Motives for Hilbert modular forms. (English) Zbl 0829.11028

In this paper the authors attach motives to collections \(\pi_1, \ldots, \pi_n\) of holomorphic Hilbert modular forms over a totally real field \(F\) of discrete series type at infinity. The method they used is based on the construction of certain sub-motives of the cohomology of fiber systems over Picard modular surfaces. Let \(U(3)\) be a unitary group in three variables defined relative to a quadratic extension \(E = KF\) that determines a Picard modular surface, where \(K\) is an imaginary quadratic extension of \(\mathbb{Q}\). Then the sub-motives of interest are attached to endoscopic automorphic forms on \(U(3)\) by passing from \(\pi_1, \ldots, \pi_n\) to Picard modular surfaces through a base change to \(GL(2)_E\), a twisting and a descent to \(U(2)_{E/F}\), an extension to \(U(2) \times U(1)\), an endoscopic transfer to quasi-split \(U(3)\), and finally a transfer to an inner form of \(U(3)\). Producing motives themselves involves the construction of algebraic correspondences, and this is achieved by verifying the Hodge conjecture for the generic fibers of the fiber systems of abelian varieties carried by the Picard modular surfaces.
As an application the authors obtain compatible systems of \(\ell\)-adic representations of \(\text{Gal} (\overline F/F)\) of degree \(2^n\) whose \(L\)-function coincides with the tensor product \(L\)-functions \(L(s, \pi_1 \otimes \cdot \otimes \pi_n)\) at almost all places.


11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
Full Text: DOI EuDML


[1] Arthur, J.: Unipotent automorphic representations: conjectures. In: Orbites unipotentes et répresentations, II. (Asterisque, vol. 171-172, pp. 13-71) Paris: Soc. Math. Fr. 1989
[2] Blasius, D., Rogawski, J.: Galois representations for Hilbert modular forms. Bull. Am. Math. Soc.21 (no. 1), 65-69 (1989) · Zbl 0684.12013 · doi:10.1090/S0273-0979-1989-15763-7
[3] Clozel, L.: Motifs et formes automorphes: applications du principe de fonctorialité. In: Clozel, L., Milne, J. (eds.) Automorphic forms, Shimura varieties, and L-functions, vols. I and II, pp. 77-159. New York London: Academic Press 1990
[4] Deligne, P.: Travaux de Shimura. Sémin. Bourbaki 1970/71, exposé 389 (Lect. Notes Math., vol. 244) Berlin Heidelberg New York: Springer 1971
[5] Deligne, P.: Formes modulaires et représentations ?-adique. Sémin. Bourbaki 1969, exposé 55. (Lect. Notes Math., vol. 179. pp. 139-172) Berlin Heidelberg New York: Springer 1969
[6] Fontaine, J.-M.: Sur certains types de représentationsp-adiques du groupe de Galois d’un corps local, construction d’un anneau de Barsotti-Tate. Ann. Math.115, 529-577 (1982) · Zbl 0544.14016 · doi:10.2307/2007012
[7] Faltings, G.: Crystalline cohomology andp-adic Galois representations. In: Igusa, J.I., (ed). Algebraic analysis, geometry, and number theory. pp. 25-80. Baltimore: Johns Hopkins University Press 1989 · Zbl 0805.14008
[8] Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. Am. J. Math.103 (no. 4), 727-776 (1981) · Zbl 0477.14024 · doi:10.2307/2374049
[9] Katz, N., Messing, W.: Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math.23, 73-77 (1974) · Zbl 0275.14011 · doi:10.1007/BF01405203
[10] Langlands, R.P.: On the zeta functions of some simple Shimura varieties. J. Can. Math.31, 1121-1216 (1979) · Zbl 0444.14016 · doi:10.4153/CJM-1979-102-1
[11] Langlands, R., Ramakrishnan, D. (eds.) Zeta-functions of Picard modular surfaces. (eds.) Publ. CRM. Université de Montréal 1992 · Zbl 0752.00024
[12] Ribet, K.: Hodge classes on certain types of abelian varieties. Am. J. Math.105, 523-538 (1983) · Zbl 0586.14003 · doi:10.2307/2374267
[13] Rogawski, J.D.: Automorphic representations of unitary groups in three variables, (Ann. Math. Stud.) Princeton: Princeton University Press 1990 · Zbl 0724.11031
[14] Rogawski, J.D.: Analytic expression for the number of points modp. In: Langlands, R., Ramakrishnan, D.: Zeta functions of Picard modular surfaces, pp. 65-109. Publ. CRM. Université de Montreal 1992
[15] Scholl, A.: Motives for modular forms. Invent. Math.100, 419-430 (1990) · Zbl 0760.14002 · doi:10.1007/BF01231194
[16] Shimura, G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math.78 (no. 1), 149-192 (1963) · Zbl 0142.05402 · doi:10.2307/1970507
[17] Shimura, G.: Moduli and fibre systems of abelian varieties. Ann. Math.83 (no. 2), 294-338 (1966) · Zbl 0141.37503 · doi:10.2307/1970434
[18] Shimura, G.: The special values of the zeta-functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5
[19] Tate, J.: Number theoretic background. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations, and L-functions, II. (Proc. Symp. Pure Math., vol. XXX, pp. 3-26) Providence, RI: Am. Math. Soc. 1979
[20] Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math.98, 265-280 (1989) · Zbl 0705.11031 · doi:10.1007/BF01388853
[21] Wiles, A.: On ordinary ?-adic representations associated to modular forms. Invent. Math.94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275
[22] Zucker, S.: Hodge theory with degenerating coefficients;L 2-cohomology in the Poincaré metric. Ann. Math.109, 415-476 (1979) · Zbl 0446.14002 · doi:10.2307/1971221
[23] Zucker, S.: Locally homogeneous variations of Hodge structure. Enseign. Math.27, 243-276 (1981) · Zbl 0584.14003
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.