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

${\displaystyle [i]:=T^{q^{i}}-T,\,}$

${\displaystyle D_{i}:=\prod _{1\leq j\leq i}[j]^{q^{i-j}}}$

and D₀ := 1. Note that the usual factorial is inappropriate here, since n! vanishes in 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

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

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

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

where we may view ${\displaystyle \tau }$ as the power of ${\displaystyle q}$ map or as an element of the ring ${\displaystyle F_{q}(T)\{\tau \}}$ of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ: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 S. (2004). Function field arithmetic. New Jersey: World Scientific Publishing. ISBN 978-981-238-839-1. MR 2091265.