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[edit]

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

and D0 := 1. Note that 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_{j = 0}^\infty \frac{x^{q^j}}{D_i}.

Relation to the Carlitz module[edit]

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.

References[edit]