L-functions of Carlitz modules, resultantal varieties and rooted binary trees, II

TL;DR

This paper explores algebraic varieties linked to L-functions of Carlitz modules, revealing their structure through finite weighted rooted binary trees and providing parametrizations and invariants, with implications for understanding the analytic rank of twists.

Contribution

It offers a conjectural complete description of resultantal varieties related to Carlitz modules using binary trees and parametrizations, advancing the understanding of their algebraic and geometric properties.

Findings

Varieties described in terms of finite weighted rooted binary trees.

Parametrizations of irreducible components and their invariants.

Potential implications for the boundedness of the analytic rank of twists.

Abstract

We continue study of some algebraic varieties (called resultantal varieties) started in a paper of A. Grishkov, D. Logachev "Resultantal varieties related to zeroes of L-functions of Carlitz modules". These varieties are related with the Sylvester matrix for the resultant of two polynomials, from one side, and with the L-functions of twisted Carlitz modules, from another side. Surprisingly, these varieties are described in terms of finite weighted rooted binary trees. We give a (conjecturally) complete description of them, we find parametrizations of their irreducible components and their invariants: degrees, multiplicities, Jordan forms, Galois actions. Proof of the fact that this description is really complete is a subject of future research. Maybe a generalization of these results will give us a solution of the problem of boundedness of the analytic rank of twists of Carlitz modules.