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

Abstract

We work out in detail the Drinfeld module over the ring A = F2 [x, y] / (y2 + y = x3 + 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 dk and lk of the exponential and logarithmic functions and relate them with the product Dk of all monic elements of A of degree k. On the Carlitz module, Dk and dk coincide, but this is not true for 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.