In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Carlitz exponential is a characteristic ''p'' analogue to the usual
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
studied in
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. It is used in the definition of the
Carlitz module – an example of a
Drinfeld module In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex ...
.
Definition
We work over the polynomial ring F
''q'' 'T''of one variable over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F
''q'' with ''q'' elements. The
completion C
∞ of an
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...
of the field F
''q''((''T''
−1)) of
formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums ...
in ''T''
−1 will be useful. It is a complete and algebraically closed field.
First we need analogues to the
factorials
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
, which appear in the definition of the usual exponential function. For ''i'' > 0 we define
:
:
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
:
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
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
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
*
*{{Cite book , last1=Thakur , first1=Dinesh S. , title=Function field arithmetic , publisher=
World Scientific Publishing
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various f ...
, location=New Jersey, isbn=978-981-238-839-1 , mr=2091265 , year=2004
Algebraic number theory
Finite fields