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 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
- [math]\displaystyle{ [i] := T^{q^i} - T, \, }[/math]
- [math]\displaystyle{ D_i := \prod_{1 \le j \le i} [j]^{q^{i - j}} }[/math]
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
- [math]\displaystyle{ e_C(x) := \sum_{i = 0}^\infty \frac{x^{q^i}}{D_i}. }[/math]
Relation to the Carlitz module
The Carlitz exponential satisfies the functional equation
- [math]\displaystyle{ e_C(Tx) = Te_C(x) + \left(e_C(x)\right)^q = (T + \tau)e_C(x), \, }[/math]
where we may view [math]\displaystyle{ \tau }[/math] as the power of [math]\displaystyle{ q }[/math] map or as an element of the ring [math]\displaystyle{ F_q(T)\{\tau\} }[/math] 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.
- Thakur, Dinesh S. (2004). Function field arithmetic. New Jersey: World Scientific Publishing. ISBN 978-981-238-839-1.
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