Carlitz exponential

<h2 id="definition">Definition</h2>
We work over the polynomial ring Fq[T] of one variable over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> Fq with q elements. The <a href="/facts/Completion_(metric_space)/lsckmU8G">completion</a> C∞ of an <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of the field Fq((T−1)) of <a href="/facts/Formal_Laurent_series/CNmaG0Vk">formal Laurent series</a> in T−1 will be useful. It is a complete and algebraically closed field.
First we need analogues to the <a href="/facts/Factorials/kkfy1Bwe">factorials</a>, which appear in the definition of the usual exponential function. For i > 0 we define

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

D
          
            i
          
        
        :=
        
          ∏
          
            1
            ≤
            j
            ≤
            i
          
        
        [
        j
        
          ]
          
            
              q
              
                i
                −
                j
              
            
          
        
      
    
    {\displaystyle D_{i}:=\prod _{1\leq j\leq i}[j]^{q^{i-j}}}

and D0 := 1. Note that the usual factorial is inappropriate here, since n! vanishes in Fq[T] unless n is smaller than the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> of Fq[T].
Using this we define the Carlitz exponential eC:C∞ → C∞ by the convergent sum

e
          
            C
          
        
        (
        x
        )
        :=
        
          ∑
          
            i
            =
            0
          
          
            ∞
          
        
        
          
            
              x
              
                
                  q
                  
                    i
                  
                
              
            
            
              D
              
                i
              
            
          
        
        .
      
    
    {\displaystyle e_{C}(x):=\sum _{i=0}^{\infty }{\frac {x^{q^{i}}}{D_{i}}}.}

<h2 id="relation-to-the-carlitz-module">Relation to the Carlitz module</h2>
The Carlitz exponential satisfies the functional equation

e
          
            C
          
        
        (
        T
        x
        )
        =
        T
        
          e
          
            C
          
        
        (
        x
        )
        +
        
          
            (
            
              
                e
                
                  C
                
              
              (
              x
              )
            
            )
          
          
            q
          
        
        =
        (
        T
        +
        τ
        )
        
          e
          
            C
          
        
        (
        x
        )
        ,
        
      
    
    {\displaystyle e_{C}(Tx)=Te_{C}(x)+\left(e_{C}(x)\right)^{q}=(T+\tau )e_{C}(x),\,}

where we may view 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 as the power of 
 
 
 
 q
 
 
 {\displaystyle q}
 
 map or as an element of the ring 
 
 
 
 
 F
 
 q
 
 
 (
 T
 )
 {
 τ
 }
 
 
 {\displaystyle F_{q}(T)\{\tau \}}
 
 of <a href="/facts/Noncommutative_polynomials/5aw6A1tw">noncommutative polynomials</a>. By the <a href="/facts/Universal_property/5XJBYMNz">universal property</a> 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.

<ul><li><a href="/facts/David_Goss/T2LteJP2">Goss, D.</a> (1996). Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 35. Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-61087-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1423131">1423131</a>.</li>
<li>Thakur, Dinesh S. (2004). Function field arithmetic. New Jersey: <a href="/facts/World_Scientific_Publishing/qJPTP0Yk">World Scientific Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-238-839-1. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2091265">2091265</a>.</li></ul>

Carlitz exponential open-in-new

Carlitz exponential