Wojciech J. Gajda

Abstract: L-functions appear as generating functions encapsulating information about various objects such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We will discuss the important conjectures, one concerning the distribution of the zeros and one concerning the distribution of the central values, and explain a general principle that any restricted result towards the first conjecture can be refined to show that most corresponding central values have the typical distribution predicted by the second conjecture. We will instanciate this general principle in the case of L-functions attached to modular forms.