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&minus;1)) of formal Laurent series in T&minus;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


 * $$[i] := T^{q^i} - T, \, $$
 * $$D_i := \prod_{1 \le j \le i} [j]^{q^{i - j}}$$

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


 * $$e_C(x) := \sum_{i = 0}^\infty \frac{x^{q^i}}{D_i}.$$

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:Fq[T]→C∞{τ}, defining a Drinfeld Fq[T]-module over C∞{τ}. It is called the Carlitz module.