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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
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##### 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
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