Teichmüller character
In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where [math]\displaystyle{ q = p }[/math] if [math]\displaystyle{ p }[/math] is odd and [math]\displaystyle{ q = 4 }[/math] if [math]\displaystyle{ p = 2 }[/math], taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section ω : k → O of the natural surjection O → k. The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character.
Definition
If x is a p-adic integer, then [math]\displaystyle{ \omega(x) }[/math] is the unique solution of [math]\displaystyle{ \omega(x)^p = \omega(x) }[/math] that is congruent to x mod p. It can also be defined by
- [math]\displaystyle{ \omega(x)=\lim_{n\rightarrow\infty} x^{p^n} }[/math]
The multiplicative group of p-adic units is a product of the finite group of roots of unity and a group isomorphic to the p-adic integers. The finite group is cyclic of order p – 1 or 2, as p is odd or even, respectively, and so it is isomorphic to (Z/qZ)×.[citation needed] The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the p-adic integers, by means of Hensel lifting, is given in the article on Witt vectors, where they provide an important role in providing a ring structure.
See also
References
- Section 4.3 of Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2
- Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3
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