# zbMATH — the first resource for mathematics

On the zeta functions of Shimura varieties and periods of Hilbert modular forms. (English) Zbl 0823.11018
Let $$F$$ be a totally real algebraic number field of degree $$n$$ and let $$B$$ be a quaternion algebra over $$F$$ such that $$B \otimes_ \mathbb{Q} \mathbb{R}\cong M_ 2 (\mathbb{R})^ r \times \mathbb{H}^{n -r}$$ with $$r>0$$, where $$\mathbb{H}$$ is the Hamilton quaternion algebra. Let $$H= \text{Gal} (\overline {\mathbb{Q}}/ F)$$, and let $$\Omega$$ be the subset of $$\text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})/ H$$ consisting of the archimedean places of $$F$$ at which $$B$$ splits. Let $H'= \{g\in \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\mid g \Omega= \Omega\},$ and let $$F'$$ be the fixed field of $$H'$$. Let $$G= \text{Res}_{F\wedge \mathbb{Q}} (B')$$, and let $$K$$ be a compact open subgroup of $$G (\mathbb{A}_ f)$$. Then there exists a Shimura variety $$S_ K$$ defined over $$F'$$ whose complex points $$S_ K (\mathbb{C})$$ form an analytic space isomorphic to a space of the form $$G (\mathbb{Q}) \setminus G (\mathbb{A})/ KK_ \infty$$. R. P. Langlands [Can. J. Math. 31, 1121- 1216 (1979; Zbl 0444.14016)] determined the Hasse-Weil zeta function $$Z(s, S_ K/ F')$$ of $$S_ K$$ over $$F'$$ whose main part is of the form $$\prod_ \pi L(s, \pi, r_ 1)^{m(\pi, K)}$$, where $$\pi$$ is an automorphic representation and $$r_ 1$$ is a $$2^ r [F': \mathbb{Q}]$$- dimensional representation of the $$L$$-group $$^ L G= GL (2, \mathbb{C})\rtimes \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})$$ of $$G$$.
In this paper, the author shows that $$L(s, \pi, r_ 1)$$ is equal to $$L(s, \bigotimes_ \Omega \text{ Ind}^{H'}_ H \sigma_ \lambda)$$, where $$\sigma_ \lambda$$ is the $$\lambda$$-adic representation associated to $$\pi$$. He also shows that the transcendental part of the critical values of $$L(s, \pi, r_ 1)$$ can be expressed as a product of a power of $$\pi$$ and the $$Q$$-invariants defined by G. Shimura [Invent. Math. 94, 245-305 (1988; Zbl 0656.10018)] if $$r\geq 2$$.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G35 Modular and Shimura varieties 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Full Text:
##### References:
  J.-F. Boutot, L. Breen, P. Gérardin, J. Giraud, J.-P. Labesse, J. S. Milne, and C. Soulé, Variétés de Shimura et fonctions $$L$$ , Publications Mathématiques de l’Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 6, Université de Paris VII U.E.R. de Mathématiques, Paris, 1979. · Zbl 0517.00003  J.-L. Brylinski and J.-P. Labesse, Cohomologie d’intersection et fonctions $$L$$ de certaines variétés de Shimura , Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 361-412. · Zbl 0553.12005  D. Blasius, On the critical values of Hecke $$L$$-series , Ann. of Math. (2) 124 (1986), no. 1, 23-63. JSTOR: · Zbl 0608.10029  D. Blasius, Appendix to Orloff: Critical values of certain tensor product $$L$$-functions , Invent. Math. 90 (1987), 181-188. · Zbl 0625.10022  D. Blasius, Period relations and critical values of $$L$$-functions , preprint, 1987. · Zbl 0625.10022  A. Borel, Automorphic $$L$$-functions , Automorphic forms, representations and $$L$$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27-61. · Zbl 0412.10017  I. N. Bernšteĭ n and A. V. Zelevinskiĭ, Representations of the group $$GL(n,F),$$ where $$F$$ is a local non-Archimedean field , Uspehi Mat. Nauk 31 (1976), no. 3(189), 5-70. · Zbl 0342.43017  H. Carayol, Sur les représentations $$l$$-adiques associées aux formes modulaires de Hilbert , Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409-468. · Zbl 0616.10025  W. Casselman, The Hasse-Weil $$\zeta$$-function of some moduli varieties of dimension greater than one , Automorphic forms, representations and $$L$$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 141-163. · Zbl 0415.14011  P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales , Inst. Hautes Études Sci. Publ. Math. (1968), no. 35, 259-278. · Zbl 0159.22501  P. Deligne, Les constantes des équations fonctionnelles des fonctions $$L$$ , Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 501-597. Lecture Notes in Math., Vol. 349. · Zbl 0271.14011  P. Deligne, Valeurs de fonctions $$L$$ et périodes d’intégrales , Automorphic forms, representations and $$L$$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313-346. · Zbl 0449.10022  S. Gelbart and H. Jacquet, A relation between automorphic representations of $$\mathrm GL(2)$$ and $$\mathrm GL(3)$$ , Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471-542. · Zbl 0406.10022  M. Harris, Period invariants of Hilbert modular forms. I. Trilinear differential operators and $$L$$-functions , Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Lecture Notes in Math., vol. 1447, Springer, Berlin, 1990, pp. 155-202. · Zbl 0716.11020  M. Harris, Period invariants of Hilbert modular forms II , · Zbl 0871.11037  M. Harris, $$L$$-functions of $$2\times 2$$ unitary groups and factorization of periods of Hilbert modular forms , preprint, 1991. JSTOR: · Zbl 0779.11023  M. Harris, $$L$$-functions and periods of polarized regular motives , preprint, 1991.  G. Henniart, Représentations $$l$$-adiques abéliennes , Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981) ed. M. J. Bertin, Progr. Math., vol. 22, Birkhäuser Boston, Boston, MA, 1982, pp. 107-126. · Zbl 0498.12012  H. Hida, On the critical values of $$L$$-functions of $$\mathrm GL(2)$$ and $$\mathrm GL(2)\times\mathrm GL(2)$$ , Duke Math. J. 74 (1994), no. 2, 431-529. · Zbl 0838.11036  G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen , J. Reine Angew. Math. 366 (1986), 53-120. · Zbl 0575.14004  H. Jacquet and R. P. Langlands, Automorphic forms on $$\mathrm GL(2)$$ , Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970. · Zbl 0236.12010  R. E. Kottwitz, Shimura varieties and $$\lambda$$-adic representations , Automorphic forms, Shimura varieties, and $$L$$-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 161-209. · Zbl 0743.14019  R. P. Langlands, Modular forms and $$\ell$$-adic representations , Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 361-500. Lecture Notes in Math., Vol. 349. · Zbl 0279.14007  R. P. Langlands, On the zeta functions of some simple Shimura varieties , Canad. J. Math. 31 (1979), no. 6, 1121-1216. · Zbl 0444.14016  R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen , Automorphic forms, representations and $$L$$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205-246. · Zbl 0447.12009  R. P. Langlands, Base change for $$\mathrm GL(2)$$ , Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton, N.J., 1980. · Zbl 0444.22007  M. Ohta, On the zeta function of an abelian scheme over the Shimura curve , Japan. J. Math. (N.S.) 9 (1983), no. 1, 1-25. · Zbl 0527.10023  A. A. Panchishkin, Motives over totally real fields and $$p$$-adic $$L$$-functions , preprint, 1991, (to appear in Ann. Inst. Fourier Grenoble).  D. E. Rohrlich, Nonvanishing of $$L$$-functions for $$\mathrm GL(2)$$ , Invent. Math. 97 (1989), no. 2, 381-403. · Zbl 0677.10020  J.-P. Serre, Représentations linéaires des groupes finis , Hermann, Paris, 1967. · Zbl 0189.02603  J.-P. Serre, Abelian $$l$$-adic representations and elliptic curves , McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0186.25701  G. Shimura, On the Fourier coefficients of modular forms of several variables , Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1975), no. 17, 261-268. · Zbl 0332.32024  G. Shimura, The special values of the zeta functions associated with Hilbert modular forms , Duke Math. J. 45 (1978), no. 3, 637-679. · Zbl 0394.10015  G. Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups , Ann. of Math. (2) 111 (1980), no. 2, 313-375. JSTOR: · Zbl 0438.12003  1 G. Shimura, On certain zeta functions attached to two Hilbert modular forms. I. The case of Hecke characters , Ann. of Math. (2) 114 (1981), no. 1, 127-164. JSTOR: · Zbl 0468.10016  2 G. Shimura, On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra , Ann. of Math. (2) 114 (1981), no. 3, 569-607. · Zbl 0486.10021  G. Shimura, Algebraic relations between critical values of zeta functions and inner products , Amer. J. Math. 105 (1983), no. 1, 253-285. JSTOR: · Zbl 0518.10032  G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms , Invent. Math. 94 (1988), no. 2, 245-305. · Zbl 0656.10018  G. Shimura, On the fundamental periods of automorphic forms of arithmetic type , Invent. Math. 102 (1990), no. 2, 399-428. · Zbl 0712.11028  G. Shimura, The critical values of certain Dirichlet series attached to Hilbert modular forms , Duke Math. J. 63 (1991), no. 3, 557-613. · Zbl 0752.11021  R. Taylor, On Galois representations associated to Hilbert modular forms , Invent. Math. 98 (1989), no. 2, 265-280. · Zbl 0705.11031  A. Weil, The field of definition of a variety , Amer. J. Math. 78 (1956), 509-524. JSTOR: · Zbl 0072.16001  H. Yoshida, Abelian varieties with complex multiplication and representations of the Weil groups , Ann. of Math. (2) 114 (1981), no. 1, 87-102. JSTOR: · Zbl 0473.14018
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.