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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

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

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

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

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

$$e_C(Tx) = Te_C(x) + \left(e_C(x)\right)^q = (T + \tau)e_C(x), \,$$

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

Goss, D. (1996). Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. 35. Berlin, New York: Springer-Verlag. ISBN 978-3-540-61087-8. MR 1423131.
Thakur, Dinesh S. (2004). Function field arithmetic. New Jersey: World Scientific Publishing. ISBN 978-981-238-839-1. MR 2091265.