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

: := T^ - T, \,
:D_i := \prod_

and ''D''₀ := 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

:$e_C(x) := \sum_^\infty \frac.$

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)\$ of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ''ψ'':F_''q'' 'T''��C_∞, defining a Drinfeld F_''q'' 'T''module over C_∞. It is called the Carlitz module.

References

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*{{Cite book , last1=Thakur , first1=Dinesh S. , title=Function field arithmetic , publisher= World Scientific Publishing , location=New Jersey, isbn=978-981-238-839-1 , mr=2091265 , year=2004

Algebraic number theory
Finite fields