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 \le j \le i} [j]^{q^{i - j}}

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

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 (2004), Function field arithmetic, New Jersey: World Scientific Publishing, ISBN 981-238-839-7, MR 2091265 |first2= missing |last2= in Authors list (help)