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L-values at Zero and the Galois Structure of Global Units

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Algebra

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Abstract

This article intends to present a comprehensive survey of the striking interplay between the Galois structure of the group of units in a number field and the values at zero of Artin L-functions. The algebraic ingredients come from integral representation theory, the ones from number theory include the Main Conjecture of Iwasawa theory. In fact, the discussion of recently defined invariants which go along with the unit group seems to propose possible generalizations of the Main Conjecture and fits very well into the framework of rather general conjectures regarding L-values by providing first affirmative answers. To begin with, we collect the principal ideas.

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References

  1. Beilinson, A., Higher regulators and values of L-functions. J. Sov. Math. 30, 2036–2070, 1985.

    Article  MATH  Google Scholar 

  2. Beilinson, A., Height pairing between algebraic cycles. Contemp. Math. 67, 1–24, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bley, W., Elliptic Curves and Module Structure Over Hopf Orders and The Conjecture of Chinburg-Stark for Abelian Extensions of a Quadratic Imaginary Field. Habilitation thesis, Augsburg, December 1997.

    Google Scholar 

  4. Bloch, S., A note on height pairing, Tamagawa numbers and the Birch and Swinnerton-Dyer conjecture. Invent. Math. 58, 65–76, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bloch, S., and Kato, K., L-functions and Tamagawa numbers of motives. “The Grothendieck Festschrift”, 1, Prog. Math. 86, 333–400, Birkhäuser Verlag, 1990.

    MATH  Google Scholar 

  6. Brown, K.S., Cohomology of Groups. Springer TGM 87, 1982.

    Book  MATH  Google Scholar 

  7. Burns, D., On multiplicative Galois structure invariants. Amer. J. Math. 117, 875–903, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  8. —, Iwasawa theory and p-adic Hodge theory over non-commutative algebras, I, II. Preprints (1997; King’s College London).

    Google Scholar 

  9. —, Equivariant Tamagawa Numbers and Galois Module Theory, I. Preprint (1997; King’s College London).

    MATH  Google Scholar 

  10. Burns, D., Flach, M., Motivic L-functions and Galois Module Structures. Math. Ann. 305, 65–102, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  11. —, On Galois structure invariants associated to Tate motives. Preprint (1997; King’s College London and Caltech).

    MATH  Google Scholar 

  12. —, Equivariant Tamagawa Numbers of Motives. Preprint (1997; King’s College London and Caltech).

    MATH  Google Scholar 

  13. Chinburg, T, On the Galois structure of algebraic integers and S-units. Invent. Math. 74, 321–349, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  14. —, Exact sequences and Galois module structure. Ann. Math. 121, 351–376, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  15. —, The analytic theory of multiplicative Galois structure. Mem. AMS 77, 1989.

    Google Scholar 

  16. Cassels, J.W.S. and Fröhlich, A., Algebraic Number Theory. Brighton-Proceedings (1965), Academic Press, 1967.

    MATH  Google Scholar 

  17. Curtis, C.W., and Reiner, I., Representation Theory of Finite Groups and Associative Algebras. John Wiley & Sons (1962).

    MATH  Google Scholar 

  18. —, Methods of Representation Theory, vol. 1. John Wiley & Sons, 1981.

    MATH  Google Scholar 

  19. —, Methods of Representation Theory, vol. 2. John Wiley & Sons, 1987.

    MATH  Google Scholar 

  20. Deligne, P., Valeurs de fonctions L et périodes d’intégrales. Proc. Symp. pure Math. 33, 313–346, 1979.

    Article  MATH  Google Scholar 

  21. Dubois, I., Structure galoisienne des S-unités et unités d’une extension cyclique de degré premier, Thesis, Bordeaux (1997).

    Google Scholar 

  22. Fontaine, J-M. and Perrin-Riou, B., Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L. Proc. Symp. pure Math. 55, 599–706, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  23. Fröhlich, A., Galois Module Structure of Algebraic Integers. Springer-Verlag, 1983.

    Book  MATH  Google Scholar 

  24. —, Classgroups and Hermitian Modules. Birkhäuser Verlag, PM 48, 1984.

    Google Scholar 

  25. Fröhlich, A., Galois module structure. In: Algebraic Number Fields, Durham-Proc. (1975) (Ed. Fröhlich) A., Academic Press, 1977.

    Google Scholar 

  26. —, Central extensions, Galois groups, and ideal class groups of number fields. Contemp. Math. 24, AMS, 1983.

    Google Scholar 

  27. Greither, C., The structure of some minus class groups, and Chinburg’s third conjecture for abelian fields. Preprint (1997; Université Laval et CICMA, Canada).

    MATH  Google Scholar 

  28. Gruenberg, K.W., Relation Modules of Finite Groups. Reg. Conf. Sr. Math. 25, AMS, 1976.

    Google Scholar 

  29. Gruenberg, K.W., Ritter, J. and Weiss, A., A Local Approach to Chinburg’s Root Number Conjecture. To appear in Proc. LMS.

    Google Scholar 

  30. Gruenberg, K.W. and Weiss, A., Galois invariants for units. Proc. LMS 70, 264–284, 1995.

    MathSciNet  MATH  Google Scholar 

  31. —, Galois invariants for S-units. Am. J. Math. 119, 953–983, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  32. Hasse, H., Über die Klassenzahl Abelscher Zahlkörper. Springer Verlag (Nachdruck der Ausgabe des Akademie-Verlags von 1952).

    MATH  Google Scholar 

  33. Kato, K., Iwasawa theory and p-adic Hodge theory. Kodai Math. J. 16, 1–31, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lang, S., Algebraic Number Theory. Addison-Wesley, 1970.

    MATH  Google Scholar 

  35. Mazur, B. and Wiles, A., Class fields of abelian extensions of ℚ. Invent. Math. 76, 179–330, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  36. Reiner, I., Maximal Orders. Academic Press (1975).

    MATH  Google Scholar 

  37. Ritter, J. and Weiss, A., On the local Galois structure of S-units. In: Algebra and Number Theory, Proc. Conf. Inst, experiment. Math. University of Essen, Germany, December 2–4, 1992 (Eds. G. Frey, J. Ritter), de Gruyter Proc. Math. 229–245, 1994.

    Google Scholar 

  38. —, A Tate sequence for global units. Comp. Math. 102, 147–178, 1996.

    MathSciNet  MATH  Google Scholar 

  39. —, Cohomology of units and L-values at zero. J. AMS 10, 513–552, 1997.

    MathSciNet  MATH  Google Scholar 

  40. —, The Lifted Root Number Conjecture for some cyclic extensions of ℚ. Submitted to Acta Arithmetica.

    Google Scholar 

  41. Serre, J.P., Corps Locaux. Hermann, Paris, 1968.

    MATH  Google Scholar 

  42. Swan, R.G., K-Theory of Finite Group and Orders (Notes by E.G. Evans). Springer LNM 149, 1970.

    Google Scholar 

  43. Tate, J., The cohomology groups of tori in finite Galois extensions of numer fields. Nagoya Math. J. 27, 709–719, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  44. —, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0. Prog. Math. 47, Birkhäuser, 1984.

    Google Scholar 

  45. Washington, L., Introduction to Cyclotomic Fields. Springer-Verlag (1982).

    Book  MATH  Google Scholar 

  46. Weiss, A., Multiplicative Galois Module Structure. Fields Institute Monographs 5, AMS (1996).

    Google Scholar 

  47. Wiles, A., The Iwasawa Conjecture for Totally Real Fields. Ann. Math. 131, 493–540, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  48. —, On a conjecture of Brumer. Ann. Math. 131, 555–565, 1990.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, D-86135, Augsburg, Germany

    Jürgen Ritter

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  1. Jürgen Ritter
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Editor information

Editors and Affiliations

  1. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India

    I. B. S. Passi

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© 1999 Hindustan Book Agency (India) and Indian National Science Academy

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Ritter, J. (1999). L-values at Zero and the Galois Structure of Global Units. In: Passi, I.B.S. (eds) Algebra. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-94-6_10

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