Duke Mathematical Journal

Low-lying zeros of dihedral L-functions

E. Fouvry and H. Iwaniec

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Abstract

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$, where $\psi$ is a character of the ideal class group of the imaginary quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text {squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the vicinity of the central point $s = 1/2$, the average distribution of these zeros (for $D\longrightarrow \infty$) is governed by the symplectic distribution. By averaging over $D$, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve techniques are applied.

Article information

Source
Duke Math. J., Volume 116, Number 2 (2003), 189-217.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598267

Digital Object Identifier
doi:10.1215/S0012-7094-03-11621-X

Mathematical Reviews number (MathSciNet)
MR1953291

Zentralblatt MATH identifier
1028.11055

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11N36: Applications of sieve methods 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Citation

Fouvry, E.; Iwaniec, H. Low-lying zeros of dihedral L -functions. Duke Math. J. 116 (2003), no. 2, 189--217. doi:10.1215/S0012-7094-03-11621-X. https://projecteuclid.org/euclid.dmj/1085598267


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References

  • E. Bombieri, The asymptotic sieve, Rend. Accad. Naz. XL (5), 1/2 (1975/76), 243--269.
  • D. A. Cox, Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication, Wiley Intersci. Publ., Wiley, New York, 1989.
  • W. Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 1995, 99--109.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
  • E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, I, Math. Z. 1 (1918), 357--376.
  • H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of $L$-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55--131.
  • N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, 1999.
  • --. --. --. --., Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 1--26.
  • E. Royer, Petits zéros de fonctions $L$ de formes modulaires, Acta Arith. 99 (2001), 147--172.
  • J.-P. Serre, ``Modular forms of weight one and Galois representations'' in Algebraic Number Fields: $L$-Functions and Galois Properties (Durham, England, 1975), Academic Press, London, 1977, 193--268.