Journal ArticleDOI
The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
Henryk Iwaniec,Peter Sarnak +1 more
TLDR
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.read more
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Analytic Number Theory
Henryk Iwaniec,Emmanuel Kowalski +1 more
TL;DR: In this paper, the critical zeros of the Riemann zeta function are defined and the spacing of zeros is defined. But they are not considered in this paper.
Journal ArticleDOI
Zeroes of zeta functions and symmetry
Nicholas M. Katz,Peter Sarnak +1 more
TL;DR: The spectral interpretation of the zeroes of the Riemann Zeta function has been studied in this paper, where the eigenvalues of Frobenius on cohomology are used.
Journal ArticleDOI
Low lying zeros of families of L-functions
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).
Journal ArticleDOI
Rankin-Selberg $L$-functions in the level aspect
TL;DR: In this paper, a Rankin-Selberg convolutional neural network (RSCNN) was used to achieve convexity-breaking moments with level aspect level aspect.
Book ChapterDOI
Perspectives on the Analytic Theory of L-Functions
Henryk Iwaniec,Peter Sarnak +1 more
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.
References
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Journal ArticleDOI
Heegner points and derivatives of L-series.
Journal ArticleDOI
Modular curves and the Eisenstein ideal
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Journal ArticleDOI
On Modular Forms of Half Integral Weight
TL;DR: In this article, the connection of modular forms with zeta functions was clarified, and a more affirmative aspect of the subject was revealed, which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases.
Book
Topics in classical automorphic forms
TL;DR: The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automomorphic $L$-functions Cusp forms associated with elliptic curves Spherical functions Theta functions Representations by quadratic forms Automomorphic functions associated with number fields Convolution$L$ -functions Bibliography.