Author: Benjamin Justus
Publisher: LAP Lambert Academic Publishing
ISBN: 9783838338224
Category :
Languages : en
Pages : 80
Book Description
The thesis studies the integer-power moments of the central values of families of modular L-functions. The two families under consideration in the thesis are those quadratic twists of a L-function associated with a cusp from and L-functions of a Hecke-basis of the space of cusp forms. Applications of the moment estimates derived in the thesis include (1) a non-vanishing result (2) a zero density estimate for quadratic twisted L-functions.
On the Moments of Central Values of Modular L-Functions
Author: Benjamin Justus
Publisher: LAP Lambert Academic Publishing
ISBN: 9783838338224
Category :
Languages : en
Pages : 80
Book Description
The thesis studies the integer-power moments of the central values of families of modular L-functions. The two families under consideration in the thesis are those quadratic twists of a L-function associated with a cusp from and L-functions of a Hecke-basis of the space of cusp forms. Applications of the moment estimates derived in the thesis include (1) a non-vanishing result (2) a zero density estimate for quadratic twisted L-functions.
Publisher: LAP Lambert Academic Publishing
ISBN: 9783838338224
Category :
Languages : en
Pages : 80
Book Description
The thesis studies the integer-power moments of the central values of families of modular L-functions. The two families under consideration in the thesis are those quadratic twists of a L-function associated with a cusp from and L-functions of a Hecke-basis of the space of cusp forms. Applications of the moment estimates derived in the thesis include (1) a non-vanishing result (2) a zero density estimate for quadratic twisted L-functions.
On the Moments of Central Values of Modular L-functions
Six Short Chapters on Automorphic Forms and L-functions
Author: Ze-Li Dou
Publisher: Springer Science & Business Media
ISBN: 3642287085
Category : Mathematics
Languages : en
Pages : 131
Book Description
"Six Short Chapters on Automorphic Forms and L-functions" treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature. Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.
Publisher: Springer Science & Business Media
ISBN: 3642287085
Category : Mathematics
Languages : en
Pages : 131
Book Description
"Six Short Chapters on Automorphic Forms and L-functions" treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature. Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.
Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms
Author: Michel Courtieu
Publisher: Springer
ISBN: 3540451781
Category : Mathematics
Languages : en
Pages : 202
Book Description
This book, now in its 2nd edition, is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The central object are L-functions of classical Siegel modular forms whose special values are studied using the Rankin-Selberg method and the action of certain differential operators on modular forms which have nice arithmetical properties. A new method of p-adic interpolation of these critical values is presented. An important class of p-adic L-functions treated in the present book are p-adic L-functions of Siegel modular forms having logarithmic growth. The given construction of these p-adic L-functions uses precise algebraic properties of the arithmetical Shimura differential operator. The book will be very useful for postgraduate students and for non-experts looking for a quick approach to a rapidly developing domain of algebraic number theory. This new edition is substantially revised to account for the new explanations that have emerged in the past 10 years of the main formulas for special L-values in terms of arithmetical theory of nearly holomorphic modular forms.
Publisher: Springer
ISBN: 3540451781
Category : Mathematics
Languages : en
Pages : 202
Book Description
This book, now in its 2nd edition, is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The central object are L-functions of classical Siegel modular forms whose special values are studied using the Rankin-Selberg method and the action of certain differential operators on modular forms which have nice arithmetical properties. A new method of p-adic interpolation of these critical values is presented. An important class of p-adic L-functions treated in the present book are p-adic L-functions of Siegel modular forms having logarithmic growth. The given construction of these p-adic L-functions uses precise algebraic properties of the arithmetical Shimura differential operator. The book will be very useful for postgraduate students and for non-experts looking for a quick approach to a rapidly developing domain of algebraic number theory. This new edition is substantially revised to account for the new explanations that have emerged in the past 10 years of the main formulas for special L-values in terms of arithmetical theory of nearly holomorphic modular forms.
Non-vanishing of L-Functions and Applications
Author: M. Ram Murty
Publisher: Springer Science & Business Media
ISBN: 3034802749
Category : Mathematics
Languages : en
Pages : 206
Book Description
This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.
Publisher: Springer Science & Business Media
ISBN: 3034802749
Category : Mathematics
Languages : en
Pages : 206
Book Description
This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.
The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions
Author: Valentin Blomer
Publisher: American Mathematical Society
ISBN: 1470456788
Category : Mathematics
Languages : en
Pages : 160
Book Description
View the abstract.
Publisher: American Mathematical Society
ISBN: 1470456788
Category : Mathematics
Languages : en
Pages : 160
Book Description
View the abstract.
Moments of Automorphic L-functions and Related Problems
Author: Ian Petrow
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of L-functions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is -1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number -1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of full-level GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions.
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of L-functions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is -1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number -1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of full-level GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions.
Spectral Means of Central Values of Automorphic L-Functions for GL(2)
Author: Masao Tsuzuki
Publisher: American Mathematical Soc.
ISBN: 1470410192
Category : Mathematics
Languages : en
Pages : 144
Book Description
Starting with Green's functions on adele points of considered over a totally real number field, the author elaborates an explicit version of the relative trace formula, whose spectral side encodes the informaton on period integrals of cuspidal waveforms along a maximal split torus. As an application, he proves two kinds of asymptotic mean formula for certain central -values attached to cuspidal waveforms with square-free level.
Publisher: American Mathematical Soc.
ISBN: 1470410192
Category : Mathematics
Languages : en
Pages : 144
Book Description
Starting with Green's functions on adele points of considered over a totally real number field, the author elaborates an explicit version of the relative trace formula, whose spectral side encodes the informaton on period integrals of cuspidal waveforms along a maximal split torus. As an application, he proves two kinds of asymptotic mean formula for certain central -values attached to cuspidal waveforms with square-free level.
Value-Distribution of L-Functions
Author: Jr̲n Steuding
Publisher: Springer Science & Business Media
ISBN: 3540265260
Category : Mathematics
Languages : en
Pages : 320
Book Description
These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.
Publisher: Springer Science & Business Media
ISBN: 3540265260
Category : Mathematics
Languages : en
Pages : 320
Book Description
These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.
L-Functions and Automorphic Forms
Author: Jan Hendrik Bruinier
Publisher: Springer
ISBN: 3319697129
Category : Mathematics
Languages : en
Pages : 367
Book Description
This book presents a collection of carefully refereed research articles and lecture notes stemming from the Conference "Automorphic Forms and L-Functions", held at the University of Heidelberg in 2016. The theory of automorphic forms and their associated L-functions is one of the central research areas in modern number theory, linking number theory, arithmetic geometry, representation theory, and complex analysis in many profound ways. The 19 papers cover a wide range of topics within the scope of the conference, including automorphic L-functions and their special values, p-adic modular forms, Eisenstein series, Borcherds products, automorphic periods, and many more.
Publisher: Springer
ISBN: 3319697129
Category : Mathematics
Languages : en
Pages : 367
Book Description
This book presents a collection of carefully refereed research articles and lecture notes stemming from the Conference "Automorphic Forms and L-Functions", held at the University of Heidelberg in 2016. The theory of automorphic forms and their associated L-functions is one of the central research areas in modern number theory, linking number theory, arithmetic geometry, representation theory, and complex analysis in many profound ways. The 19 papers cover a wide range of topics within the scope of the conference, including automorphic L-functions and their special values, p-adic modular forms, Eisenstein series, Borcherds products, automorphic periods, and many more.