Resultantal varieties related to zeroes of L-functions of Carlitz modules

TL;DR

This paper explores the relationship between resultantal varieties over complex numbers and varieties associated with twists of the Carlitz module, linking geometric objects to properties of L-functions in function fields.

Contribution

It establishes a connection between resultantal varieties and twists of the Carlitz module, providing initial results towards a broader theory on L-function zeros in function fields.

Findings

Connection between resultantal varieties and Carlitz module twists

Calculation of a non-trivial polynomial determinant

Indication of unboundedness of L-function zeros at 1

Abstract

We show that there exists a connection between two types of objects: some kind of resultantal varieties over C, from one side, and varieties of twists of the tensor powers of the Carlitz module such that the order of 0 of its L-functions at infinity is a constant, from another side. Obtained results are only a starting point of a general theory. We can expect that it will be possible to prove that the order of 0 of these L-functions at 1 (i.e. the analytic rank of a twist) is not bounded --- this is the function field case analog of the famous conjecture on non-boundedness of rank of twists of an elliptic curve over Q. The paper contains a calculation of a non-trivial polynomial determinant.