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

•Found matrix whose characteristic polynomial is L-series of a twisted Carlitz module.•Numerical evidence that the analytic rank of twisted Carlitz modules maybe unbounded.•Polynomials defining high-rank at infinity of Carlitz module twists, are dependent.

We show that there exists a connection between two types of objects: some kind of resultantal varieties over CC, 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 QQ. The paper contains a calculation of a relevant polynomial determinant.