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 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 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

$e_C(x):=\sum _{i=0}^{\mathrm{\infty }}\frac{x^{q^i}}{D_i}.$

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

$e_C(Tx)=T{e}_C(x)+{(e_C(x))}^q=(T+\tau )e_C(x),$

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. 
• Thakur, Dinesh S. (2004). Function field arithmetic. New Jersey: World Scientific Publishing. ISBN 978-981-238-839-1.