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

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 Fq. The analytic rank of a twist is the order of zero of its L-function at a point. The set of polynomials of degree m such that the analytic ranks of the corresponding twists are i is X (m, i) (Fq) where X (m, i) is an affine variety defined over Fₚ (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 Fₚ to 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.