Carlitz exponential
# 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

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

and $D_0$ := 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 $e_C:C_∞$ → $C_∞$ by the convergent sum

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

## Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

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

where we may view $\tau$ as the power of $q$ map or as an element of the ring $F_{q}(T)\{\tau \}$ 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.

