Converse theorems and the Selberg class

Abstract

In this thesis we will be investigating L functions of Selberg class type. Other than chapter 2 which shall be discussing the 'strong multiplicity one' principle, the rest of this thesis shall be discussing converse theorems for degree 2 elements of the Selberg class. Chapter 3 will discuss proving a generalization of the classical Weil's converse theorem, but allowing general root number and Euler product of Selberg class type. However for this result we need to have a fixed gamma factor, associated to holomorphic modular forms. In Chapters 4 and 5, we will prove converse theorems for holomorphic modular forms/Maass forms respectively, now allowing gamma factors of Selberg class type. For these results, we shall now no longer use information on multiplicative twists, but will use additive twist datum. These are the first converse theorems I am aware of for high conductor, degree 2 elements of the Selberg class.

Date of Award	4 Feb 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Andrew R Booker (Supervisor)