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\(\mathcal L\) invariant and \(p\)-adic special series. (Invariant \(\mathcal L\) et série spéciale \(p\)-adique.) (French) Zbl 1166.11331
Summary: To any integer \(k\geq 2\) and any \(\mathcal L\in\overline{\mathbb Q_p}\), we associate conjecturally a \(p\)-adic Banach space \(B(k,L)\) endowed with a continuous linear action of \(\text{GL}_2(\mathbb Q_p)\). We prove that \(B(k,L)\) does exist either if \(k=2\) or if \(k>2\) and \(L\) is “coming from” a weight \(k\) eigenform on \(\Gamma_0(pN)\) with \((p,N)=1\) and \(N=N^-N^+\) where \((N^-,N^+)=1\) and \(N^-\) is the product of an odd number of prime numbers. The Banach space \(B(k,L)\) should “correspond” (up to torsion by crystalline characters) to 2-dimensional non-crystalline semi-stable representations of \(\text{Gal} (\overline{\mathbb Q_p}/\mathbb Q_p)\) over \(\overline{\mathbb Q_p}\).

MSC:
11F85 \(p\)-adic theory, local fields
11F80 Galois representations
22E50 Representations of Lie and linear algebraic groups over local fields
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References:
[1] Barthel L. , Livné R. , Modular representations of \({\mathrm{GL}}_{2}\) of a local field: the ordinary, unramified case , J. Number Theory 55 ( 1995 ) 1 - 27 . MR 1361556 | Zbl 0841.11026 · Zbl 0841.11026 · doi:10.1006/jnth.1995.1124
[2] Barthel L. , Livné R. , Irreducible modular representations of \({\mathrm{GL}}_{2}\) of a local field , Duke Math. J. 75 ( 1994 ) 261 - 292 . Article | MR 1290194 | Zbl 0826.22019 · Zbl 0826.22019 · doi:10.1215/S0012-7094-94-07508-X · minidml.mathdoc.fr
[3] Bertolini M. , Darmon H. , Iovita A. , Spiess M. , Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting , Amer. J. Math. 124 ( 2002 ) 411 - 449 . MR 1890998 | Zbl 1079.11036 · Zbl 1079.11036 · doi:10.1353/ajm.2002.0009 · muse.jhu.edu
[4] Breuil C. , Intégration p -adique , in: Séminaire Bourbaki 860 , Astérisque , vol. 266 , 2000 , pp. 319 - 350 . Numdam | MR 1772678 | Zbl 1015.11028 · Zbl 1015.11028 · numdam:SB_1998-1999__41__319_0 · eudml:110263
[5] Breuil C. , p -adic Hodge theory, deformations and local Langlands , cours au C.R.M. de Barcelone, juillet 2001, disponible à l’adresse, http://www.math.u-psud.fr/ breuil/ . · www.math.u-psud.fr
[6] Breuil C. , Sur quelques représentations modulaires et p -adiques de \({\mathrm{GL}}_{2}\left({Q}_{p}\right)\) I , Comp. Math. 138 ( 2003 ) 165 - 188 . MR 2018825 | Zbl 1044.11041 · Zbl 1044.11041 · doi:10.1023/A:1026191928449
[7] Breuil C. , Sur quelques représentations modulaires et p -adiques de \({\mathrm{GL}}_{2}\left({Q}_{p}\right)\) II , J. Institut Math. Jussieu 2 ( 2003 ) 23 - 58 . MR 1955206 | Zbl 02011912 · Zbl 1165.11319 · doi:10.1017/S1474748003000021
[8] Breuil C. , Série spéciale p -adique et cohomologie étale complétée , prépublication I.H.É.S., 2003, disponible à l’adresse, http://www.ihes.fr/ breuil/publications.html . · www.ihes.fr
[9] Breuil C. , Mézard A. , Multiplicités modulaires et représentations de \({\mathrm{GL}}_{2}\left({Z}_{p}\right)\) et de \(\mathrm{Gal}({\overline{Q}}_{p}/{Q}_{p})\) en \(\ell =p\) , Duke Math. J. 115 ( 2002 ) 205 - 310 . Article | MR 1944572 | Zbl 1042.11030 · Zbl 1042.11030 · doi:10.1215/S0012-7094-02-11522-1 · minidml.mathdoc.fr
[10] Coleman R. , A p -adic Shimura isomorphism and p -adic periods of modular forms , Contemp. Math. 165 ( 1994 ) 21 - 51 . MR 1279600 | Zbl 0838.11033 · Zbl 0838.11033
[11] Colmez P. , Fontaine J.-M. , Construction des représentations p -adiques semi-stables , Invent. Math. 140 ( 2000 ) 1 - 43 . MR 1779803 | Zbl 1010.14004 · Zbl 1010.14004 · doi:10.1007/s002220000042
[12] Deligne P. , Formes modulaires et représentations de \({\mathrm{GL}}_{2}\) , in: Modular Functions of One Variable II , Lecture Notes , vol. 349 , 1973 , pp. 55 - 105 . MR 347738 | Zbl 0271.10032 · Zbl 0271.10032
[13] Deligne P. , Serre J.-P. , Formes modulaires de poids 1 , Ann. Sci. E.N.S. 7 ( 1974 ) 507 - 530 . Numdam | MR 379379 | Zbl 0321.10026 · Zbl 0321.10026 · numdam:ASENS_1974_4_7_4_507_0 · eudml:81946
[14] Emerton M. , On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, prépublication 2003.
[15] Féaux de Lacroix C.T. , Einige Resultate über die topologischen Darstellungen p -adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem p -adischen Körper , in: Schriftenreihe Math. Univ. Münster, 3. Serie, Heft 23 , 1999 , pp. 1 - 111 . MR 1691735 | Zbl 0963.22009 · Zbl 0963.22009
[16] Fontaine J.-M. , Représentations \ell -adiques potentiellement semi-stables , in: Astérisque , vol. 223 , Soc. Math. France , 1994 , pp. 321 - 347 . MR 1293977 | Zbl 0873.14020 · Zbl 0873.14020
[17] Grosse-Klönne E. , Integral structures in automorphic line bundles on the p -adic upper half plane , Math. Annalen 329 ( 2004 ) 463 - 493 . MR 2127986 | Zbl 1087.11029 · Zbl 1087.11029 · doi:10.1007/s00208-004-0512-7
[18] Iovita A. , Spiess M. , Derivatives of p -adic L -functions, Heegner cycles and monodromy modules attached to modular forms , Invent. Math. 154 ( 2003 ) 333 - 384 . MR 2013784 | Zbl 1099.11032 · Zbl 1099.11032 · doi:10.1007/s00222-003-0306-7
[19] Mazur B. , On monodromy invariants occurring in global arithmetic, and Fontaine’s theory , in: Contemp. Math. , vol. 165 , 1994 , pp. 1 - 20 . MR 1279599 | Zbl 0846.11039 · Zbl 0846.11039
[20] Mazur B. , Tate J. , Teitelbaum J. , On p -adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 ( 1986 ) 1 - 48 . MR 830037 | Zbl 0699.14028 · Zbl 0699.14028 · doi:10.1007/BF01388731 · eudml:143332
[21] Morita Y. , A p -adic theory of hyperfunctions I , Publ. R.I.M.S. 17 ( 1981 ) 1 - 24 . MR 613933 | Zbl 0457.12010 · Zbl 0457.12010 · doi:10.2977/prims/1195186702
[22] Morita Y. , Analytic representations of \({\mathrm{SL}}_{2}\) over a \(p\)-adic number field II , in: Prog. Math. , vol. 46 , Birkhäuser , 1984 , pp. 282 - 297 . MR 763019 | Zbl 0549.22008 · Zbl 0549.22008
[23] Saito T. , Modular forms and p -adic Hodge theory , Invent. Math. 129 ( 1997 ) 607 - 620 . MR 1465337 | Zbl 0877.11034 · Zbl 0877.11034 · doi:10.1007/s002220050175
[24] Schneider P. , Nonarchimedean Functional Analysis , Springer-Verlag , 2001 . MR 1869547 | Zbl 0998.46044 · Zbl 0998.46044
[25] Schneider P. , Teitelbaum J. , Locally analytic distributions and p -adic representation theory, with applications to \({\mathrm{GL}}_{2}\) , J. Amer. Math. Soc. 15 ( 2002 ) 443 - 468 . MR 1887640 | Zbl 1028.11071 · Zbl 1028.11071 · doi:10.1090/S0894-0347-01-00377-0
[26] Schneider P. , Teitelbaum J. , \(U\left(g\right)\)-finite locally analytic representations , Representation Theory 5 ( 2001 ) 111 - 128 . MR 1835001 | Zbl 1028.17007 · Zbl 1028.17007 · doi:10.1090/S1088-4165-01-00109-1
[27] Schneider P. , Teitelbaum J. , Banach space representations and Iwasawa theory , Israel J. Math. 127 ( 2002 ) 359 - 380 . MR 1900706 | Zbl 1006.46053 · Zbl 1006.46053 · doi:10.1007/BF02784538
[28] Schneider P. , Teitelbaum J. , p -adic boundary values , in: Astérisque , vol. 278 , Soc. Math. France , 2002 , pp. 51 - 125 . MR 1922824 | Zbl 1051.14024 · Zbl 1051.14024
[29] Schneider P. , Teitelbaum J. , Algebras of p -adic distributions and admissible representations , Invent. Math. 153 ( 2003 ) 145 - 196 . MR 1990669 | Zbl 1028.11070 · Zbl 1028.11070 · doi:10.1007/s00222-002-0284-1
[30] de Shalit E. , Eichler cohomology and periods of modular forms on p -adic Schottky groups , J. Reine Angew. Math. 400 ( 1989 ) 3 - 31 . Article | MR 1013723 | Zbl 0674.14031 · Zbl 0674.14031 · doi:10.1515/crll.1989.400.3 · crelle:GDZPPN002206889 · eudml:153161
[31] Shimura G. , Introduction to the Arithmetic Theory of Automorphic Functions , Princeton University Press, Mathematical Society of Japan. MR 1291394 | Zbl 0221.10029 · Zbl 0221.10029
[32] Teitelbaum J. , Values of p -adic L -functions and a p -adic Poisson kernel , Invent. Math. 101 ( 1990 ) 395 - 410 . MR 1062968 | Zbl 0731.11065 · Zbl 0731.11065 · doi:10.1007/BF01231508 · eudml:143809
[33] Teitelbaum J. , Modular representations of \({\mathrm{PGL}}_{2}\) and automorphic forms for Shimura curves , Invent. Math. 113 ( 1993 ) 561 - 580 . MR 1231837 | Zbl 0806.11027 · Zbl 0806.11027 · doi:10.1007/BF01244318 · eudml:144137
[34] Vignéras M.-F. , Arithmétique des algèbres de quaternions , Lecture Notes in Math. , vol. 800 , Springer , 1980 . MR 580949 | Zbl 0422.12008 · Zbl 0422.12008
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