On the zeta functions of Shimura varieties and periods of Hilbert modular forms

previous :: next

On the zeta functions of Shimura varieties and periods of Hilbert modular forms

Hiroyuki Yoshida

Source: Duke Math. J. Volume 75, Number 1 (1994), 121-191.

First Page: Show

Primary Subjects: 11F41

Secondary Subjects: 11F67, 11G18, 11G40, 11R39

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077287413
Mathematical Reviews number (MathSciNet): MR1284818
Zentralblatt MATH identifier: 0823.11018
Digital Object Identifier: doi:10.1215/S0012-7094-94-07505-4

back to Table of Contents

References

[BreL] 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.

Mathematical Reviews (MathSciNet): MR84m:14025

Zentralblatt MATH: 0517.00003

[BL] 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.

Mathematical Reviews (MathSciNet): MR86i:11026

Zentralblatt MATH: 0553.12005

[Bl1] D. Blasius, On the critical values of Hecke $L$-series, Ann. of Math. (2) 124 (1986), no. 1, 23–63.

Mathematical Reviews (MathSciNet): MR88i:11035

Zentralblatt MATH: 0608.10029

Digital Object Identifier: doi:10.2307/1971386

JSTOR: links.jstor.org

[Bl2] D. Blasius, Appendix to Orloff: Critical values of certain tensor product $L$-functions, Invent. Math. 90 (1987), 181–188.

Zentralblatt MATH: 0625.10022

Mathematical Reviews (MathSciNet): MR1554038

Digital Object Identifier: doi:10.1007/BF01389037

[Bl3] D. Blasius, Period relations and critical values of $L$-functions, preprint, 1987.

Mathematical Reviews (MathSciNet): MR1610835

Digital Object Identifier: doi:10.2140/pjm.1997.181.53

[Bo] 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.

Mathematical Reviews (MathSciNet): MR81m:10056

Zentralblatt MATH: 0412.10017

[BZ] 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.

Mathematical Reviews (MathSciNet): MR54:12988

[C] 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.

Mathematical Reviews (MathSciNet): MR89c:11083

Zentralblatt MATH: 0616.10025

[Ca] 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.

Mathematical Reviews (MathSciNet): MR82e:10046

Zentralblatt MATH: 0415.14011

[D1] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. (1968), no. 35, 259–278.

Mathematical Reviews (MathSciNet): MR39:5582

Zentralblatt MATH: 0159.22501

[D2] 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.

Mathematical Reviews (MathSciNet): MR50:2128

Zentralblatt MATH: 0271.14011

[D3] 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.

Mathematical Reviews (MathSciNet): MR81d:12009

Zentralblatt MATH: 0449.10022

[GJ] S. Gelbart and H. Jacquet, A relation between automorphic representations of $\rm GL(2)$ and $\rm GL(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542.

Mathematical Reviews (MathSciNet): MR81e:10025

Zentralblatt MATH: 0406.10022

[Ha1] 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.

Mathematical Reviews (MathSciNet): MR91j:11031

Zentralblatt MATH: 0716.11020

Digital Object Identifier: doi:10.1007/BFb0085729

[Ha2] M. Harris, Period invariants of Hilbert modular forms II, preprint.

Mathematical Reviews (MathSciNet): MR1302316

Zentralblatt MATH: 0871.11037

[Ha3] M. Harris, $L$-functions of $2\times 2$ unitary groups and factorization of periods of Hilbert modular forms, preprint, 1991.

Mathematical Reviews (MathSciNet): MR1186960

Zentralblatt MATH: 0779.11023

Digital Object Identifier: doi:10.2307/2152780

JSTOR: links.jstor.org

[Ha4] M. Harris, $L$-functions and periods of polarized regular motives, preprint, 1991.

Mathematical Reviews (MathSciNet): MR1431843

[He] 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.

Mathematical Reviews (MathSciNet): MR85d:11070

Zentralblatt MATH: 0498.12012

[Hi] H. Hida, On the critical values of $L$-functions of $\rm GL(2)$ and $\rm GL(2)\times\rm GL(2)$, Duke Math. J. 74 (1994), no. 2, 431–529.

Mathematical Reviews (MathSciNet): MR98f:11043

Zentralblatt MATH: 0838.11036

Digital Object Identifier: doi:10.1215/S0012-7094-94-07417-6

Project Euclid: euclid.dmj/1077288205

[HLR] G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120.

Mathematical Reviews (MathSciNet): MR87k:11066

Zentralblatt MATH: 0575.14004

[JL] H. Jacquet and R. P. Langlands, Automorphic forms on $\rm GL(2)$, Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970.

Mathematical Reviews (MathSciNet): MR53:5481

Zentralblatt MATH: 0236.12010

[K] 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.

Mathematical Reviews (MathSciNet): MR92b:11038

Zentralblatt MATH: 0743.14019

[L1] 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.

Mathematical Reviews (MathSciNet): MR50:7095

Zentralblatt MATH: 0279.14007

[L2] R. P. Langlands, On the zeta functions of some simple Shimura varieties, Canad. J. Math. 31 (1979), no. 6, 1121–1216.

Mathematical Reviews (MathSciNet): MR82h:10042

Zentralblatt MATH: 0444.14016

[L3] 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.

Mathematical Reviews (MathSciNet): MR83f:12010

Zentralblatt MATH: 0447.12009

[L4] R. P. Langlands, Base change for $\rm GL(2)$, Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton, N.J., 1980.

Mathematical Reviews (MathSciNet): MR82a:10032

Zentralblatt MATH: 0444.22007

[O] 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.

Mathematical Reviews (MathSciNet): MR85j:11067

Zentralblatt MATH: 0527.10023

[P] A. A. Panchishkin, Motives over totally real fields and $p$-adic $L$-functions, preprint, 1991, (to appear in Ann. Inst. Fourier Grenoble).

Mathematical Reviews (MathSciNet): MR1306547

[R] D. E. Rohrlich, Nonvanishing of $L$-functions for $\rm GL(2)$, Invent. Math. 97 (1989), no. 2, 381–403.

Mathematical Reviews (MathSciNet): MR90g:11062

Zentralblatt MATH: 0677.10020

Digital Object Identifier: doi:10.1007/BF01389047

[Se1] J.-P. Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1967.

Mathematical Reviews (MathSciNet): MR38:1190

Zentralblatt MATH: 0189.02603

[Se2] 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.

Mathematical Reviews (MathSciNet): MR41:8422

Zentralblatt MATH: 0186.25701

[Sh1] 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.

Mathematical Reviews (MathSciNet): MR58:5528

Zentralblatt MATH: 0332.32024

[Sh2] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679.

Mathematical Reviews (MathSciNet): MR80a:10043

Zentralblatt MATH: 0394.10015

Digital Object Identifier: doi:10.1215/S0012-7094-78-04529-5

Project Euclid: euclid.dmj/1077312955

[Sh3] G. Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. (2) 111 (1980), no. 2, 313–375.

Mathematical Reviews (MathSciNet): MR81g:10041

Zentralblatt MATH: 0438.12003

Digital Object Identifier: doi:10.2307/1971202

JSTOR: links.jstor.org

[Sh4]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.

Mathematical Reviews (MathSciNet): MR83d:10036a

Zentralblatt MATH: 0468.10016

Digital Object Identifier: doi:10.2307/1971381

JSTOR: links.jstor.org

[Sh4]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.

Mathematical Reviews (MathSciNet): MR83d:10036b

Zentralblatt MATH: 0486.10021

Digital Object Identifier: doi:10.2307/1971302

[Sh5] G. Shimura, Algebraic relations between critical values of zeta functions and inner products, Amer. J. Math. 105 (1983), no. 1, 253–285.

Mathematical Reviews (MathSciNet): MR84j:10038

Zentralblatt MATH: 0518.10032

Digital Object Identifier: doi:10.2307/2374388

JSTOR: links.jstor.org

[Sh6] G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Invent. Math. 94 (1988), no. 2, 245–305.

Mathematical Reviews (MathSciNet): MR90e:11069

Zentralblatt MATH: 0656.10018

Digital Object Identifier: doi:10.1007/BF01394326

[Sh7] G. Shimura, On the fundamental periods of automorphic forms of arithmetic type, Invent. Math. 102 (1990), no. 2, 399–428.

Mathematical Reviews (MathSciNet): MR91k:11041

Zentralblatt MATH: 0712.11028

Digital Object Identifier: doi:10.1007/BF01233433

[Sh8] G. Shimura, The critical values of certain Dirichlet series attached to Hilbert modular forms, Duke Math. J. 63 (1991), no. 3, 557–613.

Mathematical Reviews (MathSciNet): MR92e:11045

Zentralblatt MATH: 0752.11021

Digital Object Identifier: doi:10.1215/S0012-7094-91-06324-6

Project Euclid: euclid.dmj/1077296070

[T] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), no. 2, 265–280.

Mathematical Reviews (MathSciNet): MR90m:11176

Zentralblatt MATH: 0705.11031

Digital Object Identifier: doi:10.1007/BF01388853

[W] A. Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509–524.

Mathematical Reviews (MathSciNet): MR18,601a

Zentralblatt MATH: 0072.16001

Digital Object Identifier: doi:10.2307/2372670

JSTOR: links.jstor.org

[Y] H. Yoshida, Abelian varieties with complex multiplication and representations of the Weil groups, Ann. of Math. (2) 114 (1981), no. 1, 87–102.

Mathematical Reviews (MathSciNet): MR82k:14044

Zentralblatt MATH: 0473.14018

Digital Object Identifier: doi:10.2307/1971378

JSTOR: links.jstor.org

previous :: next

Duke Mathematical Journal

Turn MathJax Off
What is MathJax?