L-functions and equidistribution

Hybrid talk. For zoom details contact Christian Kaiser (kaiser@mpim-bonn.mpg.de).

Choosing a random prime p and reducing modulo an integer m one will find p is unit in the integers modulo m. Dirichlet proved that each class of units in the integers modulo m is as likely as another. That is to say, the primes are equidistributed among the units module m. Dirichlet used certain non-vanishing and holomorphicity of some associated L-functions to prove this. We will begin our discussion on generalisations of this result by considering solutions to one variable polynomials over the rationals. We will illustrate a generalisation of Dirichlet's relation between properties of L-functions and equidistribution to a class of representations of the absolute Galois group. Here our examples will be zero dimensional and next time we will turn to one dimensional examples given by elliptic curves. The Sato-Tate conjecture will then describe the analogues of the equidistribution results we will discuss.