Analytic ranks of twists of Carlitz modules -- a survey of results

Abstract

We give in this paper a survey of results obtained in our earlier papers, and state explicitly some problems of further research, for example: are the analytic ranks bounded, or not? Twists of Carlitz modules are parametrized by polynomials over finite fields ${\mathbb{F}}_q$. The analytic rank of a twist is the order of zero of its L-function at a point. The set of polynomials of degree $\le m$ such that the analytic ranks of the corresponding twists are $\ge i$ is $X(m,i)({\mathbb{F}}_q)$ where $X(m,i)$ is an affine variety defined over ${\mathbb{F}}_p$ (we do not know what is its dimension). We consider also a related invariant of a twist, namely, the behaviour of its L-function at infinity (the rank at infinity). We know much more on varieties corresponding to twists of a fixed rank at infinity and on their lifts from ${\mathbb{F}}_p$ to $\mathbb{Z}$ . For example, for $q=2$ the irreducible components of these varieties are described in terms of finite rooted weighted binary trees. A similar description for $q>2$ is not found yet.

9 pages