Non-vanishing of automorphic L-functions of prime power level

Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip

We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ(f, 1/2) 6= 0 is ≥ pk +o(1) with pk > 0 and pk = 1/2 + O(k−2), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result ...

Non-vanishing Results of Special Values of L-functions

Here are the notes I am taking for Eric Urban’s ongoing course on non-vanishing results of special values of L-functions offered at Columbia University in Spring 2015 (MATH G6675: Topics in Number Theory). As the course progresses, these notes will be revised. I recommend that you visit my website from time to time for the most updated version. Due to my own lack of understanding of the materia...

Non-vanishing of Quadratic Twists of Modular L-functions

Lectures on automorphic L-functions

PREFACE This article follows the format of five lectures that we gave on automorphic Lfunctions. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. Three of the lectures concerned the general properties of automorphic L-functions, with particular reference to questions of spectral decomposition. We have grouped these together as ...

Zeros of Families of Automorphic L-functions

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of...