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On the \(l\)-adic representations associated to Hilbert modular forms. (Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert.) (French) Zbl 0616.10025
Let \(\pi =\otimes_{v}\pi_ v\) be a cuspidal automorphic representation of \(\text{GL}_ 2(F_{{\mathbb A}})\) where \(F_{{\mathbb A}}\) is the ring of adeles of a totally real algebraic number field \(F\) of degree \(d\) over \({\mathbb Q}\), of the same type as representations corresponding to Hilbert modular forms of weight \((k_ 1,...,k_ d)\), i.e. whose local components \(\pi_ v\) for each of the \(d\) Archimedean places \(v\) of \(F\) are essentially square integrable representations of \(\text{GL}_ 2({\mathbb R})\) occurring in the induced representation \(\text{Ind}(\mu,\nu)\) (under unitary induction) with characters \(\mu\), \(\nu\) of \({\mathbb R}^*\) given by \(\mu (t):=| t|^{(k-w- 1)/2}(\text{sgn}\, t)^ k\), \(\nu(t):=| t|^{(-k-w+1)/2}\) for integral \(k\geq 2\) and \(w\equiv k\pmod 2\), all depending on \(v\). For \(d\) even, \(\pi_ v\) is taken to be a special or cuspidal representation of \(\text{GL}_ 2(F_ v)\), for at least one non-Archimedean place \(v\) of \(F\). Let \(\bar F\) be an algebraic closure of \(F\).
The main theorem proved is the following: there exists an algebraic number field \(E\) depending on the given \(\pi\) and a strictly compatible system \(\{\sigma^{\lambda}\}\) of continuous 2-dimensional \(E_{\lambda}\)-adic representations of \(\text{Gal}(\bar F/F)\) such that for every non-Archimedean place \(v\) of \(F\) and \(\lambda\) of \(E\) with residue characteristic different from that of \(v\), the restriction \(\sigma_ v^{\lambda}\) of \(\sigma^{\lambda}\) to the local Weil group \(WF_ v\) is equivalent to \(\sigma^{\lambda}(\pi_ v).\)
What is new here is that the author determines \(\sigma_ v^{\lambda}\) for every non-Archimedean place \(v\) of \(F\). Moreover, according to Ribet, \(\sigma^{\lambda}\) turns out to be irreducible and as such, is characterized entirely.
First, for a weaker version of the main theorem, the author gives a (“geometric”) proof and then he uses ”base change for \(\text{GL}(2)\)” to deduce the main theorem. As a corollary of the main theorem for \(d=1\) and weight \(k=2\), the author shows an affirmative answer to a conjecture on Weil curves over \({\mathbb Q}\).
Reviewer: S. Raghavan

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G25 Varieties over finite and local fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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References:
[1] A. BOREL et N. WALLACH , Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (Ann. of Math. Studies, Princeton University Press, 1980 ). MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[2] H. CARAYOL , Sur la mauvaise réduction des courbes de Shimura (C. R. Acad. Sc., Paris, t. 296, série I, 1983 , p. 557). MR 85c:11050 | Zbl 0599.14019 · Zbl 0599.14019
[3] H. CARAYOL , Sur les représentations l-adiques attachées aux formes modulaires de Hilbert (C. R. Acad. Sc., Paris, t. 296, série I, 1983 , p. 629). MR 85e:11039 | Zbl 0537.10018 · Zbl 0537.10018
[4] H. CARAYOL , Sur la mauvaise réduction des courbes de Shimura [Compositio Math. (à paraître)]. Numdam | Zbl 0607.14021 · Zbl 0607.14021 · numdam:CM_1986__59_2_151_0 · eudml:89787
[5] P. DELIGNE , Formes modulaires et représentations de GL2 , Antwerp II (Lecture Notes, vol. n^\circ 349, Springer-Verlag, 1973 , p. 55-106). MR 50 #240 | Zbl 0271.10032 · Zbl 0271.10032
[6] P. DELIGNE , Lettre à Piatetskii-Shapiro , 1973 .
[7] P. DELIGNE et D. MUMFORD , On the Irreducibility of the Space of Curves of a Given Genus (Publ. Math. I.H.E.S., n^\circ 36, 1968 ). Numdam | Zbl 0181.48803 · Zbl 0181.48803 · doi:10.1007/BF02684599 · numdam:PMIHES_1969__36__75_0 · eudml:103899
[8] V. G. DRINFEL’D , Elliptic Modules (Math. U.S.S.R., Sbornik, vol. 23, n^\circ 4, 1974 ). MR 52 #5580 · Zbl 0321.14014
[9] V. G. DRINFEL’D , Coverings of p-Adic Symmetric Regions (Funct. Anal. Appl., vol. 10, 1976 , p. 29). MR 54 #10281 | Zbl 0346.14010 · Zbl 0346.14010 · doi:10.1007/BF01077936
[10] S. GELBART , Automorphic Functions on Adele Groups (Ann. of Math. Studies, n^\circ 83, Princeton Univ. Press, Princeton, 1975 ). MR 52 #280 | Zbl 0329.10018 · Zbl 0329.10018
[11] H. JACQUET et R. P. LANGLANDS , Automorphic Forms on GL (2) (Lecture Notes in Math., vol. n^\circ 114, Springer-Verlag, 1970 ). MR 53 #5481 | Zbl 0236.12010 · Zbl 0236.12010 · doi:10.1007/BFb0058988
[12] H. JACQUET , I. I. PIATETSKII , SHAPIRO et J. SHALIKA , Relèvement cubique non normal (C. R. Acad. Sc., Paris, t. 292, série I, 1981 , p. 567). MR 82i:10035 | Zbl 0475.12017 · Zbl 0475.12017
[13] Ph. KUTZKO , The Local Langlands Conjecture for GL (2) (Ann. of Math., vol. 112, 1980 , p. 381). Zbl 0469.22013 · Zbl 0469.22013 · doi:10.2307/1971151
[14] R. P. LANGLANDS , Modular Forms and l-adic Representations , Antwerp II (Lecture Notes, n^\circ 349, Springer-Verlag, 1973 , p. 361-500). MR 50 #7095 | Zbl 0279.14007 · Zbl 0279.14007 · doi:10.1007/978-3-540-37855-6_6
[15] R. P. LANGLANDS , Base Change for GL (2) (Annals of Math. Studies, Princeton University Press, 1980 ). MR 82a:10032 | Zbl 0444.22007 · Zbl 0444.22007
[16] R. P. LANGLANDS , On the Zeta Functions of Some Simple Shimura Varietes (Canad. J. Math., vol. 31, n^\circ 6, 1979 , p. 1121). Zbl 0444.14016 · Zbl 0444.14016 · doi:10.4153/CJM-1979-102-1
[17] A. P. OGG , Elliptic Curves and Wild Ramification (Amer. J. of Math., vol. 89, 1967 , p. 1-21). MR 34 #7509 | Zbl 0147.39803 · Zbl 0147.39803 · doi:10.2307/2373092
[18] M. OHTA , On the Zeta Function of an Abelian Scheme Over the Shimura Curve (Japan. J. Math., vol. 9, 1983 , p. 1-26). MR 85j:11067 | Zbl 0527.10023 · Zbl 0527.10023
[19] I. I. PIATETSKII-SHAPIRO , Zeta Functions of Modular Curves (Lecture Notes in Math., n^\circ 349, Springer-Verlag, 1973 , p. 317). MR 49 #2744 | Zbl 0308.14004 · Zbl 0308.14004
[20] J. D. ROGAWSKI et J. B. TUNNELL , On Artin L-Functions Associated to Hilbert Modular Forms of Weight One (Inv. Math., vol. 74, 1983 , p. 1-42). MR 85i:11044 | Zbl 0523.12009 · Zbl 0523.12009 · doi:10.1007/BF01388529 · eudml:143060
[21] J. TATE , Number Theoretic Background, in Automorphic Forms, Representations and L-Functions (Proc. Symp. Pure Math., vol. 33, part. 2, American Mathematical Society, Providence, Rhode Island, 1979 ). Zbl 0422.12007 · Zbl 0422.12007
[22] J. B. TUNNELL , On the Local Langlands Conjecture for GL (2) (Inv. Math., vol. 46, 1978 , p. 179-200). MR 57 #16262 | Zbl 0385.12006 · Zbl 0385.12006 · doi:10.1007/BF01393255 · eudml:142559
[23] J. B. TUNNELL , Artin’s Conjecture for Representations of Octahedral Type (Bull. Amer. Math. Soc., vol. 5, n^\circ 2, sept. 1981 , p. 173). Article | MR 82j:12015 | Zbl 0475.12016 · Zbl 0475.12016 · doi:10.1090/S0273-0979-1981-14936-3 · minidml.mathdoc.fr
[24] J. L. WALDSPURGER , Quelques propriétés arithmétiques de certaines formes automorphes sur GL (2) (Compositio Math., vol. 54, n^\circ 2, 1985 , p. 121-171). Numdam | MR 87g:11061a | Zbl 0567.10022 · Zbl 0567.10022 · numdam:CM_1985__54_2_121_0 · eudml:89701
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