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On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. (English) Zbl 1090.22008

Let \(\mathbb G\) be a connected reductive linear algebraic group defined over a number field \(F\). Let \(\mathfrak{p}\) be a finite prime of \(F\). The author considers the group \(\mathbb G(\mathbb A_f)\), where \(\mathbb A_f\) is the ring of finite adeles of \(F\), and constructs a family of locally analytic representations on locally convex topological vector spaces over a finite extension of \(\mathbb Q_p\). These representations are used to obtain \(\mathfrak{p}\)-adic analytic families of systems of Hecke eigenvalues, which \(\mathfrak{p}\)-adically interpolate the systems of Hecke eigenvalues attached to automorphic representations of cohomological type [see R. Coleman and B. Mazur, Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)]. The case \(\mathbb G=GL_2\), \(F=\mathbb Q\) is considered in detail. A survey of the related literature is given.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R56 Adèle rings and groups

Citations:

Zbl 0932.11030
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[1] Amice, Y.: Duals. In: Proc. Conf. on p-adic analysis (Nijmegen, 1978), pp. 1–15
[2] Ash, A., Stevens, G.: p-adic deformations of cohomology classes of subgroups of GL(N,\(\mathbb{Z}\)): the non-ordinary case. Preprint, draft April 25, 2000
[3] Bernstein, J.N.: Le ”centre” de Bernstein. Representations of reductive groups over a local field. Deligne , P., (ed.), Travaux en cours, pp. 1–32. Hermann 1984
[4] Bourbaki, N.: Elements of Mathematics. Topological Vector Spaces, Chaps. 1–5. Springer 1987 · Zbl 0622.46001
[5] Buzzard, K.: On p-adic families of automorphic forms. In: Modular curves and abelian varieties, ed. by J. Cremona, J.-C. Lario, J. Quer, K. Ribet (Bellaterra, 2002). Progr. Math. 23–44 (2004) · Zbl 1166.11322
[6] Buzzard, K.: Eigenvarieties. To appear in Proc. 2004 Lond. Math. Soc. Durham Conf. on L-functions and arithmetic
[7] Casselman, W.: Introduction to the theory of admissible representations of p-adic reductive groups. Unpublished notes distributed by Sally, P., draft May 7, 1993 (available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html)
[8] Chenevier, G.: Familles p-adiques de formes automorphes pour GLn. J. Reine Angew. Math. 570, 143–217 (2004) · Zbl 1093.11036
[9] Chenevier, G.: Une correspondance de Jacquet-Langlands p-adique. Duke Math. J. 126, 161–194 (2005) · Zbl 1070.11016
[10] Clozel, L.: Motifs et formes automorphes: applications du principe de fonctorialité. In: Automorphic forms, Shimura varieties and L-functions, vol. I, ed. by L. Clozel, J.S. Milne. Ann Arbor, MI: Academic Press 1988, Perspect. Math. 10, 77–159 (1990)
[11] Coleman, R., Mazur, B.: The eigencurve. In: Galois representations in arithmetic algebraic geometry, ed. by A.J. Scholl, R.L. Taylor (Durham, 1996). Lond. Math. Soc. Lect. Note Ser., vol. 254, pp. 1–113. Cambridge Univ. Press 1998
[12] Deligne, P.: Formes modulaires et representations de GL(2). Modular functions of one variable II. Lect. Notes Math., vol. 349, pp. 55–105. Springer 1973
[13] Emerton, M.: Locally analytic vectors in representations of locally p-adic analytic groups. To appear in Mem. Am. Math. Soc. · Zbl 1117.22008
[14] Emerton, M.: Jacquet modules for locally analytic representations of p-adic reductive groups I. Construction and first properties. To appear in Ann. Sci. Éc. Norm. Supér. · Zbl 1117.22008
[15] Emerton, M.: p-adic L-functions and unitary completions of representations of p-adic reductive groups. To appear in Duke Math. J. · Zbl 1092.11024
[16] Emerton, M.: Locally analytic representation theory of p-adic reductive groups: A summary of some recent developments. Preprint 2004 · Zbl 1149.22014
[17] Emerton, M.: Jacquet modules for locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. In preparation · Zbl 1117.22008
[18] Franke, J.: Harmonic analysis in weighted L2-spaces. Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, 181–279 (1998) · Zbl 0938.11026
[19] Gross, B.H.: Algebraic modular forms. Isr. J. Math. 113, 61–93 (1999) · Zbl 0965.11020
[20] Hida, H.: Galois representations into GL2(\(\mathbb{Z}\)p[[X]]) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986) · Zbl 0612.10021
[21] Ihara, Y.: On Galois representations arising from towers of coverings of \(\mathbb{P}\)10,1,. Invent. Math. 86, 427–459 (1986) · Zbl 0585.14020
[22] Kassaei, P.: \(\mathcal{P}\) -adic modular forms over Shimura curves over totally real fields. Compos. Math. 140 359–395 (2004) · Zbl 1052.11037
[23] Kisin, M., Lai, K.F.: Overconvergent Hilbert modular forms. Am. J. Math. 127, 735–783 (2005) · Zbl 1129.11020
[24] Langlands, R.P.: Modular forms and -adic representations. Modular functions of one variable II. Lect. Notes Math., vol. 349, pp. 361–500. Springer 1973 · Zbl 0279.14007
[25] Lazard, M.: Groupes analytiques p-adiques. Publ. Math., Inst. Hautes Étud. Sci. 26 (1965) · Zbl 0139.02302
[26] 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
[27] Schneider, P.: Nonarchimedean functional analysis. Springer Monographs in Math. Springer 2002 · Zbl 0998.46044
[28] Schneider, P., Teitelbaum, J.: Locally analytic distributions and p-adic representation theory, with applications to GL2. J. Am. Math. Soc. 15, 443–468 (2002) · Zbl 1028.11071
[29] Schneider, P., Teitelbaum, J.: \(U(\mathfrak{g})\) -finite locally analytic representations. Represent. Theory 5, 111–128 (2001) · Zbl 1028.17007
[30] Schneider, P., Teitelbaum, J.: Banach space representations and Iwasawa theory. Isr. J. Math. 127, 359–380 (2002) · Zbl 1006.46053
[31] Schneider, P., Teitelbaum, J.: Algebras of p-adic distributions and admissible representations. Invent. Math. 153, 145–196 (2003) · Zbl 1028.11070
[32] Shimura, G.: An -adic method in the theory of automorphic forms. Goro Shimura, Collected Works, vol. 2. Springer 2002
[33] Stevens, G.: Overconvergent modular symbols and a conjecture of Mazur, Tate, and Teitelbaum. Preprint
[34] Sweedler, M.: Hopf algebras. W.A. Benjamin 1969
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