Duke Mathematical Journal

On a representation of the idele class group related to primes and zeros of L-functions

Ralf Meyer

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Abstract

Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to a variant of the Weil distribution which occurs in André Weil's explicit formula. Hence this representation encodes information about the distribution of the prime ideals of K and is a spectral interpretation for the poles and zeros of the L-function of K. Our construction is motivated by a similar spectral interpretation by Alain Connes.

Article information

Source
Duke Math. J. Volume 127, Number 3 (2005), 519-595.

Dates
First available: 18 April 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1113847338

Digital Object Identifier
doi:10.1215/S0012-7094-04-12734-4

Mathematical Reviews number (MathSciNet)
MR2132868

Zentralblatt MATH identifier
02177466

Subjects
Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 22D12: Other representations of locally compact groups
Secondary: 18H10 43A35: Positive definite functions on groups, semigroups, etc. 58B34: Noncommutative geometry (à la Connes)

Citation

Meyer, Ralf. On a representation of the idele class group related to primes and zeros of L -functions. Duke Mathematical Journal 127 (2005), no. 3, 519--595. doi:10.1215/S0012-7094-04-12734-4. http://projecteuclid.org/euclid.dmj/1113847338.


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References

  • F. Bruhat, Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $\wp$\nobreakdash-adiques, Bull. Soc. Math. France 89 (1961), 43--75.
  • J.-F. Burnol, Sur les formules explicites, I: Analyse invariante, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 423--428.
  • --. --. --. --., On Fourier and zeta(s), Forum Math. 16 (2004), 789--840.
  • A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. (N.S.) 5 (1999), 29--106.
  • P. Deligne, La conjecture de Weil, II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137--252.
  • R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer, Berlin, 1972.
  • N. Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), 109--140.
  • --. --. --. --., An imprimitivity theorem for representations of locally compact groups on arbitrary Banach spaces, Pacific J. Math. 184 (1998), 121--148.
  • A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 1955, no. 16.
  • H. Hogbe-Nlend and V. B. Moscatelli, Nuclear and Conuclear Spaces, North-Holland Math. Stud. 52, North-Holland, Amsterdam, 1981.
  • B. Keller, ``Derived categories and their uses'' in Handbook of Algebra, Vol. 1, North-Holland, Amsterdam, 1996, 671--701.
  • R. Meyer, Generalized fixed point algebras and square-integrable groups actions, J. Funct. Anal. 186 (2001), 167--195.
  • --. --. --. --., ``Bornological versus topological analysis in metrizable spaces'' in Banach Algebras and Their Applications, Contemp. Math. 363, Amer. Math. Soc., Providence, 2004, 249--278.
  • --. --. --. --., Smooth group representations on bornological vector spaces, Bull. Sci. Math. 128 (2004), 127--166.
  • --------, The cyclic homology and K\nobreakdash-theory of certain adelic crossed products, preprint.
  • A. Neeman, The derived category of an exact category, J. Algebra 135 (1990), 388--394.
  • S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Stud. Adv. Math. 14, Cambridge Univ. Press, Cambridge, 1988.
  • D. Quillen, ``Higher algebraic $K$-theory, I'' in Algebraic $K$-Theory, I: Higher $K$-Theories (Seattle, 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973, 85--147.
  • J. T. Tate, ``Fourier analysis in number fields and Hecke's zeta-functions'' in Algebraic Number Theory (Brighton, U.K., 1965), Thompson, Washington, D.C., 1967, 305--347.
  • A. Weil, Sur les ``formules explicites'' de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund 1952, 252--265.
  • --. --. --. --., Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3--18.; English translation in Math. USSR-Izv. 6, no. 1 (1972), 1--17.
  • --------, Basic Number Theory, Classics Math., Springer, Berlin, 1995.