Editing Carlitz exponential

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In [[mathematics]], the '''Carlitz exponential''' is a characteristic ''p'' analogue to the usual [[exponential function]] studied in [[real analysis|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'''<sub>''q''</sub>[''T''] of one variable over a [[finite field]] '''F'''<sub>''q''</sub> with ''q'' elements. The [[Completion (metric space)|completion]] '''C'''<sub>∞</sub> of an [[algebraic closure]] of the field '''F'''<sub>''q''</sub>((''T''<sup>&minus;1</sup>)) of [[formal Laurent series]] in ''T''<sup>&minus;1</sup> will be needed. 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''&nbsp;>&nbsp;0 we define

:<math>[i] := T^{q^i} - T, \, </math>
:<math>D_i := \prod_{1 \le j \le i} [j]^{q^{i - j}}</math>

and ''D''<sub>0</sub> := 1. Note that the usual factorial is inappropriate here, since ''n''! vanishes in '''F'''<sub>''q''</sub>[''T''] unless ''n'' is smaller than the [[Characteristic (algebra)|characteristic]] of '''F'''<sub>''q''</sub>[''T''].

Using this we define the Carlitz exponential ''e''<sub>''C''</sub>:'''C'''<sub>∞</sub>&nbsp;→&nbsp;'''C'''<sub>∞</sub> by the convergent sum

:<math>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>e_C(Tx) = Te_C(x) + \left(e_C(x)\right)^q = (T + \tau)e_C(x), \, </math>

where we may view <math> \tau </math>  as the power of <math> q </math> map or as an element of the ring <math> 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'':'''F'''<sub>''q''</sub>[''T'']→'''C'''<sub>∞</sub>{''τ''}, defining a Drinfeld '''F'''<sub>''q''</sub>[''T'']-module over '''C'''<sub>∞</sub>{''τ''}. It is called the Carlitz module.

==References==
{{reflist}}
*{{Cite book| last1=Goss | first1=D. | author-link = David Goss | title=Basic structures of function field arithmetic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-61087-8 | mr=1423131 | year=1996 | volume=35}}
*{{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}}

[[Category:Algebraic number theory]]
[[Category:Finite fields]]