Carlitz exponential

In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module.

Definition

We work over the polynomial ring F_q[T] of one variable over a finite field F_q with q elements. The completion C_∞ of an algebraic closure of the field F_q((T^−1)) of formal Laurent series in T^−1 will be useful. It is a complete and algebraically closed field.

First we need analogues to the factorials, which appear in the definition of the usual exponential function. For i > 0 we define

and D₀ := 1. Note that the usual factorial is inappropriate here, since n! vanishes in F_q[T] unless n is smaller than the characteristic of F_q[T].

Using this we define the Carlitz exponential e_C:C_∞ Ã¢Â†Â’ C_∞ by the convergent sum

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

where we may view as the power of map or as an element of the ring of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:F_q[T]→C_∞{τ}, defining a Drinfeld F_q[T]-module over C_∞{τ}. It is called the Carlitz module.

References