Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:57:28.539Z Has data issue: false hasContentIssue false
  • English
  • Français

On mean values and non-vanishing of derivatives of L-functions in a nonlinear family

Part of: Arithmetic algebraic geometry Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 April 2010

Ritabrata Munshi
Ritabrata Munshi*
Affiliation:
Department of Mathematics, Rutgers University, Hill Center, Piscataway, NJ 08854, USA (email: rmunshi@math.ias.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a mean-value result for derivatives of L-functions at the center of the critical strip for a family of forms obtained by twisting a fixed form by quadratic characters with modulus which can be represented as sum of two squares. Such a family of forms is related to elliptic fibrations given by the equation q(t)y2=f(x) where q(t)=t2+1 and f(x) is a cubic polynomial. The aim of the paper is to establish a prototype result for such quadratic families. Though our method can be generalized to prove similar results for any positive definite quadratic form in place of sum of two squares, we refrain from doing so to keep the presentation as clear as possible.

Keywords

derivatives of L-functionsnon-vanishing of central values

MSC classification

Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11M41: Other Dirichlet series and zeta functions 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 19 - 34
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Chinta, G., Friedberg, S. and Hoffstein, J., Multiple Dirichlet series and automorphic forms, in Multiple Dirichlet series, automorphic forms, and analytic number theory, Proceedings of Symposia in Pure Mathematics, vol. 75 (American Mathematical Society, Providence, RI, 2006), 3–41.CrossRefGoogle Scholar
[2]Heath-Brown, D. R., A mean value estimate for real character sums, Acta Arith. 72 (1995), 235–275.CrossRefGoogle Scholar
[3]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004), xii+615.Google Scholar
[4]Iwaniec, H. and Munshi, R., Cubic polynomials and quadratic forms, J. Lond. Math. Soc. (2) 81 (2010), 45–64.CrossRefGoogle Scholar
[5]Iwaniec, H. and Sarnak, P., The non-vanishing of central values of automorphic L-functions and Landau–Siegel zeros, Israel J. Math. 120 (2000), 155–177, part A.CrossRefGoogle Scholar
[6]Kowalski, E., Michel, P. and VanderKam, J., Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip, J. Reine Angew. Math. 526 (2000), 1–34.CrossRefGoogle Scholar
[7]Munshi, R., The level of distribution of the special values of L-functions, Acta Arith. 138 (2009), 239–257.CrossRefGoogle Scholar
[8]Munshi, R., Quadratic families of CM elliptic curves, Preprint.Google Scholar
[9]Murty, M. R. and Murty, V. K., Mean values of derivatives of modular L-series, Ann. of Math. (2) 133 (1991), 447–475.CrossRefGoogle Scholar
[10]Rankin, R. A., Sums of powers of cusp form coefficients II, Math. Ann. 272 (1985), 593–600.CrossRefGoogle Scholar