Zeta functions over curves

Abstract

In this paper we review the theory that David Goss developed, starting from 1979, to construct zeta functions around Carlitz zeta values and other remarkable formal series in local fields of positive characteristic. In the description of Goss' theory, we will see how it is primarily motivated by analogies with the classical theory of complex valued zeta and $L$-functions. We compare Goss' theory with another way of constructing zeta and $L$-functions that emerged in more recent times. The functions in the second type have as domains curves over finite fields, with the scalars extended to complete and algebraically closed fields of positive characteristic. The second type of functions interacts with Goss' functions but remains fundamentally different. We shall review a rationality theorem of Ferraro that allows, among others, to introduce some kind of analogue of the function $\xi$ of Riemann. In the path of describing Ferraro's proof, we present some essential tools useful to get into the theory: shtuka divisors and functions, special functions, Anderson motives, Drinfeld modules, among others. We discuss certain relative zeta functions that can be considered as counterparts of Dedekind zeta functions. In particular, we use methods introduced by Goss to prove that these functions extend to entire functions. The paper contains an appendix by Ferraro where the property of entireness of the above relative zeta functions is deduced (in a special case) from the conjunction of a formula by Angl\`es, Ngo Dac and Tavares Ribeiro and Ferraro's rationality formula. Ferraro also presents a conjecture on the order of vanishing of this function at the canonical point $\Xi$ and some numerical evidences.