Carlitz exponential

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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

[ i ] := T q i − T , {\displaystyle [i]:=T^{q^{i}}-T,\,}
D i := ∏ 1 ≤ j ≤ i [ j ] q i − j {\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

e C ( x ) := ∑ i = 0 ∞ x q i D i . {\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

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

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

References