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 Fq[T] of one variable over a finite field Fq with q elements. The completion C of an algebraic closure of the field Fq((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

${\displaystyle [i]:=T^{q^{i}}-T,\,}$
${\displaystyle D_{i}:=\prod _{1\leq j\leq i}[j]^{q^{i-j}}}$

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

Using this we define the Carlitz exponential eC:C → C by the convergent sum

${\displaystyle e_{C}(x):=\sum _{i=0}^{\infty }{\frac {x^{q^{i}}}{D_{i}}}.}$

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

${\displaystyle e_{C}(Tx)=Te_{C}(x)+\left(e_{C}(x)\right)^{q}=(T+\tau )e_{C}(x),\,}$

where we may view ${\displaystyle \tau }$ as the power of ${\displaystyle q}$ map or as an element of the ring ${\displaystyle F_{q}(T)\{\tau \}}$ of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:Fq[T]→C{τ}, defining a Drinfeld Fq[T]-module over C{τ}. It is called the Carlitz module.