Analytic twists of modular forms

Analytic Theory of Modular Forms

Determination of Modular Forms by Twists of Critical L-values

Contents 0 Introduction 1 1 Preliminaries 4 2 p-power twists 7 3 Quadratic twists 13 4 Generation of coeecient elds by ratios of L-values 20 5 p-adic L-functions 22 6 Forms of half integral weight 25 Bibliography 28 0 Introduction Let f be a normalized holomorphic newform (deened on the upper half plane H) of level N, weight 2k and trivial character. It is given by a Fourier expansion f(z) = P ...

Quadratic Twists of Modular Forms and Elliptic Curves

Here we summarize the results presented in the first author’s lecture at the Millenial Conference on Number Theory. These results will be appearing in [O] in full detail. In addition, we present a new result regarding the growth of Tate-Shafarevich groups of certain elliptic curves over elementary abelian simple 2-extensions. We begin by fixing notation. Suppose that F (z) = ∑∞ n=1 a(n)q n ∈ S2...

Algebraic Twists of Modular Forms and Hecke Orbits

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, ...

Mock Θ-functions and Real Analytic Modular Forms

In this paper we examine three examples of Ramanujan’s third order mock θ-functions and relate them to Rogers’ false θ-series and to a real-analytic modular form of weight 1/2.