Showing papers by "Henryk Iwaniec published in 2000"
TL;DR: This article examined the distribution of zeros which are at or neat s = 1/2 (that is the central point) for families of GL_2 automorphic L-functions, and most of the results in this paper are conditional on the Generalized Riemann Hypothesis (GRH).
Abstract: In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.
426 citations
01 Jan 2000
TL;DR: In this paper, the authors introduce L-functions and describe the central problems connected with them, as well as some problems that can be resolved by finessing these conjectures.
Abstract: To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.
260 citations
219 citations
TL;DR: In this paper, it was shown that as n → ∞ at least 50% of the values L(½, f) are positive, with f varying among the holomorphic new forms of a fixed even integral weight for Γ 0(N) and whose functional equations are even, and that any improvement of 50% is intimately connected to Landau-Siegel zero.
Abstract: We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying among the holomorphic new forms of a fixed even integral weight for Γ0(N) and whose functional equations are even, are positive. Furthermore, we show that any improvement of 50% is intimately connected to Landau-Siegel zeros. These results may also be used to show that X0(N) = Γ0(N)\ℍ has large quotients with only finitely many rational points. The results below were announced at the conference “Exponential sums” held in Jerusalem, January 1998. The complete proofs, which were presented in courses at Princeton (1997), are being prepared for publication.
187 citations
34 citations