×

zbMATH — the first resource for mathematics

Non-tempered forms for \(\mathrm U(3)\) and Bloch-Kato conjectures. (Formes non tempérées pour \(\mathrm U(3)\) et conjectures de Bloch-Kato.) (French) Zbl 1201.11051
From the text: We use \(p\)-adic families of automorphic forms for a unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch-Kato conjectures.
Let \(E\) be an imaginary quadratic field, and let \(\chi\) be an algebraic Hecke character over \(E\), such that \(\chi^\perp=\chi(-1)\) and the type at infinity has the form \(z\mapsto z^a\bar z^{1-a}\) with \(a\geq 2\). Let \(p\) be a prime which splits in \(E\) and is unramified for \(\chi\), and let \(\chi_p: \text{Gal}(\overline{E}/E)\to L^*\) be the \(p\)-adic realization of \(\chi\) over a field \(L\). Then, if \(\varepsilon(0,\chi)=\varepsilon(\chi_p,0)=-1\), \[ \dim H^1_f(\text{Gal}(\overline{E}/E),\chi_p)\geq 1, \] where \(H\) denotes the Bloch-Kato Selmer group. It other words, there exists a non-trivial extension having good reduction everywhere of the form \[ 0\rightarrow \chi_p\rightarrow U\rightarrow 1\rightarrow 0. \]
This theorem had been proven previously by K. Rubin [Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)]. The authors’ method is different involving the Fontaine-Mazur conjecture and is inspired by the method of Skinner-Urban to create congruences by using \(p\)-adic families of modular forms.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F80 Galois representations
PDF BibTeX XML Cite
Full Text: DOI Numdam arXiv EuDML
References:
[1] Arthur J. , Clozel L. , Simple Algebras, Base Change and the Advanced Theory of the Trace Formula , Ann. of Math. Stud. , vol. 120 , Princeton University Press , 1989 . MR 1007299 | Zbl 0682.10022 · Zbl 0682.10022
[2] Artin E. , Tate J. , Class Field Theory , Benjamin , 1968 . MR 223335 | Zbl 0176.33504 · Zbl 0176.33504
[3] Bellaïche J. , Congruences endoscopiques et représentations galoisiennes, Thèse de l’université Paris 11, janvier 2002.
[4] Bellaïche J. , À propos d’un lemme de Ribet , Rend. Sem. Univ. Padova 109 ( 2003 ) 47 - 62 . Numdam | MR 1997986 | Zbl 1048.20032 · Zbl 1048.20032
[5] Bellaïche J., Graftieaux P. , Représentations sur un anneau de valuation discrète complet, Math. Ann. , à paraître. MR 2207872 | Zbl 05013672 · Zbl 1178.20040
[6] Bellaïche J., Graftieaux P. , Augmentation du niveau pour \(U\left(3\right)\), Preprint de l’université de Nice. MR 2214894
[7] Bernstein I.N. , Zelevinsky A.V. , Induced representations of reductive p -adic groups I , Ann. Sci. École Norm. Sup. (4) 10 ( 1977 ) 441 - 472 . Numdam | MR 579172 | Zbl 0412.22015 · Zbl 0412.22015
[8] Blasco L. , Description du dual admissible de \(U\left(2\text{,}1\right)\left(F\right)\) par la théorie des types de C. Bushnell et P. Kutzko , Manuscripta Math. 107 ( 2 ) ( 2002 ) 151 - 186 . MR 1894738 | Zbl 1108.22011 · Zbl 1108.22011
[9] Blasius D. , Rogawski J. , Tate class and arithmetic quotient of two-ball , in: Langlands R. , Ramakhrisnan D. (Eds.), The Zeta Functions of Picard Modular Surfaces , Publications C.R.M. , Montréal , 1992 , pp. 421 - 443 . MR 1155236 | Zbl 0828.14012 · Zbl 0828.14012
[10] Blasius D., Rogawski J. , Zeta functions of Shimura varieties, in: Motives , in: Proc. Sympos. Pure Math., vol. 55 . MR 1265563 | Zbl 0827.11033 · Zbl 0827.11033
[11] Blasius D. , Rogawski J. , Motives for Hilbert modular forms , Invent. Math. 114 ( 1993 ) 55 - 87 . MR 1235020 | Zbl 0829.11028 · Zbl 0829.11028
[12] Borel A. , Some finiteness properties of adele groups over number fields , IHÉS Publ. Math. 16 ( 1963 ) 5 - 30 . Numdam | MR 202718 | Zbl 0135.08902 · Zbl 0135.08902
[13] Borel A. , Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup , Invent. Math. 35 ( 1976 ) 233 - 259 . MR 444849 | Zbl 0334.22012 · Zbl 0334.22012
[14] Borel A. , Casselman W. , Automorphic forms, representations, and L -functions , in: Proc. Sympos. Pure Math. , vol. 33 , 1977 , Corvallis. Zbl 0412.10017 · Zbl 0412.10017
[15] Bosch S., Güntzer U., Remmert R. , Non Archimedian Analysis, in: Grundlehren der mathematischen Wissenschaften , vol. 261 , Springer-Verlag. Zbl 0539.14017 · Zbl 0539.14017
[16] Bushnell C. , Smooth representations of p -adic group, ICM 1998, pp. 770-779. MR 1403977 | Zbl 0856.22021 · Zbl 0856.22021
[17] Bushnell C. , Kutzko P. , Smooth representations of reductive p -adic groups: structure theory via types , Proc. London Math. Soc. (3) 77 ( 1997 ) 582 - 634 . MR 1643417 | Zbl 0911.22014 · Zbl 0911.22014
[18] Bushnell C. , Kutzko P. , Semi-simple types , Compositio Math. 119 ( 1999 ) 53 - 117 . Zbl 0933.22027 · Zbl 0933.22027
[19] Buzzard K. , Eigenvarieties, 2002, en préparation.
[20] Cartier P. , Representations of p -adic groups: a survey , in: Proc. Sympos. Pure Math. , vol. 33 , 1977 , pp. 111 - 155 , Part I. MR 546593 | Zbl 0421.22010 · Zbl 0421.22010
[21] Casselman W. , The unramified principal series of p -adic groups. I. The spherical function , Compositio Math. 40 ( 3 ) ( 1980 ) 387 - 406 . Numdam | MR 571057 | Zbl 0472.22004 · Zbl 0472.22004
[22] Chenevier G. , Familles p -adiques de formes automorphes pour \(GL\left(n\right)\) , J. Reine Angew. Math. 570 ( 2004 ) 143 - 217 . MR 2075765 | Zbl 1093.11036 · Zbl 1093.11036
[23] Chenevier G. , Familles p -adiques de formes automorphes et applications aux conjectures de Bloch-Kato, Thèse de l’université Paris 7, 2003.
[24] Choucroun F. , Analyse harmonique des groupes d’automorphismes d’arbres de Bruhat-Tits , Mém. Soc. Math. France (N.S.) ( 58 ) ( 1994 ) 170 . Numdam | MR 1294542 | Zbl 0840.43019 · Zbl 0840.43019
[25] Clozel L. , Représentations galoisiennes associées aux représentations automorphes autoduales de \(GL\left(n\right)\) , IHÉS Publ. Math. 73 ( 1991 ) 97 - 145 . Numdam | MR 1114211 | Zbl 0739.11020 · Zbl 0739.11020
[26] Clozel L. , Labesse J.-P. , Changement de base pour les représentations cohomologiques de certains groupes unitaires , in: Astérisque , vol. 257 , SMF , 1998 , Appendice A.
[27] Coleman R. , P -adic Banach spaces & families of modular forms , Invent. Math. 127 ( 1997 ) 417 - 479 . MR 1431135 | Zbl 0918.11026 · Zbl 0918.11026
[28] Curtis C. , Reiner I. , Representation Theory of Finite Groups and Associative Algebras , Wiley , 1962 . MR 1013113 | Zbl 0131.25601 · Zbl 0131.25601
[29] Faltings G. , Cristalline cohomology and p -adic Galois representations , in: Algebraic Analysis and Number Theory, JAMI Conference , 1988 , pp. 25 - 90 . MR 1463696 | Zbl 0805.14008 · Zbl 0805.14008
[30] Fontaine J.-M. , Le corps des périodes p -adiques , in: Périodes p -adiques , Astérisque , vol. 223 , Société mathématique de France , 1994 , pp. 59 - 111 , exposé 2. MR 1293971 | Zbl 0940.14012 · Zbl 0940.14012
[31] Fontaine J.-M. , Représentations p -adiques semi-stables , in: Périodes p -adiques , Astérisque , vol. 223 , Société mathématique de France , 1994 , pp. 113 - 184 , exposé 3. MR 1293972 | Zbl 0865.14009 · Zbl 0865.14009
[32] Fontaine J.-M. , Perrin-Riou B. , Autour des conjectures de Bloch-Kato: cohomologie galoisienne et valeurs de fonctions L , in: Motives , Proc. Sympos. Pure Math. , vol. 55 , 1994 , pp. 599 - 706 , part 1. MR 1265546 | Zbl 0821.14013 · Zbl 0821.14013
[33] Gordon B.B. , Canonical models of Picard modular surfaces , in: Langlands R. , Ramakhrisnan D. (Eds.), The Zeta Functions of Picard Modular Surfaces , Publications C.R.M. , Montréal , 1992 , pp. 1 - 27 . MR 1155224 | Zbl 0756.14011 · Zbl 0756.14011
[34] Harris M. , On the local Langlands correspondence , in: Proceedings ICM 2002, vol. 2 , 2002 , pp. 583 - 597 . MR 1957067 | Zbl 01789998 · Zbl 1151.11351
[35] Harris M. , Taylor R. , The Geometry and Cohomology of Some Simple Shimura Varieties , Ann. Math. Stud. , vol. 151 , 2001 . MR 1876802 | Zbl 1036.11027 · Zbl 1036.11027
[36] Humphreys J.E. , Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics , vol. 29 . Zbl 0768.20016 · Zbl 0768.20016
[37] Katz N. , Messing W. , Some consequences of the Riemann Hypothesis for varieties over finite fields , Invent. Math. 23 ( 1974 ) 73 - 77 . MR 332791 | Zbl 0275.14011 · Zbl 0275.14011
[38] Keys D. , Principal series representations of special unitary groups over local fields , Compositio Math. 51 ( 1 ) ( 1984 ) 115 - 130 . Numdam | MR 734788 | Zbl 0547.22009 · Zbl 0547.22009
[39] Kisin M. , Overconvergent modular forms and the Fontaine-Mazur conjecture , Invent. Math. 153 ( 2003 ) 363 - 454 . MR 1992017 | Zbl 1045.11029 · Zbl 1045.11029
[40] Kottwitz R. , Points on some Shimura varieties over finite fields , J. Amer. Math. Soc. 5 ( 2 ) ( 1992 ). MR 1124982 | Zbl 0796.14014 · Zbl 0796.14014
[41] Labesse J.-P. , Cohomologie, stabilisation et changement de base , Astérisque , vol. 257 , SMF , 1998 . MR 1695940 | Zbl 1024.11034 · Zbl 1024.11034
[42] Langlands R. , Ramakhrisnan D. (Eds.), The Zeta Functions of Picard Modular Surfaces , Publications C.R.M. , Montréal , 1992 . MR 1155223 | Zbl 0752.00024 · Zbl 0752.00024
[43] Matsumura H. , Commutative ring theory , Cambridge Studies in Adv. Math. , vol. 8 , 1980 . Zbl 0603.13001 · Zbl 0603.13001
[44] Mazur B. , The theme of p -adic variation , in: Arnold V. , Atiyah M. , Lax P. , Mazur B. (Eds.), Math.: Frontiers and Perspectives , AMS , 2000 . MR 1754790 | Zbl 0959.14008 · Zbl 0959.14008
[45] Motives , Proc. Sympos. Pure Math., vol. 55 .
[46] Périodes p -adiques , Astérisque , vol. 223 , Société mathématique de France , 1994 . MR 1293969 | Zbl 0802.00019 · Zbl 0802.00019
[47] Perrin-Riou B. , Représentations p -adiques ordinaires , in: Périodes p -adiques , Astérisque , vol. 223 , Société mathématique de France , 1994 , pp. 209 - 220 , exposé 4. MR 1293973 | Zbl 1043.11532 · Zbl 1043.11532
[48] Ribet K. , A modular construction of unramified extensions of \(Q\left({\zeta }_{p}\right)\) , Invent. Math. 34 ( 3 ) ( 1976 ) 151 - 162 . MR 419403 | Zbl 0338.12003 · Zbl 0338.12003
[49] Rogawski J. , Analytic expression for the number of points mod p , in: Langlands R. , Ramakhrisnan D. (Eds.), The Zeta Functions of Picard Modular Surfaces , Publications C.R.M. , Montréal , 1992 , pp. 65 - 109 . MR 1155227 | Zbl 0821.14015 · Zbl 0821.14015
[50] Rogawski J. , The multiplicity formula for A-packets , in: Langlands R. , Ramakhrisnan D. (Eds.), The Zeta Functions of Picard Modular Surfaces , Publications C.R.M. , Montréal , 1992 , pp. 395 - 419 . MR 1155235 | Zbl 0823.11027 · Zbl 0823.11027
[51] Rogawski J. , On modules over the Hecke algebra of a p -adic group , Invent. Math. 79 ( 1985 ) 443 - 465 . MR 782228 | Zbl 0579.20037 · Zbl 0579.20037
[52] Rogawski J. , Automorphic Representations of Unitary Groups in Three Variables , Ann. of Math. Stud. , vol. 123 , Princeton University Press , 1990 . MR 1081540 | Zbl 0724.11031 · Zbl 0724.11031
[53] Rouquier R. , Caractérisations des caractères et pseudo-caractères , J. Algebra 180 ( 1996 ) 571 - 586 . MR 1378546 | Zbl 0857.16013 · Zbl 0857.16013
[54] Rubin K. , The “main conjectures” of Iwasawa theory for imaginary quadratic fields , Invent. Math. 103 ( 1 ) ( 1991 ) 25 - 68 . Zbl 0737.11030 · Zbl 0737.11030
[55] Rubin K. , Euler systems , Ann. of Math. Stud. 147 ( 2000 ). MR 1749177 | Zbl 0977.11001 · Zbl 0977.11001
[56] Serre J.-P. , Endomorphismes complètement continus des espaces de Banach p -adiques , IHÉS Publ. Math. 12 ( 1962 ) 69 - 85 . Numdam | MR 144186 | Zbl 0104.33601 · Zbl 0104.33601
[57] Sen S. , Continuous cohomology and p -adic Galois representations , Invent. Math. 62 ( 1980 ) 89 - 116 . MR 595584 | Zbl 0463.12005 · Zbl 0463.12005
[58] Sen S. , An infinite dimensional Hodge-Tate theory , Bull. Soc. Math. France 121 ( 1993 ) 13 - 34 . Numdam | MR 1207243 | Zbl 0786.11067 · Zbl 0786.11067
[59] Grothendieck A., Artin M., Verdier J.-L. , Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique IV, exposé XVI.
[60] Skinner C. , Urban E. , Sur les déformations p -adiques des formes de Saito-Kurokawa , C. R. Acad. Sci. Paris Sér. I 335 ( 2002 ) 581 - 586 . MR 1941298 | Zbl 1024.11030 · Zbl 1024.11030
[61] Tate J. , Number theoretic background , in: Automorphic Forms, Representations and L -Functions, Part 2 , Proc. Sympos. Pure Math. , vol. 33 , 1977 , pp. 3 - 26 . MR 546607 | Zbl 0422.12007 · Zbl 0422.12007
[62] Taylor R. , Galois representations attached to Siegel modular forms of low weight , Duke Math. J. 63 ( 1991 ) 281 - 332 . Article | MR 1115109 | Zbl 0810.11033 · Zbl 0810.11033
[63] Tits J. , Reductive groups over local fields, Part I , in: Proc. Sympos. Pure Math. , vol. 33 , 1977 , pp. 29 - 69 . MR 546588 | Zbl 0415.20035 · Zbl 0415.20035
[64] Zelevinsky A.V. , Induced representations of reductive p -adic groups II. On irreducible representations of \(GL\left(n\right)\) , Ann. Sci. École Norm. Sup. (4) 13 ( 1980 ) 165 - 210 . Numdam | MR 584084 | Zbl 0441.22014 · Zbl 0441.22014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.