A lower bound for the rank of $J_0(q)$

• Autorzy: E. Kowalski , P. Michel
• Języki publikacji: EN
• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2000
• Tom: 94
• Numer: 4
• Strony: 303-343

Daty

wydano

2000

otrzymano

1997-11-10

poprawiono

2000-03-02

Twórcy

autor

E. Kowalski

• Math. Dept.-Fine Hall Washington Road Princeton, NJ 08544-1000, U.S.A.

autor

P. Michel

• Mathématiques, cc 051 Université Montpellier II Place Eugène Bataillon 34095 Montpellier Cedex 5, France

Bibliografia

• [CGG] J. B. Conrey, A. Ghosh and S. M. Gonek, Simple zeros of the Riemann zeta function, Proc. London Math. Soc. (3) 76 (1998), 497-522.
• [Du] W. Duke, The critical order of vanishing of automorphic L-functions with high level, Invent. Math. 119 (1995), 165-174.
• [DFI] W. Duke, J. Friedlander and H. Iwaniec, Bounds for automorphic L-functions, II, ibid. 115 (1994), 219-239.
• [G-R] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., Academic Press, 1994.
• [GZ] B. Gross and D. Zagier, Heegner points and derivatives of L-series II, Math. Ann. 278 (1987), 497-562.
• [Iw] H. Iwaniec, On the order of vanishing of modular L-functions at the critical point, Sém. Théor. Nombres Bordeaux 2 (1990), 365-376.
• [ILS] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros for families of L-functions, preprint, 1998.
• [IS] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic L-functions and Siegel's zeros, preprint, 1997.
• [Jut] M. Jutila, Lectures on a Method in the Theory of Exponential Sums, Tata Lectures on Math. and Phys. 80, Springer, 1987.
• [KS] N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, 1999.
• [Kow] E. Kowalski, The rank of the jacobian of modular curves: analytic methods, thesis, Rutgers University, May 1998 (available at http://www.princeton.edu/ekowalsk/These/these.html).
• [KM1] E. Kowalski and P. Michel, The analytic rank of $J_0(q)$ and zeros of automorphic L-functions, Duke Math. J. 100 (1999), 503-542.
• [KM2] E. Kowalski and P. Michel, Explicit upper bound for the rank of $J_0(q)$, preprint, Université Montpellier II, 1998.
• [Miy] T. Miyake, Modular Forms, Springer, 1989.
• [Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton Univ. Press, 1971.
• [Sou] K. Soundararajan, Non-vanishing of quadratic Dirichlet L-functions at s=1/2, preprint, 1999.
• [VdK] J. VanderKam, The rank of quotients of $J_0(N)$, preprint, 1997.