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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>−1</sup>)) of [[formal Laurent series]] in ''T''<sup>−1</sup> 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>[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 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> → '''C'''<sub>∞</sub> by the convergent sum | |||
:<math>e_C(x) := \sum_{j = 0}^\infty \frac{x^{q^j}}{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}} | |||
*{{Citation | last1=Goss | first1=D. | authorlink = 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}} | |||
*{{Citation | last1=Thakur | first1=Dinesh | first2=S. | title=Function field arithmetic | publisher=[[World Scientific Publishing]] | location=New Jersey| isbn=981-238-839-7 | mr=2091265 | year=2004}} | |||
[[Category:Algebraic number theory]] | |||
[[Category:Finite fields]] | |||
Revision as of 11:03, 13 April 2013
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
![{\displaystyle [i]:=T^{q^{i}}-T,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf9149c14138365b030f55d2b4f8e3edc39b458)
![{\displaystyle D_{i}:=\prod _{1\leq j\leq i}[j]^{q^{i-j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/967e398d254e8f7c6264b1925faf4f3d6de865f4)
and D0 := 1. Note that 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
Relation to the Carlitz module
The Carlitz exponential satisfies the functional equation
where we may view as the power of map or as an element of the ring 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
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