Cohomology and $L$-values

Cohomology and $L$-values

Hiroyuki Yoshida

Source: Kyoto J. Math. Volume 52, Number 2 (2012), 369-432.

Abstract

In a paper published in 1959, Shimura presented an elegant calculation of the critical values of $L$-functions attached to elliptic modular forms using the first cohomology group. We will show that a similar calculation is possible for Hilbert modular forms over real quadratic fields using the second cohomology group. We present explicit numerical examples calculated by this method.

First Page: Show

Primary Subjects: 11F41, 11F75

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.kjm/1335272984
Digital Object Identifier: doi:10.1215/21562261-1551003
Zentralblatt MATH identifier: 06047794
Mathematical Reviews number (MathSciNet): MR2914881

back to Table of Contents

References

[B] D. Blasius, “Hilbert modular forms and the Ramanujan conjecture” in Noncommutative geometry and number theory, Aspects Math. 37, Vieweg, Wiesbaden, Germany, 2006, 35–56.

Mathematical Reviews (MathSciNet): MR2327298

Zentralblatt MATH: 1183.11023

Digital Object Identifier: doi:10.1007/978-3-8348-0352-8_2

[BW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd. ed., Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, 2000.

Mathematical Reviews (MathSciNet): MR1721403

[CE] H. Cartan and S. Eilenberg, Homological Algebra, with an appendix by D. A. Buchsbaum, reprint of the 1956 original, Princeton Landmarks in Math., Princeton Univ. Press, Princeton, 1999.

Mathematical Reviews (MathSciNet): MR1731415

[DG] K. Doi and K. Goto, On the L-series associated with modular forms (in Japanese), Mem. Inst. Sci. Engrg. Ritsumeikan Univ. 52 (1993), 1–19.

Mathematical Reviews (MathSciNet): MR1267009

Zentralblatt MATH: 0972.11501

[DHI] K. Doi, H. Hida, and H. Ishii, Discriminant of Hecke fields and twisted adjoint L-values for GL(2), Invent. Math. 134 (1998), 547–577.

Mathematical Reviews (MathSciNet): MR1660929

Zentralblatt MATH: 0924.11035

Digital Object Identifier: doi:10.1007/s002220050273

[DI] K. Doi and H. Ishii, Hilbert modular L-values and discriminant of Hecke’s fields, Mem. Inst. Sci. Engrg. Ritsumeikan Univ. 53 (1994), 1–12.

Mathematical Reviews (MathSciNet): MR1332246

Zentralblatt MATH: 0972.11502

[E] B. Eckmann, Cohomology of groups and transfer, Ann. of Math. (2) 58 (1953), 481–493.

Mathematical Reviews (MathSciNet): MR58600

Zentralblatt MATH: 0052.02002

Digital Object Identifier: doi:10.2307/1969749

[Ha] G. Harder, Eisenstein cohomology of arithmetic groups: The case GL₂, Invent. Math. 89 (1987), 37–118.

Mathematical Reviews (MathSciNet): MR892187

Zentralblatt MATH: 0629.10023

Digital Object Identifier: doi:10.1007/BF01404673

[Hi1] H. Hida, On abelian varieties with complex multiplication as factors of the abelian variety attached to Hilbert modular forms, Japanese J. Math. (N.J.) 5 (1979), 157–208.

Mathematical Reviews (MathSciNet): MR614697

[Hi2] H. Hida, p-ordinary cohomology groups for SL(2) over number fields, Duke Math. J. 69 (1993), 259–314.

Mathematical Reviews (MathSciNet): MR1203228

Zentralblatt MATH: 0941.11024

Digital Object Identifier: doi:10.1215/S0012-7094-93-06914-1

Project Euclid: euclid.dmj/1077293571

[Hi3] H. Hida, On the critical values of L-functions of GL(2) and GL(2) × GL(2), Duke Math. J. 74 (1994), 431–529.

Mathematical Reviews (MathSciNet): MR1272981

Zentralblatt MATH: 0838.11036

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

Project Euclid: euclid.dmj/1077288205

[HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.

Mathematical Reviews (MathSciNet): MR2445243

[JL] H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer, Berlin, 1970.

Mathematical Reviews (MathSciNet): MR401654

[K] A. G. Kurosh, The Theory of Groups, Vols. 1, 2, trans. and ed. K. A. Hirsch, Chelsea, New York, 1960.

Mathematical Reviews (MathSciNet): MR109842

Zentralblatt MATH: 0111.02502

[KS] M. Kuga and G. Shimura, On vector differential forms attached to automorphic forms, J. Math. Soc. Japan 12 (1960), 258–270.

Mathematical Reviews (MathSciNet): MR133480

Zentralblatt MATH: 0097.28704

Digital Object Identifier: doi:10.2969/jmsj/01230258

Project Euclid: euclid.jmsj/1261147830

[Mac] A. M. Macbeath, Groups of homeomorphisms of a simply connected space, Ann. of Math. (2) 79 (1964), 473–488.

Mathematical Reviews (MathSciNet): MR160848

Zentralblatt MATH: 0122.17503

Digital Object Identifier: doi:10.2307/1970405

[Man] Y. I. Manin, Periods of parabolic forms and p-adic Hecke series, Math. USSR Sb. 21 (1973), 371–393.

[MM] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365–416.

Mathematical Reviews (MathSciNet): MR153028

Zentralblatt MATH: 0125.10702

Digital Object Identifier: doi:10.2307/1970348

[MS] Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2) 78 (1963), 417–449.

Mathematical Reviews (MathSciNet): MR155340

Digital Object Identifier: doi:10.2307/1970534

[PARI] PARI/GP, Version 2.3.4, Université Bordeaux I, France, 2008, available from http://pari.math.u-bordeaux.fr/

[Sc] O. Schreier, Die Untergruppen der freie Gruppen, Abh. Math. Semin. Univ. Hambg. 5 (1927), 161–183.

[Se1] J-P. Serre, Corps locaux, 2nd ed., Publ. Univ. Nancago 8, Hermann, Paris, 1968.

Mathematical Reviews (MathSciNet): MR354618

[Se2] J-P. Serre, “Cohomologie des groupes discrets” in Prospects in Mathematics (Princeton, 1970), Ann. of Math. Stud. 70, Princeton Univ. Press, Princeton, 1971, 77–169.

Mathematical Reviews (MathSciNet): MR385006

[Shi] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33–71.

Mathematical Reviews (MathSciNet): MR145106

Zentralblatt MATH: 0218.10045

Digital Object Identifier: doi:10.2307/1970201

[Sh1] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311.

Mathematical Reviews (MathSciNet): MR120372

Digital Object Identifier: doi:10.2969/jmsj/01140291

Project Euclid: euclid.jmsj/1261148160

[Sh2] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), 783–804.

Mathematical Reviews (MathSciNet): MR434962

Zentralblatt MATH: 0348.10015

Digital Object Identifier: doi:10.1002/cpa.3160290618

[Sh3] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637–679.

Mathematical Reviews (MathSciNet): MR507462

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

Project Euclid: euclid.dmj/1077312955

[Sh4] G. Shimura, The critical values of certain Dirichlet series attached to Hilbert modular forms, Duke Math. J. 63 (1991), 557–613.

Mathematical Reviews (MathSciNet): MR1121146

Zentralblatt MATH: 0752.11021

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

Project Euclid: euclid.dmj/1077296070

[Sh5] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, reprint of the 1971 original, Publ. Math. Soc. Japan. 11, Princeton Univ. Press, Princeton, 1994.

Mathematical Reviews (MathSciNet): MR1291394

[Sh6] G. Shimura, Eisenstein series and zeta functions on symplectic groups, Invent. Math. 119 (1995), 539–584.

Mathematical Reviews (MathSciNet): MR1317650

Zentralblatt MATH: 0845.11020

Digital Object Identifier: doi:10.1007/BF01245192

[Sh7] G. Shimura, Arithmeticity in the Theory of Automorphic Forms, Math. Surveys Monogr. 82, Amer. Math. Soc., Providence, 2000.

Mathematical Reviews (MathSciNet): MR1780262

Zentralblatt MATH: 0967.11001

[Si1] C. L. Siegel, Discontinuous groups, Ann. of Math. (2) 44 (1943), 674–689.

Mathematical Reviews (MathSciNet): MR9959

Digital Object Identifier: doi:10.2307/1969104

[Si2] C. L. Siegel, Lectures on Advanced Analytic Number Theory, Tata Inst. Fund. Res. Lectures Math. 23, Tata Inst. Fund. Res., Bombay, 1965.

Mathematical Reviews (MathSciNet): MR262150

Zentralblatt MATH: 0278.10001

[Su] M. Suzuki, Group Theory, I, Grundlehren Math. Wiss. 247, Springer, Berlin, 1982.

Mathematical Reviews (MathSciNet): MR648772

Zentralblatt MATH: 0472.20001

[Sw] R. W. Swan, Generators and relations for certain special linear groups, Adv. Math. 6 (1971), 1–77.

Mathematical Reviews (MathSciNet): MR284516

Zentralblatt MATH: 0221.20060

Digital Object Identifier: doi:10.1016/0001-8708(71)90027-2

[V] L. N. Vaserštein, On the group SL₂ over Dedekind rings of arithmetic type, Math. USSR Sbornik 18 (1972), 321–332.

[Y1] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, Duke Math. J. 75 (1994), 121–191.

Mathematical Reviews (MathSciNet): MR1284818

Zentralblatt MATH: 0823.11018

Digital Object Identifier: doi:10.1215/S0012-7094-94-07505-4

Project Euclid: euclid.dmj/1077287413

[Y2] H. Yoshida, On a conjecture of Shimura concerning periods of Hilbert modular forms, Amer. J. Math. 117 (1995), 1019–1038.

Mathematical Reviews (MathSciNet): MR1342839

Zentralblatt MATH: 0841.11024

Digital Object Identifier: doi:10.2307/2374957

[Y3] H. Yoshida, Absolute CM-Periods, Math. Surveys Monogr. 106, Amer. Math. Soc., Providence, 2003.

Mathematical Reviews (MathSciNet): MR2011848

[Y4] H. Yoshida, Cohomology and L-values, preprint, arXiv:1012.4573v1 [math.NT]

arXiv: 1012.4573v1

previous :: next

Kyoto Journal of Mathematics

Turn MathJax Off
What is MathJax?