Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1999 | 88 | 1 | 51-66
Tytuł artykułu

Certain L-functions at s = 1/2

Treść / Zawartość
Warianty tytułu
Języki publikacji
Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields.
   In this paper we study the central zeros of the following types of L-functions:
   (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ),
   (ii) the Rankin-Selberg convolution for a pair of Hecke eigenforms for SL₂(ℤ),
   (iii) the Dedekind zeta functions.
  The paper is organized as follows. In Section 1, the Mellin transform L(s,f) of a holomorphic Hecke eigenform f for SL₂(ℤ) is studied. We note that every L-function in this paper is normalized so that it has a functional equation under the substitution s ↦ 1-s. In Section 2, we study some nonvanishing property of the Rankin-Selberg convolutions at s=1/2. Section 3 contains Kurokawa's result asserting the existence of number fields such that the vanishing order of the Dedekind zeta function at s=1/2 goes to infinity.
Słowa kluczowe
  • Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan
  • [Br] R. Brauer, A note on zeta-functions of algebraic number fields, Acta Arith. 24 (1973), 325-327.
  • [De] P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1973), 273-307.
  • [Den1] C. Deninger, Motivic L-functions and regularized determinants. I, in: Proc. Sympos. Pure Math. 55, Part I, Amer. Math. Soc., 1994, 707-743.
  • [Den2] C. Deninger, Motivic L-functions and regularized determinants. II, in: Arithmetic Geometry, F. Catanese (ed.), Symposia Math. 37, Cambridge Univ. Press, 1997, 138-156.
  • [E-M-O-T1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, 1953.
  • [E-M-O-T2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, 1954.
  • [F-H] S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL(2), Ann. of Math. 142 (1995), 385-423.
  • [Frö] A. Fröhlich, Artin root numbers and normal integral bases for quaternion fields, Invent. Math. 17 (1972), 143-166.
  • [Ko1] W. Kohnen, Modular forms of half-integral weight on Γ₀(4), Math. Ann. 248 (1980), 249-266.
  • [Ko2] W. Kohnen, Nonvanishing of Hecke L-functions associated to cusp forms inside the critical strip, J. Number Theory 67 (1997), 182-189.
  • [Ko-Za] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175-198.
  • [Ma] H. Maass, Lectures on Modular Functions of One Complex Variable (revised edition), Tata Inst. Fund. Res. Lectures on Math. and Phys. 29, Springer, 1983.
  • [Mi] S. Mizumoto, Eisenstein series for Siegel modular groups, Math. Ann. 297 (1993), 581-625; Corrections, Math. Ann. 307 (1997), 169-171.
  • [Rad] H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169, Springer, 1973.
  • [Ran] R. A. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions, Proc. Cambridge Philos. Soc. 35 (1939), 351-372.
  • [Se] A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47-50.
  • [Sha] F. Shahidi, Third symmetric power L-functions for GL(2), Compositio Math. 70 (1989), 245-273.
  • [Sh] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), 783-804.
  • [St] J. Sturm, The critical values of zeta functions associated to the symplectic group, Duke Math. J. 48 (1981), 327-350.
  • [W1] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484.
  • [W2] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219-307.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.