Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T14:58:21.630Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Error Term in Duke's Estimate for the Average Special Value of L-Functions

Published online by Cambridge University Press:  20 November 2018

Jordan S. Ellenberg
Jordan S. Ellenberg*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08844, U.S.A. e-mail: ellenber@math.princeton.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.

Let $\mathcal{F}$ be an orthonormal basis for weight 2 cusp forms of level $N$. We show that various weighted averages of special values $L\left( f\otimes \text{ }\!\!\chi\!\!\text{ ,1} \right)$ over $f\in \mathcal{F}$ are equal to $4\text{ }\!\!\pi\!\!\text{ }c+O\left( {{N}^{-1+\in }} \right)$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error term of $O\left( {{N}^{-1/2}}\log N \right)$.

Keywords

11F6711F11
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 535 - 546
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Atkin, A. O. L. and Li, W. C. W., Twists of newforms and pseudo-eigenvalues of W-operators. Invent. Math. 48(1978), 221–243.Google Scholar
[2] Duke, W., The critical order of vanishing of automorphic L-functions with large level. Invent. Math. 119(1995), 165–174.Google Scholar
[3] Ellenberg, J., Galois representations attached to ℚ-curves and the generalized Fermat equation A4 + B2 = Cp . Amer. J. Math. 126(2004), 763–787.Google Scholar
[4] Iwaniec, H., Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17, American Mathematical Society, Providence, RI, 1997.Google Scholar
[5] Rohrlich, D., On L-functions of elliptic curves and cyclotomic towers. Invent. Math. 75(1984), 409–423.Google Scholar