2025/09/19 by A. Grishkov, Grishkov, A., Dmitry Logachev +1
Mathematics · #05C05 #05E99 #11C20 #11G09 #13C40 #13P15 #14M10 #14M12 #14Q15 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.16160
openalex publication_date 2025/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \Bbb 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)(\Bbb Fq) where X(m,i) is an affine variety defined over \Bbb 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 \Bbb Fp to \Bbb 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.