Carlitz exponential

From Wikipedia, the free encyclopedia

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

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

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

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

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.