(k')}$$ of weight $$k'$$ in such a way that $$f_{(k)}=f$$, and Fourier coefficients $$a_n(f_{(k')})$$ are given by certain $$p$$-adic analytic functions $$k'\mapsto a_n(f_{(k')})$$ (Hida, Coleman).
The author constructs a two variable $$p$$-adic $$L$$-function attached to a family $$\{f_{(k')}\}$$ of cusp eigenforms of a fixed positive slope $$\sigma=v_p(\alpha_p)>0$$, where $$\alpha_p=\alpha_p(k)$$ is an eigenvalue of the Atkin operator $$U_p$$ (Theorem 0.5). Such a $$p$$-adic $$L$$-function interpolates the special values $$L_{f(k')}(s_s\chi)$$ at points $$(s_sk')$$ where $$s=1,\dots,k'-1$$ and $$\chi$$ is an arbitrary Dirichlet character. The resulting $$p$$-adic $$L$$-functions come from certain Eisenstein distributions with values in Banach modules of families of overconvergent forms (Theorem 0.3).

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F85 $$p$$-adic theory, local fields
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##### References:
 [1] Amice, Y., Vélu, J.: Distributions p-adiques associées aux series de Hecke. Astérisque 24–25, 119–131 (1975) · Zbl 0332.14010 [2] Böcherer, S., Schmidt, C.-G.: p-adic measures attached to Siegel modular forms. Ann. Inst. Fourier 50, 1375–1443 (2000) · Zbl 0962.11023 [3] Coates, J.: On p-adic L-functions. Sem. Bourbaki, 40ème année, 1987–88, 701, Astérisque 177–178 (1989) [4] Coates, J., Perrin-Riou, B.: On p-adic L-functions attached to motives over Q. Adv. Stud. Pure Math. 17, 23–54 (1989) · Zbl 0783.11039 [5] Colmez, P.: Fonctions L p-adiques. Séminaire Bourbaki, 51ème année, 1998–99, 851. Novembre 1998 [6] Coleman, R.F.: p-adic Banach spaces and families of modular forms. Invent. Math. 127, 417–479 (1997) · Zbl 0918.11026 [7] Coleman, R., Edixhoven, B.: On the semisimplicity of the U p operator on modular forms. Math. Ann. 310, 119–127 (1998) · Zbl 0902.11020 [8] Coleman, R., Mazur, B.: The eigencurve, ed. by A.J. Scholl et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9–18, 1996. Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998). Cambridge: Cambridge University Press [9] Coleman, R., Stevens, G., Teitelbaum, J.: Numerical experiments on families of p-adic modular forms, in: Computational Perspectives on Number Theory, ed. by D.A. Buell, J.T. Teitelbaum. Am. Math. Soc. 1998 · Zbl 0990.11029 [10] Deligne, P.: Valeurs de fonctions L et périodes d’intégrales. Proc. Symp. Pure Math 33, 313–342 (1979) [11] Deligne, P., Ribet, K.A.: Values of Abelian L-functions at negative integers over totally real fields. Invent. Math. 59, 227–286 (1980) · Zbl 0434.12009 [12] Hida, H.: A p-adic measure attached to the zeta functions associated with two elliptic cusp forms. I. Invent. Math. 79, 159–195 (1985) · Zbl 0573.10020 [13] Hida, H.: Galois representations into GL 2(Z p [[X]]) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986) · Zbl 0612.10021 [14] Hida, H.: Le produit de Petersson et de Rankin p-adique. Séminaire de Théorie des Nombres, Paris 1988–1989, 87–102, Prog. Math. 91. Boston, MA: Birkhäuser 1990 [15] Hida, H.: On p-adic L-functions of GL(2){$$\times$$}GL(2) over totally real fields. Ann. Inst. Fourier 40, 311–391 (1991) · Zbl 0725.11025 [16] Hida, H.: Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts, 26, ix, 386 p. Cambridge: Cambridge University Press 1993 · Zbl 0942.11024 [17] Iwasawa, K.: Lectures on p-adic L-functions. Ann. Math. Stud. 74. Princeton: Princeton University Press 1972 · Zbl 0236.12001 [18] Katz, N.M.: p-adic interpolation of real analytic Eisenstein series. Ann. Math. 104, 459–571 (1976) · Zbl 0354.14007 [19] Katz, N.M.: The Eisenstein measure and p-adic interpolation. Am. J. Math. 99, 238–311 (1977) · Zbl 0375.12022 [20] Katz, N.M.: p-adic L-functions for CM–fields. Invent. Math. 48, 199–297 (1978) · Zbl 0417.12003 [21] Kitagawa, K.: On standard p-adic L-functions of families of elliptic cusp forms, ed. by B. Mazur et al., p-adic monodromy and the Birch and Swinnerton-Dyer conjecture. A workshop held August 12–16, 1991 in Boston, MA, USA. Providence, RI: Amer. Math. Soc. Contemp. Math. 165, 81–110, 1994 [22] Kubota, T., Leopoldt, H.-W.: Eine p-adische Theorie der Zetawerte. J. Reine Angew. Math. 214/215, 328–339 (1964) · Zbl 0186.09103 [23] Lang, S.: Introduction to Modular Forms. Springer 1976 · Zbl 0344.10011 [24] Manin, Yu.I.: Periods of cusp forms and p-adic Hecke series. Mat. Sb. 92, 378–401 (1973) (in Russian) · Zbl 0293.14007 [25] Manin, Yu.I.: Non-Archimedean integration and p-adic L-functions of Jacquet–Langlands. Usp. Mat. Nauk 31, 5–54 (1976) (in Russian) · Zbl 0348.12016 [26] Manin, Yu.I., Panchishkin, A.A.: Convolutions of Hecke series and their values at integral points. Mat. Sb. 104, 617–651 (1977) (in Russian) · Zbl 0392.10028 [27] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986) · Zbl 0699.14028 [28] Miyake, T.: Modular forms. Transl. from the Japanese by Yoshitaka Maeda, viii, 335 p. Berlin etc.: Springer 1989 · Zbl 0701.11014 [29] Panchishkin, A.A.: Convolutions of Hilbert modular forms and their non-Archimedean analogues. Math. USSR, Sb. 64, 571–584 (1989); translation from Mat. Sb., Nov. Ser. 136(178), 574–587 (1988) · Zbl 0677.10019 [30] Panchishkin, A.A.: Non-Archimedean automorphic zeta-functions, 166p. Moscow: Moscow University Press 1988 · Zbl 0667.10017 [31] Panchishkin, A.A.: Motives over totally real fields and p-adic L-functions. Ann. Inst. Fourier 44, 989–1023 (1994) · Zbl 0808.11034 [32] Panchishkin, A.A.: Non-Archimedean L-functions of Siegel and Hilbert modular forms, 166p. Lect. Notes Math. 1471 (1991) · Zbl 0732.11026 [33] Panchishkin, A.A.: 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, vol. 2, 251–292. Seattle, Providence, R.I. 1994 · Zbl 0837.11029 [34] Panchishkin, A.A.: Non-Archimedean Mellin transform and p-adic L Functions. Vietnam J. Math. 3, 179–202 (1997) · Zbl 0907.11018 [35] Panchishkin, A.A.: On the Siegel–Eisenstein measure. Isr. J. Math. 120, 467–509 (2000) · Zbl 0977.11021 [36] Panchishkin, A.A.: On p-adic integration in spaces of modular forms and its applications. Preprint of IAS 2000 (to appear in VINITI, Moscow 2003) [37] Rankin, R.A.: Contribution to the theory of Ramanujan’s function {$$\tau$$}(n) and similar arithmetical functions. I.II. Proc. Camb. Phil. Soc. 35, 351–372 (1939) · JFM 65.0353.01 [38] Rankin, R.A.: The scalar product of modular forms. Proc. Lond. Math. Soc. 2, 198–217 (1952) · Zbl 0049.33904 [39] Rankin, R.A.: The adjoint Hecke operator. Automorphic functions and their applications. Int. Conf., Khabarovsk/USSR 1988, 163–166 (1990) · Zbl 0744.11022 [40] Schmidt, C.-G.: The p-adic L-functions attached to Rankin convolutions of modular forms. J. Reine Angew. Math. 368, 201–220 (1986) · Zbl 0585.10020 [41] Serre, J.-P.: Endomorphisms completement continus des espaces de Banach p-adiques. Publ. Math., Inst. Hautes Étud. Sci. 12, 69–85 (1962) · Zbl 0104.33601 [42] Serre, J.-P.: Formes modulaires et fonctions zêta p-adiques. Lect. Notes Math. 350, 191–268 (1973). Springer [43] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton: Princeton Univ. Press 1971 · Zbl 0221.10029 [44] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. 31, 79–98 (1975) · Zbl 0311.10029 [45] Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976) · Zbl 0348.10015 [46] Shimura, G.: On the periods of modular forms. Math. Ann. 229, 211–221 (1977) · Zbl 0363.10019 [47] Shimura, G.: Arithmeticity in the theory of automorphic forms. Mathematical Surveys and Monographs. 82, x, 302 p. Providence, RI: American Mathematical Society (AMS) 2000 [48] Scholl, A.J.: An introduction to Kato’s Euler systems, ed. by A.J. Scholl et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9–18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 379–460 (1998) · Zbl 0952.11015 [49] Stevens, G.: Overconvergent modular symbols and a conjecture of Mazur, Tate and Teitelbaum. To appear [50] Tate, J.: Rigid analytic spaces. Invent. Math. 12, 257–289 (1971) · Zbl 0212.25601 [51] Tilouine, J., Urban, E.: Several variable p-adic families of Siegel-Hilbert cusp eigenforms and their Galois representations. Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 499–574 (1999) · Zbl 0991.11016 [52] Visik, M.M.: Non-Archimedean measures connected with Dirichlet series. Math. Sb. 28(1976), 216–228 (1978) · Zbl 0369.14010 [53] Vishik, M.M., Manin, Yu.I.: p-adic Hecke series of imaginary quadratic fields. Math. Sb. 24(1974), 345–371 (1976) · Zbl 0329.12016 [54] Wiles, A.: On ordinary {$$\lambda$$}-adic representations associated to modular forms. Invent. Math. 94, 529–573 (1988) · Zbl 0664.10013 [55] Wiles, A.: The Iwasawa conjecture for totally real fields. Ann. Math. (2) 131, 493–540 (1990) · Zbl 0719.11071 [56] Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem. Ann. Math. (2) 141, 443–455 (1995) · Zbl 0823.11029 [57] Weil, A.: On a certain type of characters of the idèle-class group of an algebraic number-field. Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, pp. 1–7. Tokyo: Science Council of Japan 1956 [58] Yager, R.I.: On two variable p-adic L-functions. Ann. Math. (2) 115, 411–449 (1982) · Zbl 0496.12010
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