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.


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


Zbl 0932.11030
Full Text: DOI


[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 Unpublished notes distributed by Sally, P., draft May 7, 1993 (available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html)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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.