A description of a Drinfeld module with class number $h=1$ and rank $1$

TL;DR

This paper provides a detailed analysis of a specific Drinfeld module over a quadratic imaginary field with class number one, deriving formulas for exponential and logarithmic coefficients and relating them to algebraic invariants.

Contribution

It develops explicit formulas for coefficients of exponential and logarithmic functions of a particular Drinfeld module and relates these to product invariants, extending known results from the Carlitz module.

Findings

Derived formulas for $d_k$ and $\,ell_k$ coefficients.

Established relations between invariants $D_k$ and $d_k$ for the example.

Proved a theorem relating combinatorial symbols in soliton analysis.

Abstract

We work with detail the Drinfeld module over the ring $$A=F_2[x,y]/(y^2+y=x^3+x+1).$$ The example in question is one of the four examples that come from quadratic imaginary fields with class number $h = 1$ and rank one. We develop specific formulas for the coefficients $d_k$ and $\ell_k$ of the exponential and logarithmic functions and relate them with the product $D_k$ of all monic elements of $A$ of degree $k$. On the Carlitz module, $D_k$ and $d_k$ coincide, but this is not true in general Drinfeld modules. On this example, we obtain a formula relating both invariants. We prove also using elementary methods a theorem due to Thakur that relate two different combinatorial symbols important in the analysis of solitons.