Cyclotomic character

Quality 0.50 · 1 view · Updated 2 months ago

title: "Cyclotomic character" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-number-theory"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Cyclotomic_character" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → AutR(R(1)) ≈ GL(1, R)).

''p''-adic cyclotomic character

Fix p a prime, and let G_\mathbf{Q} denote the absolute Galois group of the rational numbers. The roots of unity \mu_{p^n} = \left{ \zeta \in \bar\mathbf{Q}^\times \mid \zeta^{p^n} = 1 \right} form a cyclic group of order p^n, generated by any choice of a primitive p**nth root of unity ζpn.

Since all of the primitive roots in \mu_{p^n} are Galois conjugate, the Galois group G_\mathbf{Q} acts on \mu_{p^n} by automorphisms. After fixing a primitive root of unity \zeta_{p^n} generating \mu_{p^n}, any element \zeta\in\mu_{p^n} can be written as a power of \zeta_{p^n}, where the exponent is a unique element in \mathbf{Z}/p^n\mathbf{Z}, which is a unit if \zeta is also primitive. One can thus write, for \sigma\in G_\mathbf{Q},

\sigma.\zeta := \sigma(\zeta) = \zeta_{p^n}^{a(\sigma, n)}

where a(\sigma,n) \in (\mathbf{Z}/p^n \mathbf{Z})^\times is the unique element as above, depending on both \sigma and p. This defines a group homomorphism called the mod *pn* cyclotomic character**:

\begin{align}{\chi_{p^n}}:G_{\mathbf{Q}} &\to (\mathbf{Z}/p^n\mathbf{Z})^{\times} \ \sigma &\mapsto a(\sigma, n), \end{align} which is viewed as a character since the action corresponds to a homomorphism G_{\mathbf Q} \to \mathrm{Aut}(\mu_{p^n}) \cong (\mathbf{Z}/p^n\mathbf{Z})^\times \cong \mathrm{GL}_1(\mathbf{Z}/p^n\mathbf{Z}).

Fixing p and \sigma and varying n, the a(\sigma, n) form a compatible system in the sense that they give an element of the inverse limit \varprojlim_n (\mathbf{Z}/p^n\mathbf{Z})^\times \cong \mathbf{Z}p^\times,the units in the ring of p-adic integers. Thus the {\chi{p^n}} assemble to a group homomorphism called p-adic cyclotomic character:

\begin{align} \chi_p:G_{\mathbf Q} &\to \mathbf{Z}p^\times \cong \mathrm{GL_1}(\mathbf{Z}p) \ \sigma &\mapsto (a(\sigma, n))n \end{align} encoding the action of G{\mathbf Q} on all p-power roots of unity \mu{p^n} simultaneously. In fact equipping G{\mathbf Q} with the Krull topology and \mathbf{Z}_p with the p-adic topology makes this a continuous representation of a topological group.

As a compatible system of {{math|ℓ}}-adic representations

By varying ℓ over all prime numbers, a compatible system of ℓ-adic representations is obtained from the ℓ-adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol ℓ to denote a prime instead of p). That is to say, is a "family" of ℓ-adic representations

:\chi_\ell:G_\mathbf{Q}\rightarrow\operatorname{GL}1(\mathbf{Z}\ell)

satisfying certain compatibilities between different primes. In fact, the χℓ form a strictly compatible system of ℓ-adic representations.

Geometric realizations==

The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group scheme Gm,Q over Q. As such, its representation space can be viewed as the inverse limit of the groups of p**nth roots of unity in .

In terms of cohomology, the p-adic cyclotomic character is the dual of the first p-adic étale cohomology group of Gm. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of H2ét(P1 ).

In terms of motives, the p-adic cyclotomic character is the p-adic realization of the Tate motive Z(1). As a Grothendieck motive, the Tate motive is the dual of H2( P1 ).Section 3 of {{Citation | last=Deligne | first=Pierre | author-link=Pierre Deligne | contribution=Valeurs de fonctions L et périodes d'intégrales | contribution-url=https://www.ams.org/online_bks/pspum332/pspum332-ptIV-8.pdf | title=Automorphic Forms, Representations, and L-Functions | editor-last=Borel | editor-first=Armand | editor-link=Armand Borel | editor2-last=Casselman | editor2-first=William | publisher=AMS | location=Providence, RI | series=Proceedings of the Symposium in Pure Mathematics | isbn=0-8218-1437-0 | mr=0546622 | zbl=0449.10022 | year=1979 | volume=33 | issue=2 | page=325 | language=French}}

Properties

The p-adic cyclotomic character satisfies several nice properties.

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

algebraic-number-theory